PARTIALLY ORDERED
TOPOLOGICAL
VECTOR SPACES
BY
YAU-CHUEN WONG
AND
KUNG-FU NG
MATH EM ATSSCH 1N STITU Ш
WjKSUNiViRSnTJT ТЕ U /
BIBLIOTHEEK
RIJKSUNIVERSITEIT UTRECHT
1110 0926
CLARENDON PRESS • OXFORD
1973
Oxford University Press, Ely House, London W, 1
GLASGOW NEW YORK TORONTO MELBOURNE WELLINGTON
САРИ TOWN IBADAN NAIROBI DAB ES SALAAM LUSAKA ADDIS ABABA
DELHI BOMBAY CALCUTTA MADRAS KARACHI LAHORE DACCA
KUALA LUMPUR SINGAPORE HONG KONG TOKYO
ISBN 0 19 8&3523 6
© OXFORD UNIVERSITY PRESS 1973
PRINTED IN NORTHERN IRELAND
AT
THE UNIVERSITIES PRESS, BELFAST.
TO OUR TEACHERS
J. D. Weston
S. T. Tsou
H. P. Rogosinski

PREFACE
The duality theory is one of the most important and fruitful
theories in the study of topological vector spaces. This book gives
an account of the duality theory of partially ordered topological
vector spaces. Through the pioneering works of many great mathe-
maticians including Riesz, Frendenthal, Birkhoff, Kakutani, Kan-
torovitch, Krein, and Nakano, the theory of Riesz spaces (vector
lattices), and in particular the theory of Banach lattices, has been
well developed; there are many excellent books containing informa-
tion on the classical theory of Banach lattices (for example, Day
(.1962), Kelley-Namioka (1963)). The theory was further developed
by Luxemburg and Zaanen through their systematic and extensive
studies carried out in a series of papers entitled 'Notes on Banach
Function Spaces’ published in the 60’s.
The classical duality theory of Banach lattices has been generalized
and developed in the following two divergent directions:
(a)	Partially ordered Banach spaces. Krein seems to be the man
who initiated this subject; in particular, he shows how the order
properties of the Banach space E determine the dual order properties
of the Banach dual space E'. Ando, Edwards, and Ellis initiated the
attack on the (much more difficult) converse problem; they show
that some of the order properties of E are completely determined by
the dual properties of E'. Studies of this kind have been carried
further by Davies, Ng, Asimow, Perdrizet, and others. The develop-
ment of the abstract theory of partially ordered Banach spaces has
much been influenced and motivated by other branches of mathe-
matics (e.g. G'*-algebra theory and Choquet boundary theory); on
the other hand, the abstract theory has also enriched our understand-
ing in other branches (cf. for example, the works of Effros, Stormer,
Alfsen, and others).
(b)	Locally convex Riesz spaces. The study of such spaces has
been strongly influenced by the general theory of topological vector
spaces and abstract integration theory (Kothe spaces theory in
particular). Roberts seemed to be the first to investigate the duality
theory for locally convex Riesz spaces. The theory has then been de-
veloped rapidly through the works of Namioka, Schaefer, Kawai,
Peressini, Goffman, Wong, and Fremlin. In most of their investigations,
a remarkable theorem of Nakano (asserting that for topological
viii
PREFACE
Riesz spaces topological completeness follows from certain order
completeness assumptions) plays an important role. By utilizing
this powerful theorem, Schaefer proved in 1960 that each reflexive
locally convex Riesz space is topologically complete. This result is
remarkable in view of Kdmura’s example of a non-complete locally
convex reflexive space, and indicates that order properties may
substantially ‘influence’ the topological structure. In 1969, Wong-
extended Schaefer’s theorem to semi-reflexive spaces by considering
the .Dieudonne topology ots (and applying Nakano’s theorem); he
also showed that a8 is relevant for establishing a converse of Nakano’s
theorem..
The present book is an attempt to provide a unifying and balanced
treatment of the above two seemingly unrelated subjects (a) and (b).
Accordingly, the book is roughly divided into three parts. The first
part consisting of the first eight chapters is mainly on the theory of
partially ordered locally convex spaces, that is, the ‘common’ theory
which is applicable to both (a) and (b). Apart from the work of those
mentioned above, we have also included some work by Bonsall and
by Weston, and some recent results of Jameson and Duhoux. The
second part consists of one single (long) chapter and is a brief account
of the results described in (a) with emphasis on those duality results
peculiar to normed spaces. The third part consists of the last nine
chapters, in which is given not only the theory described in (b) but
also Wong’s work on barrelled, order-infrabarrelled, infrabarrelled,
and bornological locally convex Riesz spaces.
At the end of the book we have included ‘Notes on the Bibliography’.
These, together with some comments and remarks distributed through,
the text, should be sufficient to lead the reader to the relevant literature.
In most cases we have not attempted to trace the origins of the results
but instead to refer the reader to standard reference books whenever
possible. Thus, for example, when we say that certain results are due to
Schaefer (cf. Schaefer (1966)), we mean that the results and some related
material can be found in his book published in 1966 even though he
published his results in a much earlier paper. Likewise, when we say that
by Schaefer (1966, p. 126), we mean ‘by a result appearing on page 126
of his book,’ even though the result may not be due to Schaefer.
The book has evolved from lecture notes prepared by the authors
for seminars conducted in the Chinese University of Hong’ Kong.
These seminars were attended primarily by advanced undergraduated
students (in their fourth year) and also by some of our colleagues. We
PREFACE
ix
believe that this book should be quite accessible to any student who
is acquainted with the elementary theory of topological vector spaces.
The authors’ interest in the subject was cultivated when they did
their research at University College, Swansea. It gives them great
pleasure to thank Professor J. D. Weston, the Department Head, and
his staff members, in particular Dr A. J. Ellis, Dr H. P. Rogosinski,
and Professor G. M. Peterson (who is now at the University of
Canterbury) for their guidance and stimulation. We would also like
to thank Professor Weston for suggesting that we write the book
and for his help in arranging for its publication. The manuscript
was written in Hong Kong; we should like to take the opportunity
to thank the Chinese University of Hong Kong in general and the
United College in particular for financial and moral support. We
wish in particular to thank Dr 8. T. Tsou who constantly encourages
and helps us in many ways, not only as the Department Head but
also as a teacher and as a friend. We are also grateful to the staff of
the Clarendon Press for their assistance.
United College
The Chinese University of Hong Kong
May 1973
Y. C. W.
К. E. N.

CONTENTS
l.	Fundamentals of ordered vector spaces	1
2.	Cones in topological vector spaces	1 8
3.	Locally decomposable spaces	30
4.	^-Cones and local ^-cones	43
5.	Locally o-convex spaces	48
6.	Locally solid spaces	60
7.	The order-bound topology	67
8.	Metrizable ordered topological vector spaces	77
9.	Ordered normed vector spaces	.	85
10,	Elementary theory of Riesz spaces	113
11.	Topological Riesz spaces	136
]2. Locally o-convex Riesz spaces	155
13.	Completeness for the Dieudon nd topology	160
14.	Reflexivity for locally convex Riesz spaces	173
15.	Bornological and infrabarrellcd Riesz spaces	178
16.	The structure of order-infrabarrelled Riesz spaces and its simple
properties	187
17.	Permanence properties of order-infrabarrellcd Riesz spaces	194
18.	Relationship between barrelled, order-infrabarrelled, and infrabarrelled
Riesz spaces	197
NOTES ON THE BIBLIOGRAPHY	206
BIBLIOGRAPHY	210
INDEX
215

1
FUNDAMENTALS OF
ORDERED VECTOR SPACES
In this chapter we review some basic facts in linear algebra and ordered
vector spaces, which we shall need in what follows. The following
notation is used: <f> denotes the empty set, and A\B denotes the com-
plement of В relative to A, where A and В are sets.
Throughout this book we shall restrict our attention to vector spaces
over the real field R. If A and В are subsets of a vector space E and if
Я, у are real numbers, we define
kA-\-yB ~ {flaAyb'.a e A, b g В].
The expression {ж}4~А will be abbreviated by x-\-A, ( — 1)4 by —A,
and A. +(— B) by A — B. Let К be a subset of E. К is said to be convex
if ЯА-[-(1 — X)K с К whenever 0 < Я < 1, К is said to be symmetric
if —К = K, and К is circled if ЛК £: К whenever |Л| < 1. If В is a
subset of E, the smallest convex set containing B, denoted by co B, is
called the convex hull of B; and the smallest convex circled set con-
taining В (denoted by ГВ) is called the convex circled hull (or ab-
solutely convex hull) of B. Let A and В be subsets of E. We say that A
absorbs В if there exists Я > 0 such that В s yA for all у with
|/z| > Я. If A absorbs every finite subset of E, then A is said to be
absorbing. If A is absorbing, the functional pA defined by
pA(x) “ inf{2 > 0:«g AA} for any ж in В
is called the gauge (or Minkowski functional) of A. A functional p on
E is said to be sublinear if
7?(ж+у) <p(x)Ap{y} and р(Я«) — Я^(ж)
for all x, у in E and Я > 0. A sublinear functional p on E is called a
semi-norm if p(yx) = p(x) for all x in E and у in R.
A non-empty convex subset C of E is called a cone if ЯС cz C for all
Я > 0. Clearly a cone G in E determines a transitive and reflexive
relation ‘ ’ by	. r
J	x < у it y—x e 6;
2 FUNDAMENTALS OF ORDERED VECTOR SPACES
moreover this relation is compatible with the vector structure, i.e,
(a)	if ж > 0 and ?/ > 0 then x -\-y > 0,
(6)	if x > 0 then ax > 0 for all Я > 0.
The relation determined by the cone G is called the vector {partial)
ordering oiE, and the pair {E, G) (or {E, < ))is referred to as '^{partially)
ordered vector space. Conversely if *< ’ is a relation in E which is tran-
sitive, reflexive, and compatible with, the vector structure of E and if
we define	„	, w
G = {ж e E: x > 0},
then G is a cone in E, and is exactly the vector ordering of E
induced by C.
A cone G in E is said to be proper if С C\ {—C) = {0}. The vector
ordering ‘ ’ of E, induced by a cone G, is antisymmetric if and only if
G is proper.
It is easily seen that the intersection of a family of cones in E is
a cone. The smallest cone containing a given set A is denoted by pos A.
Clearly
(n
2 ed, > 0 for all i ;= I, 2,..., n
/=1
We see that pos A is proper if and only if all = 0 {i = 1, 2,..., n)
n
whenever 2	where a{ g A\{0} and > 0 {i = 1, 2,..., n). If
«=1
A is convex, then pos A has a simpler expression as shown in the
following proposition.
(1.1)	Proposition. If A is a non-empty convex subset of a vector
space E, then	pos Л = и {U: Л 0}.
Furthermore, if 0 ф A then pos A is proper.
Proof. Suppose that
P U {£4:Л > 0}.
It is clear that pP £ P for all у > 0. From the convexity of A, it is
easy to show that P is convex, and so P is a cone containing A. Further,
if IT is a cone containing A then
M c W for all Л > 0,
and so P £ W. Therefore P is the smallest cone containing A, i.e.
pos A — P.
FUNDAMENTALS OF ORDERED VECTOR SPACES 3
Finally, we show that pos A is proper whenever 0 ф A. Suppose, on
the contrary, that there exist a, b in A and 2, у > 0 such that
2a = —-[Ab.
Then the convexity of A entails that
0 =------a 6 e Л,
2 ft 2 -(* ft
which gives a contradiction. This completes the proof.
A cone C in E is said to be generating if E = C--C.
(1.2)	Proposition. Let C be a cone in E. Then the following state-
ments are equivalent:
(a)	C is generating;
(b)	for any x e E, there exists ueC such that и > x;
(c)	the vector ordering in E is directed in the sense that for any x, у
in E there exists z e E such that x < z and у < z.
Proof. Straightforward.
Let (E, <) be an ordered vector space. The vector ordering < is
said to be Archimedean if x < 0 whenever nx < y, or almost-Archi-
medean if x = 0 whenever —y < nx < y, for all positive integers n
and some у e E. It should be noted that if the vector ordering is proper
and Archimedean then it is almost-Archimedean, but the word ‘proper’
cannot be dropped in the above conclusion as shown by the following
examples, (a) In R2, suppose that
L = {(ж, y):x > 0}
and that ‘ <3 is the vector ordering determined by the (improper) cone
C. Then ‘ ’ is Archimedean, but not almost-Archimedean.
(b)	{(а, Д) g R2: a > 0,	> 0} U {(0, 0)} is a proper almost-Archi-
medean cone in R2, but it is not Archimedean.
In the definition of almost-Archimedean. ordering, the element у
involved in the inequalities —y < nx < у must be positive (that is, in
the positive cone of E). In the definition of Archimedean ordering we
can also consider only positive elements y. More precisely, if (E, <)
satisfies the property that x < 0 whenever nx < y' for all positive
integers n and some yr e E with y' > 0, then ‘ < ’ is Archimedean. (In
fact, suppose nx < у for all positive integers n and for some у e E,
4 FUNDAMENTALS OF ORDERED VECTOR SPACES
where у is not necessarily positive. Let y' = y—x. Then у’ > 0 and
nx < y'.) This observation makes the following two propositions clear.
(1.3)	Proposition. For any ordered vector space (E, <), the
following statements are equivalent:
(a)	the vector ordering ‘ <’ is Archimedean;
(b)	if x, у in E are such that x < Лу for all Я > 0 then x < 0;
(c)	if x, у are in E and e is a positive real number such that x < Лу for
all Я with 0 < Л < s, then x < 0.
(1.4)	Proposition. For any ordered vector space (E, <), the
following statements are equivalent:
(a)	the vector ordering < is almost-Archimedean;
(b)	if x, у are in E such that —Лу < x < Лу for all Я > 0 then x -= 0;
(c)	if x, y, z are in E and s is a positive real member such that
Лу < x < Az for all Я with 0 < Я < e, then x -- 0.
Let F be a vector subspace of an ordered vector space (E, G), where
C is a cone in E. Then F П G is a cone in F and the vector ordering
induced by F Г\ C is called the relative ordering. The ordering for a
subspace will always be assumed to be defined in this manner. It
should be noted that if G is a proper cone then so is F П G. However,
it may happen that F C\ G is a proper cone while G is not. It is clear
that the relative ordering of an Archimedean ordering is again Archi-
medean and the relative ordering of an almost-Archimedean ordering
is almost-Archimedean.
Examples
(a)	The usual ordering for R is that induced by the proper cone R+
of all non-negative real numbers.
(b)	If E is a vector space of real-valued functions defined on a set 8
then the usual ordering for E is defined pointwise:
f < g in E <=>f(s} < gr(s) for all s in
In particular, if 8 is a topological space then the ordering for the space
of all continuous real-valued functions on 8 is defined in this manner.
Another example of pointwise-defined ordering is that in sequence
spaces.
(c)	The usual ordering for the vector space co of all sequences {Я„} of
real numbers is that induced by the cone E consisting of all sequences
FUNDAMENTALS OF ORDERED VECTOR SPACES
5
{/Q, where each Яп > 0. The following subspaces of co are endowed
with the relative ordering:
m:	the space of all bounded sequences of real numbers;
c:	the space of all convergent sequences of real numbers;
c0: the space of all null sequences (that is, sequences converging to
zero);
lv: the space of all Zi’-summable sequences of real numbers.
(d	) Let (Q, (U) be a measure space and E the vector space of all
measurable real-valued functions. Here, as usual, functions which are
equal almost everywhere are identified. For/, g in E, we define
f уоДж) < р(ж) for almost every x in Q.
The subspace of all A25-summable functions in E is endowed, with the
relative ordering, where p is a positive real number.
In all the examples (a)-(d), the vector orderings are Archimedean.
Let В be a subset of an ordered vector space (E, С). В is said to be
majorized if there exists x in E such that Ъ < x for all b in В; В is
said to be minorized if there exists x in E such that x < b for all b in В;
В is said to be positive if В и C. For a pair of elements x, у in E with
let	[x, y] ~ {z e Eix z < y}.
The sets of the form [x, y\ are called order-intervals. It is easily verified
that order-intervals are convex; the converse is, of course, inexact. A
subset in (E, C) is said to be order-bounded if it is contained in some
order-interval in E. A subset A of E is said to be order-convex (or full) if
[cq, a2] ~ whenever a13 a2 e A and cq < a2. An order-convex set
is not necessarily convex. For instance, consider R2 with the cone C
defined by c _ {(a> e R2 . a >
Then R2\(7 is order-convex but not convex. In fact, we have the
following proposition.
(1.5) Proposition. Let (E, C) be an ordered vector space. Then the
following statements are equivalent:
(a)	each order-convex set in E is convex;
(b)	the vector ordering induced by G is total, that is, E ~ G U — (7.
Proof, (a) => (b): Clearly E\C is order-convex, so must be convex
by (a). If G is not total in E, let x e E\(C и — C). Then both x and -~x
2
6
FUNDAMENTALS OK ORDERED VECTOR SPACES
are in the convex set E\G; hence
0 - ММ
contrary to the fact that the cone G contains 0.
(b) => (a): Let В be an order-convex set. Let &15 Ь2 e В, and Лх, Л2 > 0
with Ях + Я2 = 1. Since the vector ordering in E is total, we have either
&x < bz or Ьг > &a. Without loss of generality, suppose 6X < &2. Then
&x < Лх&х + Я262 < so Лх&х+Л26а g [&XJ 62] — В by the order
convexity of B.
Simple examples in the plane show also that convex sets are not
necessarily order-convex. Thus the concepts of convexity and order
convexity are quite distinct. A subset A in an ordered vector space is
said to be o-convex if it is both convex and order-convex. The
o-convexity must not be confused with the order-convexity.
It is clear that the intersection of order-convex sets is order-convex
and the intersection of o-convex sets is o-convex. Given a subset В in
(E, C), the smallest order-convex set in E containing В is called the
order-convex hull of В and is denoted by [В]. Notice that
[B] = g E:b1 < x < &2 for some b2 in B} = (BdC) r> (B — C).
The order-convex hull [ B] will sometimes be referred to as the full hull
of В and will be alternatively denoted by E(B).
A subset A of (E, C) is called:
(a)	positive-order-convex if [0, a] £ A whenever a g A and a > 0
(thus A contains 0 if it contains a positive element);
(b)	absolute-order-convex if [—a, a] £ A whenever a g A and a > 0.
If A is a symmetric set then the order-convexity implies (b) which in
turn implies (a). Simple examples in the plane show that the three
order-convexities are distinct.
A semi-norm p on (E, G) is called:
(a') monotone if p(x) < p(y) whenever 0 < x < у in E;
(b') absolute-monotone ifp(«) < p(y) whenever — у < x < у in E.
The following result demonstrates the relationship between these
concepts.
(1.6)	Proposition. Let p be a semi-norm on an ordered vector space
(E, C) and let V = (x g E:p(x) < I}. Then the following statements hold:
(a)	p is absolute-monotone if and only if V is absolute-order-convex;
(b)	p is monotone if and only if V is positive-order-convex;
(с)	V is order-convex if and only if p satisfies the following implication:
x у < z in E p(y) < тах{р(ж), р(г)}.	(1.1)
FUNDAMENTALS OF ORDERED VECTOR SPACES 7
(// V is replaced by the closed ball S = {z g E:p(x) < 1}, the proposition
remains true.)
Proof. Straightforward.
(If condition (1.1) is satisfied, G will be referred to as a 1-normal cone
in E with respect to p.)
An important concept dual to the order-convexity is that of de-
composable sets. A set A in an ordered vector space (A', G) is said to
be decomposable if for each a in A there exist a2 in A C\G such that
a =	f°r some Ях, Л2 > 0 with ЛхрЛ2 -= I. Note that a sym-
metric, decomposable set may not be convex and that a symmetric
convex set is not necessarily decomposable. Observe, however, that the
symmetric convex hull of a decomposable set is decomposable. Also
the union of a family of decomposable sets is decomposable, but the
intersection of decomposable sets may not be decomposable.
Given any set A in (E, C), the union of all decomposable sets in E
contained in A, which is the largest decomposable set contained in A,
will bo called the decomposable kernel of A, and will be denoted by
D(A). If A is circled and convex then D(A) can be explicitly expressed
as D(A) = {2xux — Л2а2:а1; ай e A n C, Ях, X2 > 0, ЯХ4~/1Й = 1}
= Г(А n G) =- co( — (A n G) и (A n G)).
A subset В in (E, C) is said to be positively generated if for each b
in В there exist bls b2 in В Г\ C such that &x —	= b. Thus E itself is
positively generated if О is a generating cone. It is also clear that any
symmetric convex and decomposable set is positively generated.
(1.7)	Proposition. Let p be a semi-norm on an ordered vector space
(E, G) and let V = {x g E:p(x) < 1}. Then the following statements
are equivalent:
(a)	V is decomposable;
(b)	for each x in E and each positive real number s there exist two
positive elements жх, x2 withptxf) pp(x2) < p(x) -j-s such that x = x1—x2.
Proof. (a)=>(b): Note that x g (/)(«) +e)F. Since V is decom-
posable, there exist vlf v2 in V Ci G and Ях, Л2 > 0 with Ях ТЛ2 = 1 such
that	x = Ш + еЖ'Ш
Let xx = hfp(x) + e)vx and x2 = h2(p(x) -fs)v2. Then (b) is easily
verified.
8 FUNDAMENTALS OF ORDERED VECTOR SPACES
(b)	=> (a): Let v e V. Then there exists e > 0 such thatp(v)-]-e < 1.
Lor these v and. e, by (b), there exist xlt x2eC with
2>Ы+р(^) < .рИ+б < i
such that v = x1-—x2. Take positive real numbers <i2 such that
0 < p(Xi) < (% for each i, and +	— 1. Then x-Jd^ and«2/da are in V
and v =	— ^й(жй/^а) e D(V). Therefore V is decomposable.
A subset В of (E, O') is said to be:
(i)	absolutely dominated if for each b in В there exists ж in В such that
—b, b < x',
(ii)	positively dominated if for each b in В there exists ж in В such that
О, Ъ < ж.
Clearly the condition (i) implies (ii). Also a decomposable convex set
A is always absolutely dominated. In fact,	if a —	where
ax, aa e А О С, a e А, Л1} Я2 > 0, and 4-A2	= L then let
ж ” ^1®1+Я2а2з
since A is convex, ж e A. It is clear that	< ж. Simple	examples
show that an absolutely dominated and convex set may	not	be de-
composable. A set which, is both absolutely dominated and absolute
order-convex is said to be solid. Solid sets may not be convex and are not
necessarily decomposable.
Example. Let R2 be the plane with the usual ordering ((a, /3) > 0 if
and only if a > 0 and /? > 0). Let A = {(a, /3) e R2: |a] < 2, |/3| < 2}
and В ~ {(a, /3): |a| +|/3| < 3}. Then A is a non-decomposable solid
set, and В is a solid set but not order-convex. The union A U В is
solid but neither circled nor convex nor order-convex.
The proof of the following proposition is similar to that given for the
preceding proposition and therefore will be omitted.
(1.8)	Proposition. Let p be a semi-norm on an ordered vector space
(В, C) and let V ~ {x e E\p{x) < 1}. We consider the following state-
ments:
(а,) V is absolutely dominated;
(a') for each x in E and each e > 0 there exists у in E with
p(y) <p[x)+s
such that —ж, ж < у;
(b) V is positively dominated;
FUNDAMENTALS OF ORDERED VECTOR SPACES
9
(b') for each x in E and each e > 0 there exists у in E with
p(y} <
such that 0, x y.
Then the statements (a) and (a') are equivalent, and statements (b) and
(b') are equivalent.
An ordered vector space (E, C) is said to have the Riesz decomposition
property if [0, u] + [0, r] = [0, и |-r] whenever и, v are elements in C.
It is easily verified that (E, G) has the Riesz decomposition property
if and only if [«1; г/J + |>а= У2] = Oi • жй> 2/i +2/aJ f°r alt xo Vi in ® with
хг < yi for each i. The following proposition is well known; the reader
is referred to Bourbaki (1965) or Fuchs (1966) for its proof.
(1.9)	Proposition. For an ordered vector space (E, G) the following
statements are equivalent:
(a)	(E, C) has the Riesz decomposition property;
(b)	if xit y^ e E and xi < y^ for i = 1, 2,..., n, and j =
there exists z in E such that xt < z < y$ for all i, j;
(c)	if xi} y} eC for all i = 1, 2,... n and j = 1, 2,... m, and if
n	m	m	n
2^ = 2 y^ then there exist c^ e G such that xi = ci3- and yj -= съ-.
г=1	j—1	i=l
A vector lattice (Riesz space) is defined to be an ordered vector space
(E, G) with proper cone G such that, for any pair of elements x, у in
E, sup (x, y) (the supremum of x and y) and inf(«, y) (the infimum of
x and y) exist in E. It is well known and elementary that each vector
lattice has the Riesz decomposition property, but the converse is false.
Next let us turn to a discussion of ‘dual orderings’. A functional f
(not necessarily linear) on an ordered vector space (E, G) is said to be
order-bounded if f is bounded on each order-bounded subset of E and is
said to be positive if f(x) > 0 for all x e G. Let E* denote the algebraic
dual of E consisting of all linear functionals on E. Let Eb denote the
set of all order-bounded linear functionals on E and C* the set of
positive linear functionals on E. Then C* £ Eb Q E*, and C* is a
cone in E* while Eb is a vector subspace of E*. The space Eb will be
conveniently referred to as the order-bound dual of E. C* will be called
the dual cone in E* and the induced vector ordering will be referred to
as a dual ordering. Unless an explicit statement is made to contrary, the
ordering in E* will always be the one induced by G*, and the ordering
in any subspace F of E* will be the relative ordering induced by
C* C\ F. In particular, the order-bound dual Eb has the dual ordering
10 FUNDAMENTALS OF ORDERED VECTOR SPACES
induced by С* Гу Eb = C*. Let E# denote the linear hull of C* in E*,
that is, E# — C* — C*. Then E# £ Eb c L?#} where the inclusions
may be strict (cf. Namioka (1957)). The ordered vector space (E#, C*)
will be referred to as the order dual of E.
(1.10)	Theorem (Riesz). Let (L?, C) be an ordered vector space with
the Riesz decomposition property and E = C—C. Then (Eb, C*) is a
vector lattice and Eb = E#.
Proof. Let f g Eb. Define for each и in C that
g(u) sup{/(a?):rr g [0, u]}.	(1.2)
Clearly g is positively homogeneous. Also, if u, v g C then
g(uj+g(v) = вир{Д«):ж G [0, w]}+sup{/(^):^ g [0, <]}
= sup{/(x) +f(y):x g [0, u\, у g [0, г?]}
= swp{f(x+y}:x G [0, u\, у G [0, yj}
= sup{/(^):s G [0,
where the last equality holds since E has the Riesz decomposition
property. Therefore g is additive on C. If x — u — v g E = C—C,
where u, v eC, then we define g(x) — g(u)—g(v). It is easy to verify
that g is a well-defined linear functional on E and agrees with the
formula (1.2) on C. It is easy to see that g eL*. By formula (1.2),
0,f < g so g g C* 22 Eb; in fact it is not difficult to verify that
g = sup{0,/}. Consequently, Eb is a vector lattice. Note also that
f = g--{g-f) g	—- E#. This shows that Eb c E^; hence
Eb - E#.
(1,11)	Proposition. Let (E, C) be an ordered vector space, and let
f g Eb. Then for each и in C we have
sup/[0, u\ 4-inf/[0, u\ = f(u).	(1.3)
Proof. If и = 0, the equality (1.3) is trivial. We may therefore
suppose that и g C and и Ф 0. Let у g [0, и]. Then u—y g [0, u] and
inf Д0, u\ <.f(u-y) —,f(u)—f{y) < sup ДО, и],
hence
f(y) + inf ДО, и] < f(u) < f(y) + sup ДО, и], for all у g fO, u].
The equality (1.3) follows easily.
FUNDAMENTALS OF ORDERED VECTOR SPACES 11
We shall now turn our attention to a problem concerned with
positive extensions of linear functionals. The following theorem is due,
independently, to Namioka (1957) and Bauer (1957).
(1.12)	Theorem. Let F be a vector subspace of an ordered vector space
(E, C), and letf be a linear functional on F. Then the following statements
are equivalent:
(a)	f can be extended to a positive linear functional on E;
(b)	there exists a convex and absorbing subset V of E such that
f(x) < 1 for all x g F Г\ (V~~C).
Proof. The implication (a) => (b) is clear; in fact, if g is a positive
extension of f on E, then the set
V - {x еЕ:д(х) < 1}
has the desired property. To prove the implication (b) => (a), we note
that V-- G is convex and absorbing because of V £ V — C. Suppose
that p is the gauge of V—G; then p is a sublinear functional on E and
/(a?) < p(x) for all x g F.
By making use of the Hahn-Banach theorem, there exists a linear
functional g on E which is an extension of f and
Note that
< Pky} for all у e E.
G £ V~C c= {хеЕ:р(х) < 1}
and that G is a cone; we conclude that g must be positive.
Let (E, G) be an ordered vector space. A subspace E of A1 is said to
be cofinal if G £ F—G.
(1.13)	Corollary. Let F be a, cofinal subspace of an ordered vector
space (E, C). Then every positive, linear functional on F can be extended
to a positive linear functional on E.
Proof. Without loss of generality we can assume that G is generating.
Let f be a positive linear functional on F, and let
Suppose that
U - {ж g Ftf(x) < 1}.
V - U-G.
12
FUNDAMENTALS OF ORDERED VECTOR SPACES
Then it is not hard to verify that V is convex and absorbing because F
is cofinal. Since C is a cone, it follows that
F r\ (V—C) - F n (Г7-С)
and that /(«)<! for all x g F Л (F—C').
The result now follows from the preceding theorem.
An element e in an ordered vector space (E, C) is called an order-unit
if the order-interval [—e, e] is absorbing in E, that is, if e g C and for
each x in E there exists Я > 0 such that — Яе < x < Яе. More generally,
a net {ел, Я g Л, <} in Е is called an approximate order-unit if the
following conditions are satisfied:
(a)	each ел is in C;
(b)	for any pair of elements Я15 Я2 in the directed set A with Ях < Я2,
it is true that < e2g;
(c)	for each x in E there exist Я e A and a positive real number a
such that —- aeA < x < аел.
Thus, if {e2} is an approximate order-unit then the set
Ал = U [—ел, e/J = {x e E-. —ex < x < eA for some Я e A}
is circled, convex, and absorbing in E. Thus the Minkowski functional
of Ал is a semi-norm on E and will be referred to as the approximate
order-unit semi-norm defined by {ez}. If e is an order-unit in E then the
Minkowski functional of [--e, e] is called an order-unit semi-norm defined
by e.
If e is an order-unit in (E, C) then the vector ordering is Archimedean
if and only if x < 0 whenever x < ae for all а > 0 and is almost-
Archimedean if and only if x — 0 whenever — ае < x < ae for all
а > 0.
(1.14)	Coeollaby, Let (E, C) be an ordered vector space with an
order-unit (or, more generally, an approximate order-unit {ел}), and let F
be a vector subspace of E containing e (or {eA}). Then each positive linear
functional on F can be extended to a positive linear functional on E.
Let (E, C) be an ordered vector space. A functional q, defined on 0,
is said to be superlinear if —q is sublinear, i.e.
q(hu) = hq(u) for all Я > 0, и e C, and
fpm1)) > Q.(E)-fq(oo) for all u, co in (7.
FUNDAMENTALS OF ORDERED VECTOR SPACES
13
The following generalization of the Hahn-Banach theorem was proved
by Bonsall (1955) and will be very useful in our subsequent discussions.
(1.15)	Theorem (Bonsall). Let (E, G) be an ordered vector space and
p a sublinear functional on E, and suppose that q is a superlinear func-
tional on C such that
q(u) < p(u) for all и in C.
Then there exists a linear functional f on E such that
f(x) < P(x) for x e
an^	q(u) < f(u) for all и g C.
Proof. Define, for every x e E, that
r(x) — inf{j?(a?+u)—-q{u)\u G C}.
Since p(u) < p(^+u)+p( —x), it follows that r(x) —p(—x'), and
hence that r is finite on E. Clearly r is a sublinear functional on E,
r(x} < р(ж) for all x in E, and r(—-u) < — q(u) for all и in G. By the
Hahn-Banach theorem, there exists a linear functional / on E such that
f(x) < r(x) for all xeE, and so /(a?) < p{x) for all x g E. Since
/( — u) < r(— u) < — q(u) for all и g G,
we conclude that/(u) > q(u) for all и g G. This completes the proof.
Bonsall’s theorem wifi be useful in our investigation of the duality
problems. To facilitate its applications, we introduce the following
notation. For any subset U of an ordered vector space (F, 0), we define
= u {[-u, u]:u gU n G}.
Then U is absolutely dominated if and only if 8(U) 2 U; U is
absolute order-convex if and only if 8(U) 2 U, and U is solid if and
only if 8(U) = U.
If A с E the polar of A, taken in the algebraic dual E*, will be
denoted by A” and defined by
A* = {fGE*:f(a) < 1, Va g A}.
For example, Cv —	C*.
(1.16)	Lemma. Let V be a subset of an ordered vector space (E, C).
Then we have	и/тгял icu-trwv
14 FUNDAMENTALS OF ORDERED VECTOR SPACES
Consequently, if V is absolutely dominated then the polar W of V, taken
in E*, is absolute order-convex in E*.
Proof. .Let f be in ^(F17). Then there is 0 < g e V” such that
Pf < 9- Let x e S(F) and suppose that < v for some v e V. Then
0 < (g~f)(vPx) = 9&) +^И
and	0 < (gPf)(v-x) = g(v)-g(x)Pf(v)-f(x).
Summing up, it follows that
0 < 2g(v) — 2/(ж).
Since v e V and g e V77, we then have f(x) < g(v) < 1, valid for all x in
$(F). This shows that f e (S(V)y and hence that ^(F77) £ (S(V))V.
.Further, if V is absolutely dominated, then V £ &(F); hence
^(F77) c (&(F))ff <= F";
that is, F* is absolute-order-convex.
(1.17)	Theorem (Jameson). Let (E, C) be an ordered vector space,
and let V be a convex and absorbing subset of E. Then
GS'(F))®' = A7(F").
Consequently, if F is absolute-order-convex then Vv is absolutely dominated,
and if V is solid then Vv is solid.
Proof. In view of the preceding lemma, we have only to show that
($(F))ff £ /S(Fff). Let g be in (/S(F))77. We have to find an f with
0 < f g V77 such that \ g < f. To do this, we define for each x e C that
= swp{g{yy.y e E, ±y < ж}.
Since g e (£(F))r and since F is absorbing, it is easy to see that q is a
well-defined superlinear functional on C. Further, let p be the gauge
of V. Notice that
and hence that
±V < X E F g(y) < 1
q(v) < p(v) for all v e C.
By theorem (1.15), there exists / e E* such that
and
/(?/) < p(y) for all у e E,
q(v) < f(v) for all v e C.
FUNDAMENTALS OF ORDERED VECTOR SPACES
15
Notice that < q(v) for all vgG; hence < f(v) for all
v G G and ffir? < f- Notice also that f(y) < p(y) < 1 for all у g V so
that f g This completes the proof for the first assertion of the
theorem. Furthermore, if V is absolute-order-convex then $(P) с V
and, by the first part of the proof,
7я £= $(7)я -£(7Я);
i.e. 7я is absolutely dominated. This, together with lemma (1.16),
implies that if V is solid then so is 7я.
There do not always exist non-zero positive linear functionals on an
arbitrary ordered vector space (cf. Jameson (1970), p. 2 ). By virtue of
the preceding theorem we'are now in a position to derive a condition
which is necessary and sufficient for the existence of a non-zero positive
linear functional on an ordered vector space E.
(1.18)	Proposition. Let (IS, C) be an ordered vector space. Then the
following statements are equivalent:
(a)	there exists a non-zero positive linear functional on E;
(b)	there exists a non-zero absolute-monotone semi-norm on E;
(c)	there exists a non-zero monotone semi-norm on E.
Proof, (a)	(b): If f is a non-zero positive linear functional on E
then the semi-norm p, defined by
p(x) = |/(ж)| (xeEf
is a non-zero absolute-monotone semi-norm, on E.
(b) => (c): Trivial.
(c) => (a): Let p be a non-zero monotone semi-norm on E. Then
there exists x0 in E such that т?(ж0) E 0. By the Hahn-Banach theorem,
there exists a linear functional g on E such that t/(«0) ~ p(x0) and
g(x)<p(x) (xgE).
Notice that g g IE, where U = {x eE:p(x) < 1}. Also, since p is
monotone, S(U) £ 3U; in fact, if —x < у < x and x G XJ, then
0 < x У у < 2x; hence
p(y) < p(x-Py)+p(—x) < p(x-]-y)Ep(x') < 24-1 = 3.
By the preceding theorem, (S(U)}V = 8(11^), and it follows that
IIP <= (S(U)Y =S(U*f
16 FUNDAMENTALS OF ORDERED VECTOR SPACES
In particular, g g 3 • £(IA); hence < 3/ for some f e Un with
f > 0. Let g1 = W+бО and g2 = i(3/-gr). Then gr, g2 are positive
linear functionals on E and g = gi~gz- Since дг(ж0) "A 0> one °f 9i an(i
g2 must be non-zero at x0.
Suppose that (E, C) is an ordered vector space. A sequence {xn: n e N}
in E is said to be increasing;, and we write xn], if xn < xm whenever
n < m. The sequence is said to be decreasing, and we write xn[, if
x.m < xn whenever n < m. If and x = sup xn exists in E, we then
write xnfx. Similarly for a decreasing sequence. The index set {хт: т g ,D]
(abbreviated to {ay}) of E is said to be directed upwards, denoted by
xT], if for every pair rx and r2 in D there exists r in D such that
xr < ay and xT^ < xT. It is said to be directed downwards, denoted by
жД, if for every pair ту and ra in D there exists r .in D such that
xr > xT and xT > ay. If {ay: r g j9} is a directed upwards subset of E,
and we define a relation 4 < ’ in D by
< r2 if хТ1 < xT2 ,
then (D, <) is a directed set, and hence {ау:т g D} is a net with the
property that жГ1 < жТа whenever r1 < r2; in this case, {жт:т G D} is
called an increasing net. If {жг:т g D} is a directed downwards subset
of E, and we define a relation ' < ’ in D by < 72 if xTi > xv then
(D, <) is a directed set, and hence {xr: r g D} is a net with the property
that xT xTz whenever тх < t2; in this case, {xT\r eD} is called a
decreasing net. If жД and if x — sup xT exists in E, we write жДж.
Similarly for sets directed downwards. A subset 7? of 77 is said to bo
order-complete if every increasing net in В that is majorized in E has a
supremum which belongs to B, and is a-order-complete if every increasing
sequence in В that is majorized in E has a supremum in B. In particular,
if E itself is order-complete, we then say that E is an order-complete
vector space or if E is a-order-complete, then E is an a-order-complete
vector space. A subspace off? which is order-convex is called an o-ideal.
A solid subspace В of E is called a normal subspace of E if it follows
from xT f x in E with xT in В for all r that x belongs to В; it is called
a a-normal subspace if it follows from xn f x in E with xn in В for all n
that x belongs to B. It is clear that each solid subspace of E which is
order-complete must be a normal subspace of E, and that if E is an
order-complete vector space, then each normal subspace of E is an
order-complete subset of E.
If {[Ea,	e!} is a family of ordered vector spaces then Ca
ael
is a cone in the product space JJ Ea. Notice that J J Ca is a proper cone
ctel	asl
FUNDAMENTALS OF ORDERED VECTOR SPACES 17
if and only if each Ca is proper. If (E, C) is an ordered vector space
and if E is the algebraic direct sum of subspaces	(i = 1, 2,..., n) of
E, we say that E is the ordered direct sum of E* (i = 1, 2,..., n) if
n
C — PJ where = C C\ Et (i = 1, 2,..., n). It is easily seen that
i==i
0 is a proper cone if and only if each is a proper cone in Fi, that G is
generating if and only if each is generating, and that the vector
ordering determined by G is Archimedean if and only if each vector
ordering determined by Ci is Archimedean.
CONES IN TOPOLOGICAL VECTOR SPACES
It is easily seen that the closure of a cone in a topological vector space
J? is a cone. But the closure of a proper cone in E need not be proper.
By way of example, consider R2 with a cone Cz defined by
Of = {(#, y)-x > 0} U {(0, y):y > 0},
then G\ is a proper cone in R2, and the closure Сг of C} with respect to
the usual topology is rz .
r ы	{(x,y)\x > 0},
but Cz is not proper.
A topological vector space with a cone is called an ordered topological
vector space and a locally convex space with a cone is referred to as an
ordered convex space. It should be noted that in an ordered topological
vector space the vector ordering and the vector topology need not
have any connection. However, if there exists a closed cone C in a
topological vector space (E, then the vector topology 0s must be
Hausdorff because С П — C — {0} is ^-closed; but the converse is, in
general, not true; for instance, consider R2 with a cone Co defined by
G) “ {(ж> У}'-х > 0 and У > 0} U {(0, 0)},
then CQ is a cone in R2, but it is not closed with respect to the usual
topology.
In what follows, all topological vector spaces will be assumed to be
Hausdorff, unless a statement is made to the contrary.
If (E, C, is an ordered convex space, throughout this book E'
will denote the topological dual of E, Eib will denote the set of all
bounded linear functionals on E, and C will denote the cone consisting
of all positive ^-continuous linear functionals on E, that is,
C' E’ C\ C*. Eib is referred to as the topologically bounded dual of E.
(2.1)	Proposition. Let (E, C, be an ordered topological vector
space and let < be the vector ordering determined by C. Then we have:
(a)	the vector ordering < is Archimedean if C is ^-closed;
(b)	the vector ordering < is almost-Archimedean if the &-closure C of
C is proper;
CONES :IN TOPOLOGICAL VECTOR SPACES
19
(c)	if C is tP-closed and if {хт: r g D} is an increasing net in E which
converges to x with respect to S?, then x = sup xT; if C is tP-closed and
if {xT: r g D} is a decreasing net in E which converges to x with respect to
<P, then x = inf xT.
Proof, (a) Suppose that nx < у for all positive integers n; we wish.
1	1
to show that x < 0. Since — у 0 with, respect to (P, and - y--x g G,
n	n
we conclude from the closedness assumption that
/1	\
—x = lim I —y —x g G,
n \n	/
and hence that x < 0.
(b) Suppose that —у nx у for all positive integers n. Then
~x — lim -- y-xf,
n \П /
x lim I- yPx
n \n
1	1
On the other hand, since - n — x g G and since - у 4-х g G, it follows that
n	nu
—xgG, x g G, and hence that x g G П --C. We conclude, from the
fact that (7 is proper, that x 0; therefore the vector ordering < is
almost-Archimedean.
(c) Suppose that xf\. We first show that xT < x for all т e D. For any
r in D, let	{a;v:v > r, w e D}.
Then AT c xT-\--G, and so the ^-closure AT of AT is contained in xrpG
because xTpG is ^-closed. Since xr converges to x, it follows that
x g Ar, and. hence that x g xtPG or, equivalently, x—xr > 0 for all r
in D. This shows that x is an upper bound of {ж/ r g D}. If у in E is
such that xT < у for all т g D, then y—-xr g О for all r g D. Since G is
^-closed and since y—xT converges to y~x> we conclude that y — x g C,
and hence that x < y. Therefore x ~ sup xT.
Finally if then {—-xT'.r g .D} is directed upwards and -—xT con-
verges to —x with respect to therefore ~x = sup(— xT) = — inf xT3
consequently x = inf xT. This completes the proof.
Remark. It is easily seen that if (E, G, P) is an ordered topological
vector space and if G is ^-closed, then each order-interval in E is
^-closed. Furthermore, G is ^-closed if and only if the vector ordering
< induced by C is ‘continuous’ in the following sense: Whenever
20
CONES TN TOPOLOGICAL VECTOR, SPACES
{жи}, {ym} are two convergent nets in (A?, ^) with limits x and у such
that xn < ym for all m, n, then it is true that x < y.
(2.2)	Proposition. Let (E, G, tP) be an ordered topological vector
space. Then e is an interior point of C if and only if [— e, e] is a &-
neighbourhood of 0; in this case, e is an order-unit element.
Proof, (a) Necessity. Let e be an interior point of G. There exists a
circled ^-neighbourhood V of 0 such that e + F C; it follows that
V S IG—e) П (e — C) since V is circled. It is clear that
H,e] = (G—e) n (e~G),
consequently [—e, e] is a ^-neighbourhood of 0.
(b) Sufficiency. Suppose that [—e, e] is a ^-neighbourhood of 0.
We conclude from	e] £ C that e is an interior point of G.
Finally, e is an order-unit element because [- e, e] is absorbing.
(2.3)	Corollary. Let (E, G, £P} be an ordered topological vector
space. If the interior of C is non-empty, then Еь с E'.
Proof. Let e be an interior point of (7, and let f be in Еъ. By the
preceding result, [—e, e] is a ^-neighbourhood of 0. Since f is bounded
on [-e, e], there exists Л > 0 such that
f(y) < 2 for all у e [—e, e].
It then follows that [—e, e] £ {y e E:f(y) < 2}, and hence that / e E'.
(2.4)	Corollary. Let (E, C, .P) be an ordered topological vector
space, and let the interior of G be non-empty. If e is an order-unit element,
then e is an interior point of C.
Proof. Let x be an interior point of G. By proposition (2.2),
[—- x, ж] is a ^-neighbourhood of 0. Since e is an order-unit element in
E, there exists 2 > 0 such, that
— 2e < x < 2e.
Clearly e-J-2~1 [~-x, ж] с G, and so e is an interior point of G.
For a barrelled space with a dosed cone, the condition that the
interior of G is non-empty in the preceding result can be dropped.
CONES IN TOPOLOGICAL VECTOR SPACES 21
(2.5)	Corollary. Let (E, C, PF} be an ordered convex space, and
let C be ^-closed. If (E, ^) is barrelled, then each order-unit element is an
interior point of C.
Proof. Let e be an order-unit element. Since C is ^-closed,
[—-e, e] is ^-closed, and so [ — e, e] is a barrel, consequently [—e, e] is a
^-neighbourhood of 0. By proposition (2.2), e is an interior point of C.
This completes the proof.
(2.6)	Proposition. Let (E, C, be an ordered convex space and let
Еьъ be the set of all A bounded linear functionals on E. Then Etbc. Eb if
and only if each order-bounded subset of E is tP-bounded.
Proof. The sufficiency is clear. We use the Mackey-Arens theorem
(of. Schaefer (1966)) to verify the necessity. Let В be an order-bounded
subset of E, and f e E'. Then f e Etv> £ Еъ. Hence f is bounded on B.
This shows that В is a(E, F')-bounded, hence ^-bounded.
In theorem (1.12) we considered a positive linear extension problem.
The following result deals with continuous and positive linear extensions;
the proof is exactly that given for theorem (1.12).
(2.7)	Proposition. Let F be a vector subspace of an ordered convex
space [E, C, <P). Then a linear functional f defined on F has a
&-continuous, positive linear extension to E if and only if there exists
a convex &-neighbourhood V of 0 such that
f(x) < 1 for all x g F Ci ( V— C).
(2.8)	Proposition. Let (E, C, IF) be an ordered topological vector
space with the topological dual E' and let f g E'. Then the following
statements are equivalent:
(a)	fGC'-C;
(b)	there exists a convex &-neighbourhood V of 0 and a positive constant
a such that f(x) < a for all x e С C (F—C);
(c)	there exists an o-convex circled Sd-neighbourhood W of 0 and a
positive constant such that f(x) < /3 for all x e W.
Proof, (a)	(b): Suppose that f — g—h where g, h are in O'. Let
F — {x eE:g(x) < 1}. Then F is a convex ^-neighbourhood of 0.
Further, if x e C C\ (F—C) then 0 < x < v for some v in F. Hence
f{x) g(x)~-h(x) < g(x) < 9^) < 1.
3
22 CONES IN TOPOLOGICAL VECTOR SPACES
(b)=>(c): We can further assume that V is symmetric. Let
U = {ж e V: \f{x) | < a} and W the order-convex hull of (7, i.e.
W = {U-j-C) n (U—C). Since/e E', it follows from
U - V П {хеЕ:Ц(х)\ < a}
that U is a ^-neighbourhood of 0, consequently W is an o-convex
^-neighbourhood of 0. Let w e IF and assume that и < w < v for
someiqv e L.Then|/(?z)| < a. Also, since0 < {w~-u)l2 < {v — u)j2 e V,
we have (w-~u)/2 e С П (F —C) so that
Consequently
f{w) ^f{w-u)Ef{u)	+	< 3a,
valid for all го e W.
(c)	=> (a): Since W is a neighbourhood, the polar W° of IF, taken in
E', is the same as that taken in E*. Let g fjfi. Then g e IF0. Since W
is order-convex, A( IF) £ IF; it follows from the duality theorem (1.17)
that	g e £ £(1FF = ^(IFff) - A(IF°).
Therefore there exists h e IF0 such that Eg < h. Then
g -= |(A+^)-|(A-g) eC-C'.
This completes the proof of the proposition.
Next we shall give a duality theorem parallel to the duality theorem
(1.17). We first prove a simple computation rule for polars. Recall that
if J. is a subset of a topological vector space E, the polar of A, taken in
the topological dual E', is denoted by A0; thus A0 = An П E'.
(2.9)	Lemma. Let 8 and T be two convex subsets of a locally convex
space E containing the origin 0. Then the following propositions hold:
(a)	if 8 П T — 8 П T {for example, 8 and T are closed) then
{8 (~\ T)° = co($° U T°), where the polars are taken in the topological
dual E' and go{8° U T°) denotes the o{E', Efclosed convex hull of
8° u T°:
(b)	if 0 is an interior point of 8 and of T then 8 П T = 8 C\ T.
(Thus, roughly speaking, if 8 and T are both open or both closed
convex subsets of E containing 0, then the polar of the intersection
equals to the closed convex hull of polars.)
CONES IN TOPOLOGICAL VECTOR SPACES 23
Proof. Notice that 8 and T are closed convex subsets of E con-
taining 0. By the bipolar theorem, we have
($ П T)° = co($° U T°f
Since 8 П T = 8 П T and the polar (in E') of a set is the same as the
polar of its closure, we have
(Я П T)° (ТГгГТ)0 =3 ($ П Ту = cb($° и T°) = hb(£° U T°),
proving (a).
To prove (b), let x g 8 ГЛ T. Since 8 is a convex set containing the
origin as an interior point, it is easily seen that Ax g 8 for each
0 < 2 < 1 (Ax is in fact an interior point of 8, cf. Schaefer (1966, p. 38)).
Similarly Ax g T. Letting A —> 1 in Ax e 8 Г) T, we have x g 8 П T.
This shows that 8 П T cz 8 n T and consequently 8 Г\ T = 8 T
since the opposite inclusion is obvious.
Remark. If we assume one of the sets 8, T to be a neighbourhood
of 0 and the other set is closed, then (b) remains true. If both sets are
neighbourhoods of 0 then $°, T0 are a(E', E)-compact by the Alaoglu
theorem (cf. Schaefer (1966)); hence co($° U T°) is o(E', JE')-closed and
(S n Ту - co(A° U T°f
(2.10)	Corollary. Let 8 and T be two convex absorbing subsets of
a vector space E containing the origin 0. Then, for polars taken in the
algebraic dual E*, we have
(8 n Ту - co(/Sir и T’).
Proof. Let r denote the Mackey topology in E with respect to the
duality (E, E*). Then all convex absorbing sets ini? are neighbourhoods
of 0 in (E, t). Hence the corollary follows from the remark preceding
the statement of the corollary.
(2.11)	Theorem. Let V be a circled, convex, absorbing subset of an
ordered vector space (E, Cf Then the following propositions hold:
(a)	(H(F)f = L>(F7r);
(b)	(П(У)Г - F(V*y,
where DIV'1'} and И(УЯ) respectively denotes the decomposable kernel and
the order-convex hull of V77 in the algebraic dual E*.
24 CONES IN TOPOLOGICAL VECTOR SPACES
Proof, (a) We first recall that (P — —C*. Next we show that
(7+Cy = 7я П Gv = -(7я n G*).	(2.1)
In fact, since 7, G contain 0, we have VpG э V, C; hence (Ffi-Cy cz
V" П C". On the other hand, if f g 7я П Cv then —f e F1 C\ C*
(since 7 is circled); hence ( —f)(c) > 0 for all cgC. Consequently
f(vpc) = /(v)+/(c) < f(v) <1 (v g 7, e g C),
showing that/G (7 + С)я. Therefore formula (2.1) holds. Similarly, we
can show that (7 -Gy = 7я П (7*. On applying corollary (2.10), we
then have
(7(7)F - ((7+0) n (7-+7)Г = со((7+СУ и (7-С)я)
= со(-(7я п С*) и (7я п С*)) - П(7Я).
(b)	Let / g (Р(7))я. Then/(ж) < p(x) for all х g G, where р denotes
the gauge of 7. By Bonsall’s generalization of the Hahn-Banach
theorem (theorem (1.15)), there exists a linear functional g on E such
that /(ж) < g(x) and g(y) < p(y) for all x g C and у e E. Then g g V*
and f < g. Similarly, considering -f instead of /, we can find A e 71
such that h < /. Hence / g 7(7я). This shows that (7)(7))я £ 7(7Я).
Conversely, let f' e F(Vvf and suppose x = Л1х1 — Л2х2, h' < /' < д',
where Л1) Л2 > О, -7Л2 = 1, хг, х2 g 7 п G, and h', д' е 7я. То
complete the proof we have to show that f'(x) < 1. To this end, we
notice that f'(xf) < </(aq) < 1 and f'(—x2) < h'{ -.ra) < 1 (since 7 is
circled, — x2 e 7). Hence
f'(x) = Л1/'(+)+Я2/'(-ж2) < z( za -= 1.
The proof of the theorem is completed.
(2.12)	Corollary (Jameson). Let V be a circled, convex, absorb-
ing subset of an ordered vector space (E, G), and suppose that V is order-
convex. Let f be a linear functional on E such that sup/(7) < go. Then
there exist positive linear functionals g, h such that
f^=g—h and sup ^(7) + sup A( 7) = sup/(7).
Proof. Without loss of generality, we can assume that sup /(7) = 1.
Then f e 7я = (#(7))я. By the preceding theorem, f e D(VV); thus
there exist positive linear functionals/!, f2 e 7я such that
/ = V1--V2
CONES IN TOPOLOGICAL VECTOR SPACES 25
for some Л2 > 0 with +	= 1- Let g = and h = Л2/2. Then
/ = g—h and, for each v in the circled set F, it is true that
f{v) ~g{v)--h{v) = g{v)-\~h{ — v) < sup gr(F)-f sup A(F);
hence 1 = sup /(F) < sup <?(F) -)-sup h{ F). On the other hand, since
S' “ and A e V\ we know that sup g{ F) < Similarly,
sup A(F) < Я3. Consequently,
sup (?(F)+sup A( Г) < A + /l2 ~ 1 < sup/(F).
Therefore	/(F) — sup g( V) + sup h( F),
(2.13)	Corollary. Let {E, C, IP) be an ordered convex space and V
a circled convex IP-neighbourhood of О-in E. If the topological dual E' is
order-convex in E* then (D(F))° = _F(F°).
Proof. Since Ef is order-convex in E*, the order-convex hull in E'
of a subset A in E' is the same as the order-convex hull of A in E*;
Ee- J-(J) = (Л+С") П (А—С") = (Л+С*) П (A-C*).
By (b) of theorem (2.Il),
(D(F)f - #(7”-) = F(F°)
and the above sets are contained in E'; thus the equalities can be re-
written as	(D(F))° = #(F°).
The following result is dual to (b) of theorem (2.11).
(2.14)	Theorem. Let {E, C, ^) be an ordered convex space with a
^-closed cone C, and let E' be the topological dual space with the dual
cone C'. Let F be a circled convex IP-closed neighbourhood of 0 in E, and
let Е(У^) denote the order-convex hull of F° in (Ef, C'). Then
(#(F°))° =Ж)>
where the polars are taken with the duality (E, E') and closure relative to IP.
{Remark. Since P(F) is convex, the ^-closure D(F) of D(F) is the
o{E, $')-closure of D(F).)
Proof. As in formula (2.1) in the proof of theorem (2,11), we have
(F°+C")° = F00 n (C")° - - F00 n C00 :- . - F о C,
26 CONES IN TOPOLOGICAL VECTOR SPACES
since V and d are closed. Also, by the Alaoglu-Bourbaki theorem, V°
is сфЕ', A)-compact, hence VOjrd' is a(E', A)-closed (and convex).
By lemma (2.9), we have
(A(P))° = ((7« + C") n (7«-C"))° = to((7°+C/)° и (У0 -С')0)
= ch(-(7 nC) u(F nd) - A(F).
The following separation theorem will be useful in our further
investigation.
(2.16) Theorem. Let (E, C, be an ordered convex space with the
topological dual E'. If К is a ^-compact convex subset of E, then there
exists fed' such that	„
J	swpf(K) < 0
if and only if К r\ d = ф, where C is the ^-closure of d.
Proof, (a) Necessity. If К П d ф, take x g К C\ d- then there
exists a net {жт: r g D} in d such that xT converges to x with respect to
because fed'; consequently
sup/(A) > sup{/(x):« g К cC)>()
which gives a contradiction.
(b) Sufficiency. Since C is ^-closed and convex, it follows from the
strong separation theorem, that there exists f e E' such that
sup/(A) < inf f(d) == inf f(d).
We claim that / is positive. In fact, if и g d then nu Ed for all natural
numbers n, and it follows from
sup/(A) <f(nu) — nf(u) (n = 1, 2,...)
that/(u) > 0; this shows that/is positive. Hence inf f(d) = 0 and so
sup/(A) < 0. This completes the proof.
(2.16)	Corollary. Let (E, d, be an ordered convex space, and
let d be &-closed. Then d is proper if and only if d' is total over E.
Proof. By making use of theorem (2.15), we have
d = {xeE:f(x) > 0, V/eC"),
CONES IN TOPOLOGICAL VECTOR SPACES
27
&o С n -C ^{xeE: f(x) = О, V/ e O'};
consequently the result follows immediately.
We are now in a position to give a similar result to the Bonsall
theorem (theorem (1.16)) with topological properties involved.
(2.17)	Proposition. Let(E,C, tP) be an ordered convex space with a
closed cone C. Let К be a convex compact subset of E containing the origin
and let F be the cone generated Ъу K, that is, F = pos K. Let p be a sub-
linear functional on F and let q be a superlinear functional on C with the
following properties:
(a)	q is upper semi-continuous on C;
(b)	p is lower semi-continuous on К and s = sup |p(W)| < d~oo;
(c)	q(x) < p(x) for all x e F C\ C.
Then, for each e > 0, there exists f e E' such that
q(x}<f(x) and f(y)<p(y)+e
for all x eC and у e K.
Proof. Consider the space E X R with the product topology. Let
H = {(a, £) g E xR:a e K, p(a) < £ —s < s}
ld	j9 = {(a, £) g E x R:a g C, S, < q(a)}.
Then H is a compact, convex subset of E x R containing (a, p(a) d-s) for
each a in ,K (in particular, containing (0, e)), and D is a closed cone in
E X R. Further, by (с), H and D are disjoint. Theorem (2.16) shows
that there exists a continuous linear functional ip on E X R such that
0 ~ sup ^(D) < inf ip(H).
Since (0, e) g 11, 0 < ip(0, e) = s. ip(0, 1) so ip(0, 1) > 0. Define p on E
by the rule	,
vdo, i)
The у is a continuous linear functional on E such that
ip(a, y(a)) =0 (a e E).
If у e К then (y, p(y) -|-e) g H and so
since y;(0, 1) > 0, it follows that p(y) < p(y) + e for all у e K. Similarly,
28
CONES IN TOPOLOGICAL VECTOK SPACES
since {x, д'(ж)) G D, we can show that q{x) < /z(x) for all x g G. Therefore
у is a required functional/, and the proof of the proposition is complete.
If {E, С, ёё?) is an ordered convex space, then O' is a o{E', 77)-closed
cone in E'. .Recall also that the topological dual of {E', cf{E', E)) is E.
Thus we arrive at the following proposition.
(2.18)	Proposition. Let {E, С, be an ordered convex space. Let
К be a convex o{Er, E)-compacl subset of E' containing the origin, and let
E be the cone generated by K. Let p* be a sublinear functional on F and let
g* be a superlinear functional on Gr with the following properties:
(a)	g* is upper semi-continuous on G' with respect to the relative,
a[E', Eftopology;
(b)	p* is lower a{Er, E)-semi-continuous on К and
s — sup |p*(7<) | <4-00
(c)	q*{f) < p*(f)for all fe F П G'.
Then, for each s > 0 there exists a o{Ef, E)-conlinuous linear functional
{and hence an element x in E) such that
£*(/)< Ж <™d д(ж)<р*(д)+«
for all f eC and g g K.
Remark. Let {E, ё?} be a locally convex space. Recall (cf. Baker
(1968)) that a subset A of E' is said to be almost o{E', Enclosed if its
intersection with each a{E', 7<7)-closed equicontinuous subset of E' is
a{E', 7?)-closed. This is the case if and only if А П F° is a{E', 7?)-closed
for each ^-neighbourhood V of 0 in E. Thus any a{E', 7?)-closed set is
certainly almost cr{E', .^-closed. Following Baker (1968), a locally
convex space {E, ёР) is called a hypercomplete space if each convex,
almost o{E', 7?)-closed subset of E’ is o{E', 7£)-closed. In such a space,
the condition (a) in the preceding proposition may be replaced by the
following equivalent condition:
(a') q* is upper semi-continuous on each equicontinuous subset of
O' with respect to the relative o{E', E)-topology.
In fact, (a) certainly implies (a'). Conversely, if g* satisfies (a') and
if Л is a real number, then let
ЛМ/еС^<д*(/)}.
Since g* is superlinear on O', Ax is convex. Further, if В is a a{E', E)-
closed and equicontinuous subset of G' then, by (a'), g*|5 is upper
CONES IN TOPOLOGICAL VECTOR SPACES
29
semi-continuous and hence
A, n В = {fA < (<?*U)(f)}
is a o(E', j&)-closed subset of B. Therefore ЛА П В is cr(7£', .^-closed in
E', whenever В is и(2?', J£)-closed and equicontinuous. Since E is
hypercomplete, Ал is о^', J5)-closed and so q* is upper semi-continuous
on C. This shows that (a')	(a). Therefore in proposition (2.18),
(a) may be replaced by (a') provided that E is hyper complete. By the
Krein-Smulian theorem (cf. Schaefer (1966)), any Banach space is
hypercomplete. Thus we arrive at the following corollary (by an
ordered Banach space we mean a Banach space with a partial ordering
induced by a cone).
(2.19)	Corollary. Let (E, (7, ||. ||) be an ordered Banach space. Let
К be a convex o(E', Efcompact subset of E' containing the origin, and
let F be the cone generated by K. Ijetp* be a sublinear functional on F and
let q* be a superlinear functional on C with the following properties:
(a') is upper semi-continuous on O' C\ S' with respect to the relative
a(E', Ef topology, where S' denotes the closed unit ball in (E1, ||. ||);
(b) p* is lower a(E', Efsemi-continuous on К and
s = sup |p*(7T)[ < -f oo
(c) q*(f) < p*(f) for all f g F ГТ C".
Then for each e > 0, there exists x in E such that
q*(f) < f(x) and g(x) < p*(g) -)-e
for all f g Cf and all g g K.
Remark. In most applications, К is taken to be the closed unit ball
S' or S' П (У. If К S' then the property
g{x)<p*{g)Te
implies that	....	. .	x .	.
r	||ж|5 = sup (/(ж) < supp'I=(gr) he — «s-f-e.
j/eE'	gelV
3
LOCALLY DECOMPOSABLE SPACES
In this chapter we study in detail the class of ordered topological
vector spaces (E, G, satisfying the property that each V gG — V G C
is a ^-neighbourhood of 0 whenever V is a ^-neighbourhood of 0.
Such a property is called an open decomposition property and, in this
case, we shall say that G gives an open decomposition in (E, £F). Recall
that a subset В of an ordered topological vector space is positively
generated if В с В G G — В G G. Thus G gives an open decomposition
in (E, SE) if and only if {E, SE) admits a neighbourhood-base at 0 con-
sisting of positively generated ^-neighbourhoods.
An ordered topological vector space (S', G, 3E) is said to have a
nearly-open decomposition property if V G G—V H G is a ^-neighbour-
hood of 0 whenever V is a ^-neighbourhood of 0. In such a case G is
said to give a nearly-open decomposition in (E,
Recall that if V is a subset of an ordered vector space (E, G) then
jD(V) denotes the decomposable kernel of F. If V is circled convex, we
have the following inequalities:
i(7 n G—V G G) cz B(V) £ F nC-F n C;
thus V G G—V H (7 is a ^-neighbourhood of 0 if and only if D(V) is
a ^-neighbourhood of 0.
A semi-norm p on (E, G) is said to be semi-decomposable if there exists
a positive constant M (depending, in general, on p) such that the
following condition holds: for any x e E and any e > 0 there exist
ж15 x.2 in C such that
x — x1—x2 and p(x1')Ji-p(x2) < Л/(р(ж)+е).
By a similar argument to that of proposition (1.7), p is semi-decom-
posable if and only if F £	where F — {x e E:p(x) < 1}. Thus
the equivalence (a) <=> (d) in the following theorem is clear.
(3.1)	The о кем. Let (E, G, ^F) be an ordered convex space. Then the
following statements are equivalent:
(a)	G gives an open decomposition in (E, &);
LOCALLY DECOMPOSABLE SPACES	31
(b)	& admits a neighbourhood-base at 0 consisting of circled convex
and decomposable sets;
(c)	admits a neighbourhood-base at 0 consisting of circled convex
and absolutely dominated sets;
(d)	P is defined by a family of semi-decomposable semi-norms
feaC Г}.
Proof. By the remarks preceding the theorem, it is clear that
(a) <> (b) <=> (d). It is easy to verify that a circled, convex, de-
composable set is absolutely dominated and that a circled, convex,
absolutely dominated set is positively generated. Thus (b)	(с)	(a).
liemark. Similarly we can show that C gives a nearly-open de-
composition if and only if the sets of the form 7)(F), where 7 is a
circled, convex ^-neighbourhood of 0, constitute a neighbourhood-base
at 0.
As suggested by part (b), an ordered convex space with the open
decomposition property will be referred to as a locally decomposable
space and, if it has the nearly-open decomposition property, it will be
referred to as a locally near-decomposable space. Specializing in ordered
normed spaces (that is, normed spaces with a vector ordering), we have
the following corollary.
(3.2)	Corollary. Let (E, G, |].||) be an ordered normed space and
P the vector topology induced by the norm || . || . Then the following state-
ments are equivalent:
(a)	(E, G, P) is locally decomposable;
(b)	there exists a norm ||. ||x on E and a positive real number a such
that ||. id is equivalent to ||. || and G is ^.-generating in (E, j|. ||x) in the
following sense:
Уж gE,3x1,x2eG such that || || x + ||tr21| x < a||x|[i
and
—— /у „____________________________, a
ел/ •--	*^2^
(c)	there exists a norm || ||2 on E which is equivalent to ||. || and has the
following property:
Уе > 0, Уж g E, Зу E G with 1Ы|3< IML + e
such that
—x, x < y.
Proof, (a) => (b): By the corresponding implication in theorem
(3.1), there exists a circled, convex, and decomposable ^-neighbourhood
32
LOCALLY DECOMPOSABLE SPACES
V of 0. Let ||. || j be the Minkowski functional of V. Then ||. || and ||. ||x
are equivalent. Also, since V is decomposable, in view of proposition
(1.7) (more precisely the proof of (a) => (b) in proposition (1.7)), we
conclude that C is (1 -fe)-generating in (E, ||. ||x) for arbitrary e > 0.
1
Conversely, if (b) holds, then - Sx c Sx П C-—Sx H C, where
a
Sx = {ж eE: ||ж||х < 1}. Hence Sx n C-~SX П C is a neighbourhood
of 0 in (E, || ||x) and hence in (E, &). This shows that (b) -> (a). There-
fore (a) о (b). Similarly we can show (а) о (c).
A cone C in a normed space (or semi-normed space) (E, ||. ||) is said
to be ^-generating if a is a positive constant such that for each x in E
there exist x1} x2 in G with ||жх|| +||ж2|| < а ||ж|| such that x жх—ж2.
This is the case if and only if S £ aD(S), where S — {x e E\ ||x|| < 1}
and 2)(S) is the decomposable kernel of S in (E, С). C is said to be
strictly generating if it is а-generating in (E, ||. ||) for some a. It is
obvious that a strictly generating cone must be generating. However a
generating cone in (E, ||. ||) may not be strict (of. example (3.7) below).
A cone G in a normed space (or semi-normed space) (E, ||. ||) is said to
be a nearly ^-generating if for each x in E there exist two sequences
{?/„}, {zn} in G such that ||«/n||-|~ИМ < а||ж|| and {yn~z^ converges to
x. Clearly this is the case if and only if S £ a D(S). Similarly to
corollary (3.2), the following result can be verified without any diffi-
culty.
(3.3)	Proposition. Let (E, C, ||. ||) be an ordered normed space and
TP the vector topology induced by the norm ||. ||. Then the following state-
ments are equivalent:
(a)	C gives a nearly-open decomposition in (E, TP);
(b)	there exists a norm || (f on E which is equivalent to ||. j| and is such
that G is nearly ^.-generating in (E, ||. ||') for some a > 0.
The following result, due to Nachbin (1965), implies that the open
decomposition property, in an ordered metrizable topological vector
space, is equivalent to a decomposition property for null sequences.
(3.4)	Proposition. Let (E, G, ^) be an ordered topological vector
space. Consider the following statements:
(a)	if {жа} is a net convergent to 0, then, for each a, there exist aa and
ba in C such that xa = a7-ba and the nets {aa}, {ba} converge to 0;
(b)	statement (a) with ‘net’ replaced by 'sequence';
(c)	G gives an open decomposition in (E, ^).
LOCALLY DECOMPOSABLE SPACES
33
Then (a) implies (c), and (b) is equivalent to (c) whenever the topology
is metrizable.
Proof. Suppose that C does not give an open decomposition in
(E, P}. Then there exists a ^-neighbourhood U of 0 such that
V Ф С n U—G n U
for every ^-neighbourhood V of 0. Let be the family of all
neighbourhoods of 0. Tor any V g there exists xv such that
xr g V and xv ф G И U—С И U.
It is clear that the net {xv: V g $1} converges to 0. If xv = av—bv,
where av, br are in C, then one of av> bv is not in U; hence neither of
the nets {aF: V g <$'}, {bv: VgP] converges to 0.
If the topology P is metrizable, then it is clear that (c) implies (b).
This completes the proof.
Given an ordered topological vector space (E, G, P), if E — C — G,
then we can construct a new vector topology in E with the open
decomposition property. In fact, let be a neighbourhood-base at 0
in {E, P) consisting of circled sets. For each d g % the set
U П G — U Odisa circled and absorbing set in E. Then the family
{U PiC-U C\G:U g determines uniquely a vector topology,
denoted by in E, for which the family is a neighbourhood-base at
0 for Clearly Фф is the greatest lower bound of all vector topologies
which are finer than P and have the open decomposition property.
The topology will be conveniently referred to as the vector topology
with the open decomposition property associated withP. It is clear that &
gives an open decomposition if and only if P = ^D. Notice that a
subset В of G is ^D-bounded if and only if it is ^-bounded. If P is
locally convex then so is and Ръ will also be called the locally
decomposable topology associated with P.
(3.6)	Proposition. Let (E, C, P) be an ordered topological vector
space such that E = C - G, and consider the topology PV) associated with
P. Then the following statements hold:
(a)	a positive subset В of E is P-bounded if and only if it is P^-
bounded;
(b)	if C is P-closed then a monotone increasing net {xn} P-converges
to x in E if and only if it P^-converges to x.
34
LOCALLY DECOMPOSABLE SPACES
Proof, (a) is easy to verify, and the sufficiency of (b) is trivial
since < ^D. Conversely, suppose {«„} ^-converges to x. Then, by
proposition (2.1), x > xn for each n; and for any circled ^-neighbour-
hood V of 0, there exists nQ such that x—xn e V whenever n > n0.
Consequently, for all n > n0, it is true that
x—xn eV П G с V П G — V nC.
This shows that {«„} converges to x in E with respect to
(3.6)	Theorem (Klee). Let P be a metrizable vector topology in an
ordered vector space {E, C) such that E =^= G—G. Then is also
metrizable. If, in addition, C is tP-complete, then {E, ^D) is complete.
In particular, if G gives an open decomposition in {E, tP) {and if C is
^-complete}, then {E, tP) is complete.
Proof. Let {Vf.n = 1, 2,...} be a countable base at 0 in {E,
consisting of closed, circled sets such that Vn+1pVn+1 £ Vn for each n.
Then the family of all sets
F„. П G-Vn nG (n ~ .1, 2,...)
is a neighbourhood-base at 0 in (E, ^D). Therefore is metrizable.
Next we show that {E, is complete. In fact, let (wn) be a Cauchy
sequence in (E, lPD). Then, for each Vk, there exists Mle > 0 such that
wri---wm e Vk П C—Vk П G whenever m, n > Mk,
therefore, there exists a subsequence of {wn} such that
П n G {k = 1, 2,...).
For each k, let xk, yk&Vk C\ G be such that
27c+] —= xk yk.
Then	n	n n
Zn+1' ri 2 (ht+l ‘ 2^7e' 2 У1С
7i=l	fc^l.
To show the convergence of the ^D-Cauchy sequence {wn}, it is suffi-
cient to show the convergence of the subsequence {z„}, and in turn it is
sufficient to show the convergence of the formal series 2 Lc and 2 У к
in {E, ^D). Let un = J xk, for each positive integer n. Then each
un e G and
un^4—un — xn^I-... ~Ixn_N e f^+1 П Gp...-j-Vn+(i H G c Vn nd;
LOCALLY DECOMPOSABLE SPACES
35
this shows that {un} is a ^-Cauchy sequence in C. Since G is F-
complete, {un} converges to an element, say u, in G. Further, since Vn
and G are ^-closed, passing to the limit as д' -> co in the last displayed,
formula we see that u—un e Vn Л G £ Vn C\ C--Vn C\ 0. This shows
GO
that the series xk converges in (E, .0^) to ад. Similarly, we can show
co	V-1
that	У к converges. The proof of the theorem is complete.
We have noted earlier that an ordered topological vector space
(E, G, F) has the open decomposition property if and only if F — 8?^,
and that this property certainly implies G is a generating cone. How-
ever, the converse is incorrect as the following example shows.
(3.7)	Example. Consider the Banach space C[0, 1] of all real-
valued continuous functions defined on [О, I]. Let C be the cone in
C[0, 1] consisting of all non-negative and convex functions, and let
E = G — C. By the Stone-Weierstrass theorem, E is dense in the Banach
space C*[0, 1]. Also, since any function in G must be differentiable on.
[0, Ij except at, at most, a countable number of points, any function /
in (7[0, 1] which is not differentiable at an uncountable subset of [0, 1]
is not in E. Therefore E is a proper dense subspace of (7[0, i], and E is
a non-complete normed space in its own right. Further, G is a generating
and norm-complete cone in E. By the preceding theorem of Klee, the
cone C does not give an open decomposition in (E, |j. ||).
The following result implies however that in a .Banach space, a
generating and complete cone always gives an open decomposition.
(3.8)	Theorem. Let (E, G, F) be a metrizable and ordered topo-
logical vector space, and suppose that C is F-complete. Then the following
statements (a) and (b) are equivalent:
(a)	(E, G, F) has the open decomposition property;
(b)	(E, G, F) has the nearly-open decomposition property.
Further, if E is of the second category, then each of the statements is
equivalent to
(с)	C is generating in E,
and, in this case, E must be F-complete.
Proof. It is trivial that (a) => (b). Conversely, suppose (b) holds.
Let F = G—C and let be the vector topology in F with the open
decomposition property associated with F. By Klee’s theorem
(theorem (3.6)), (F, FB) is metrizable and complete. Let i be the
identity map from (F, Fv) into (E, F). Then i is continuous. Further,
36
LOCALLY DECOMPOSABLE SPACES
take a ^^-neighbourhood of 0 in F of the form V C\ G—V И C,
where V is a ^-neighbourhood of 0 in E. Then, by (b), i( V C} G—V Ci G)
is a neighbourhood of 0 in (E, tF). This shows that the continuous map
i is ‘nearly-open’. By the Banach open-mapping theorem (cf. Schaefer
(1966, p. 76)), i is open and hence г is a homeomorphism from (F, iFV)
onto (E, 3F). In particular, & — and C gives an open decom-
position in (E, tF). Therefore (b) => (a) and statements (a) and (b) are
equivalent.
It is trivial that (a) => (c). Conversely, we show that (c)	(a) for
the case when (E, tF) is of the second category. By (c), E = C—C = F,
so г is a continuous linear mapping from the complete metrizable space
(E, ^D) onto the space (E, tF) of the second category. By the open-
mapping theorem (cf. Schaefer (1966, p. 76)), i is in fact a homeo-
morphism; hence & — and G gives an open decomposition in
{E, tF). Further, since E is ^D-comp]etc, so is ^-complete. This
completes the proof of the theorem.
Let us consider the case when is a locally convex topology in an
ordered vector space (E, C) such that E — G—G. Let Ft be a neighbour-
hood-base at 0 consisting of circled convex sets. Since, for all V g
КV n G — V nd) £= 77(F) s V n G — V n G,
the family 77(^<) — {79(F): F e F/} is a neighbourhood-base at 0 in E
with respect to the associated locally decomposable topology <FO. In
particular, if £F is the vector topology induced by a norm ||. || and if
2 = {x g E‘. ||ж|| < I}, then {e77(2): e > 0} — {I7(s2): e >0} is a
neighbourhood-base at 0 in (E, tFv). Thus is precisely the vector
topology induced by the Minkowski functional of 77(2). Specializing in
this normable case, we have the following numerical version of the
preceding theorem.
(3.9)	Theorem. Let (E, G, |j.||) be an ordered normed space and
suppose that G is || .\\-complete. Let a be a positive constant. Then the
following statements are equivalent:
(a)	C is (ad-е)-generating for each s > 0;
(b)	G is nearly ^.-generating.
Further, if (a) or (b) holds, then (E, [j. li) is complete.
Proof. Let 2 denote the closed unit ball in (E, ||. ||), and let 77(2)
denote the decomposable kernel of 2. Define a new norm ||.||' in
F — C—C to be the Minkowski functional of 77(2). Then (#, ||. ||') is a
LOCALLY DECOMPOSABLE SPACES
37
Banach, space, by Klee’s theorem. Also, the identity map i from
(E, ||. Г) into (E, ||.[|) is continuous. By (b), S 2 a .D(E), so the
closure of the image of Z)(S) under i is a neighbourhood of 0 in (E, |j. |().
By a well-known result (с/. Schaefer (1966, p. 76)),
whenever /3 > у > 0. In particular, in the space (E, ]|. )|),
(ad-r)Z>(E) 2 a I)(Xj 2 S (e > 0),
which implies that C is (aAs)-generating for each e > 0. This shows
that (b) => (a). Conversely, if (a) holds then
E 2 аВД-АВД <= ah(S)H-£S (e >0);
hence E 2 a I)(E), proving (b).
Returning to the case when (E, C> ^) is an ordered convex space,
the following theorem gives a dual characterization of the ^p-topology.
We first recall (cf. theorem (2.11)) that if V is a circled convex
neighbourhood of 0 then the polar F\ taken in E*, is the same as the
polar F° taken in E’, and (P(F))’r = E(F°), where -F(F°) denotes the
order-convex hull of F° in E*. Since D(%) = {D(F):F is a
neighbourhood-base at 0 in (E, bPD), it follows that is the topology
of uniform, convergence on all sets of the form #(F°), i.e. on the order-
convex hulls in E* of ^-equicontinuous subsets of E'. Thus we arrive
at the following theorem.
(3.10)	Theorem. Let (E, C, be an ordered convex space such that
E = C —(7. Then the locally decomposable topology associated with
& is the topology of uniform convergence on the order-convex hulls in the
algebraic dual (i£*, O*) of a,ll &-equicontinuous subsets of E'. Conse-
quently & is locally decomposable (that is & — ^D) if and only if the
order-convex hull in E* of each E^-equicontinuous subset is &-equi~
continuous.
Similarly, we have the following dual characterization (Duhoux
1972b) for (E, C, ^) to have the nearly-open decomposition property.
(3.11)	Theorem. Let (E, C, be an ordered convex space. Then C
gives a nearly-open decomposition in (E, if and only if the order-
convex hull in E' of each 0-equicontinuous subset of E' is &-equi-
continuous.
4
38
LOCALLY DECOMPOSABLE SPACES
Proof, (a) Necessity. Let A be a ^-equicontinuous subset of E'.
Then there exists a circled convex ^-neighbourhood У of 0 such that
A s 7° = 7ff. By part (a) of the duality theorem (2.11), we have
- (D(7))°	(П(7)Г n E' = 7(7°) n E' F(A) n E'
where F(A) denotes the order-convex hull in E* of A (so F(A) (~\ E' is
the order-convex hull in E' of Л). Since G gives a nearly-open de-
composition, .D( V) is a ^-neighbourhood of 0, it follows that the order-
convex hull in E' of A is ^-equicontinuous.
(b) Sufficiency. Let U be a circled convex ^-neighbourhood of 0.
Then U° is ^-equicontinuous and hence F(U°) C\ E' is ^-equicon-
tinuous by assumption; i.e., (Z)(?7))0 is ^-equicontinuous by theorem
(2.11). Hence D(U) is a ^-neighbourhood of 0. This shows that G
gives a nearly-open decomposition.
Theorems (3.10) and (3.11) have many important applications; we
mention a few below.
(3.12)	Corollary. Lei (E, G> be an ordered convex space and
the locally decomposable topology in E associated with 0. Then the
topological dual (E, tPfff of (E, hPff) is equal to the order-convex hull in
E* of E' = (E, &)'. In particular, if Sfl is locally decomposable then E' is
order-convex in E*.
Proof. Apply theorem (3.10).
The following result should be compared with theorem (3.8).
(3.13)	Corollary. For an ordered convex space (E, G, tP), the
following statements are equivalent:
(a)	C gives an open decomposition in (E, tF);
(b)	G gives a nearly-open decomposition in (E, SF) and E' is order-
convex in E*.
Proof. In view of the preceding corollary it is clear that (a) => (b).
Conversely, suppose (b) holds. Then, by theorem (3.11), the order -
convex hull in E' of each ^-equicontinuous subset A of E’ is equi-
continuous. Since E’ itself is order-convex in jS*, the order-convex hull
of A in E' is the same as the order-convex hull in E*. Thus the order-
convex hull in E* of each ^-equicontinuous subset of E' is ^-equi-
continuous. By theorem (3.10), tP must be locally decomposable;
this shows that (b) => (a).
LOCALLY DECOMPOSABLE SPACES
39
(3.14)	Corollary. Let (E, C, tP) be an ordered convex space with the
topological dual E' and suppose that E = C—C. Let т(Е, Ef s r be
the Mackey topology in E. Then т{Е, Ef is locally decomposable if and
only if E' is order-convex in E*. In particular, if (E, C, is locally de-
composable then so is (E, C, r{E, E')).
Proof. We recall that т(Е, E') is the strongest locally convex
topology in E with the dual E'. Thus, if r is locally decomposable
then E' is order-convex in E* by corollary (3.12). Conversely, suppose
E' is order-convex in E*, and let be the locally decomposable
topology associated with r. By corollary (3,12) again, the dual of
(E, rD) coincides with Ef. Since r < td it follows from the definition of
the Mackey topology that r •— rD; hence т is locally decomposable.
The last assertion of this corollary follows from the first and from
corollary (3.12).
We now give an example of an ordered convex space (E, C, tP) for
which E' is order-convex in E* but tP is not locally decomposable
(though the Mackey topology r(E, E') is locally decomposable). This, in
particular, shows that the open decomposition property is not invariant
with respect to all topologies for a dual pair (E, E'f
(3.15)	Example. Let L1 = i?[0, 1] be the ordered Banach space of
all Lebesgue integrable real-valued functions on [0, 1] with the usual
Td-norm ||. || and the usual ordering (we identify, as usual, the func-
tions which are equal almost everywhere). Then the topological dual
of L1 is L00 = L“[0, 1]. It is well known that the norm topology in L1
is precisely the Mackey topology t(Lx, L™), and is strictly finer than
the weak topology cfL1, L™). Let V be the polar in L1 of the constant
function 1 (as an element of L“). Then V is a <y{Lf Lro)-neighbourhood
of 0. Let S be the closed unit ball in L1. Then it is easy to verify that
V П C-V П C <= 2S,
where C denotes the positive cone in L1. Since S is not a tr(Lx, L00)-
neighbourhood ofO, V C—V П C must not be a neighbourhood of
0. Therefore C does not have the open decomposition property in
(L1, а(Е\ L00)).
Finally, we consider some permanence property of locally decom-
posable spaces. Let {(Ea, Cai : a g Г} be a family of ordered convex
spaces. Let E = jQ Ea denote the Cartesian product of Ea, ordered by
«еГ
40	LOCALLY DECOMPOSABLE SPACES
the product cone G = Ga. Let F = © be the algebraic direct
«сГ	«еГ
sum of the Ею ordered by relative cone C C\ F. For each a g Г, ja
denotes the natural canonical embedding map from Ea into F. The most
important topology for F is perhaps the so-called locally convex direct-
sum topology © which is the finest locally convex topology for
асГ
which is continuous. A neighbourhood-base at 0 for this topology is
provided by the family of all sets of the form V = co(U Fa), where
each Va is a neighbourhood of 0 in (Ea, SPfg. Here, as usual, we write
Fa instead ofyfiFJ; thus each E^ is considered as a subspace of F.
(3.16)	Theorem. Let be the locally decomposable topology
associated with and let (© be the locally decomposable topology
аеГ
associated with © ^a. Then
аеГ
© A	(3.1)
аеГ / D йеГ
Consequently the following assertions hold:
(a)	if each B°a is locally decomposable then so is ©
яеГ
(b)	the order-convex hull in F* of the product JJ (Ea, &f)' is precisely
aef
the product of order-convex hulls in E* of (Ea,
Proof. For each a in Г, let 17 be a circled convex neighbourhood
of 0 in (Ea, Shf). Let
A = co f U Faj and В = co ( U-Ж)
\абГ /	\аеГ
To prove the formula (3.1), it is sufficient to show that D(A) = B,
because D(A) and В are ‘typical’ neighbourhoods of 0 in F with respect
to (© and © respectively.
аеГ	аеГ
It is easy to see that В is a decomposable set; hence to show D(A) = В
it suffices to show that А П G = В C\ G. Since A 2 B, we have only
to show that A C\ G В C\ C. Let x g А П G and suppose that
71	П
x = 2 where 2 К = L each > 0, xt g Уя., and g Г. By the
i=l	*
definition of the product cone G, it follows from x e G that each
xi g Ca<. Hence xt g J7. П Ca. 2 D(K.)> and x g B. This shows that
x g В Pi G and hence that A n C 2 В (~\ G. Therefore formula. (3.1)
LOCALLY DECOMPOSABLE SPACES
41
is proved, and assertion (a) follows immediately. The assertion (b) also
follows from formula (3.1) and corollary (3.12) together with the
following well-known result:
(#, ® ^.V = II С®., ^)' and (f, ®	= II
\ аеГ / аеГ	\ аеГ / аеГ
The assertion (a) in the preceding theorem can be further generalized
in the following form. (A map from an ordered vector space into another
space is positive if it sends positive elements into positive elements.)
(3.17)	Theorem. Let (E, G) be an ordered vector space, {(Ea, Ca, tPfp.
a g Г} a family of locally decomposable spaces, and t№ a positive linear
map from Ea into E. Suppose that E is the linear hull of (J t^Lf). Then
аеГ
the inductive topology IP on E with respect to {(Ea, Ca; : a g Г} and
{ta: a g Г} is locally decomposable.
Proof. Recall that a neighbourhood-base at 0 for P is provided
by the family of all sets of the form V — col (J £a(K) )> where each
\аеГ /
Va is a neighbourhood of 0 in (Ea, .Pf). Now, since Рл is locally de-
composable, we can take to be circled, convex, and. decomposable.
Since ta is positive, HVf) must also be decomposable. Consequently V,
as the convex hull of decomposable sets, must be decomposable. This
implies that P is locally decomposable.
(3.18)	Corollary. Let (E, C, P) be a locally decomposable space,
J an order-convex subspace of E, and let q be the quotient map from E onto
E/J. Then the quotient topology is locally decomposable in (E[J, CjJ).
The following result may be regarded as a dual to theorem (3.16).
(3.19)	Theorem. Let {(Ea, Ga, ^):аеГ} be a family of ordered
convex spaces, and let E be the product space ordered by the product cone C.
Let JJ be the product topology and IJJ I the locally decom-
аеГ	\аеГ /В
posable topology associated with JJ ^a. Then
аеГ
П C.D
аеГ
(3.2)
42	LOCALLY DECOMPOSABLE SPACES
Consequently the following assertions hold:
(a)	if each is locally decomposable then so is JJ &a;
(b)	the order-convex hull in E* of the direct sum © (Ea, &ff is pre-
«ег
cisely the direct sum of order-convex hulls in E* of (Ea, &f}'.
Proof. For each a in Г, let Va be a circled convex neighbourhood
of 0 in (Ea, &f), and suppose all Va except for a finite number of a in Г
are equal to Ea. Then the formula (3.2) follows from an easily verified
fact:
(пк) Л с -(nd) П с = П (F„ n c-K n C),
because the set on the left-hand side is a ‘typical’ neighbourhood of 0
with respect to IJJ 0s A -topology and the set on the right-hand side
\аеГ /В
is the same with respect to IJJ D I-topology.
\аеГ /
Assertion (a) follows immediately from formula (3.2); and assertion
(b) follows also from formula (3.2) and corollary (3.12) together with the
following well-known result:
(e, n
\ аеГ / аеГ
and	,	\,
\	Л/ZlF	/	Afd"’
4
^-CONES AND LOCAL ^-CONES
Consider a non-empty set X and let be a family of subsets of X.
A subfamily Я of is called a fundamental system (or a fundamental
subfamily) for if each, member of is contained in some member of
Я 5 in other words, if Я is cofinal in under the set inclusion relation.
Let (E, C, &) be an ordered topological vector space, and & the
family of all ^-bounded subsets of E. Let £ 3d. The positive cone C
is called an ^-cone if the family
: A e Я
is a fundamental system for , and C is called a strict Чо-cone if the family
{J. П C—A CxC'.Ae^} is a fundamental system for In par-
ticular, G is an 0d-cone if each ^-bounded subset of E is contained in a
set of the form А П G—А П C where A is a ^-bounded set in E, and
C is a strict 3d-cone if each ^-bounded subset of E is contained in a set
of the form A C\ G—A П G. The former is the case if and only if the
family of all closures of all positively generated ^-bounded subsets
is fundamental in and the latter is the case if and only if the family
of all positively generated ^-bounded subsets is fundamental in 3d.
In view of the Mackey-Arens theorem, if 3?2 are two locally convex
topologies in E with the same topological dual, then a cone G in E is an
^-cone (or strict ^-cone) with respect to if and only if it is so with
respect to ^2.
Let V be a neighbourhood of 0 in an ordered topological vector space
(E, G, 3d). A subset В of E is said to be locally bounded with respect
to V if В is absorbed by V, i.e. if there exists Л > 0 such that В 2.V.
The positive cone G is called a locally strict 3d-cone if, for any ^-bounded
subset В of E and any ^-neighbourhood V of 0, there exist two
positive subsets Bx, B2 which are locally bounded with respect to V such
that В £ Bx-B2. C is called a local 3d-cone if, for any ^-bounded
subset В of E and any ^-neighbourhood V of 0, there exist two
positive subsets Bt, B2 which are locally bounded with respect to V
such that В ^.B1 —B2. Actually in the decomposition we can always
44	^-CONES AND LOCAL ^-CONES'
take Bt — B2 (replace Bx U B2 for Bx, B2 if necessary). Moreover, since
each ^-bounded set is locally bounded, with respect to each neighbour-
hood of 0, it is obvious that each ^-cone must be a local ^-cone and
each strict ^-cone must be a locally strict ^-cone. In the case when IP
is normable then the converse also holds.
(4.1)	Proposition. Let (E,C, IP) be an ordered topological vector
space. If G gives an open decomposition then G must be a locally strict
Id-cone. Conversely, in the case when (E, IP) is a bornological locally
convex space, any locally strict Id-cone (and, in particular, any strict
Id-cone) in E gives an open decomposition.
Proof. Suppose (E, C, IP) has the open decomposition property.
Let В be a ^-bounded subset of E, and V a ^-neighbourhood of 0.
Then V П G — V C\ G is also a ^-neighbourhood of 0 and thus absorbs
B‘s i.e. there exists Я > 0 such that В s Я(Р AC— V Л G). Let
A — A(V C\ G). Then A is a positive subset of E, absorbed by F and
В £ A—A, showing that C is a locally strict ^-cone. Conversely,
suppose G is a locally strict ^-cone in (E, IP) and that (E, IP) is a
bornological (locally convex) space. Let F be a circled convex IP-
neighbourhood of 0. We have to show that F И С — V AC is a
^-neighbourhood of 0. To this end, let us take a ^-bounded set В in
E. Then there exist positive subsets Bx, B2 of E such that В s B1—B2
and B1} В2 are absorbed by F. Hence В is absorbed by F П C—- F П C.
It is now clear that F H C—V П G is a circled convex set in E and
absorbs all .^-bounded subsets. Since IP is bornological, it follows that
F Г\ G—V ACisa ^-neighbourhood of 0.
Remark. By example (3.15), the bornological assumption in the
preceding proposition cannot be dropped.
Similarly, we can show the following proposition to be true.
(4.2)	Proposition. Let (E, C, IP) be an ordered topological vector
space. If C gives a nearly-open decomposition then C must be a local
Id-cone. Conversely, in the case when (E, IP) is an infrabarrelled locally
convex space, any local Id-cone (and, in particular, any Id-cone) in E
gives a nearly-open decomposition.
Proof. Similar to that given in the proof of proposition (4.1).
(4.3)	Proposition. Let (E, G, Id) be an ordered topological vector
space and suppose that E = C—G. Let be the topology in E associated
Ц
all
в
1
1||
II
11
11
я
ALL
'Mg'
fig
.If
ж
il
f
iff ;
О
I!
Ii

Si
I
SCONES AND LOCAL .^-CONES
4.5
with dt with the open decomposition property. Then the following state-
ments are equivalent:
(u) any subset В of E is dt^-bounded if (and only if) it is St-bounded;
(b) C is a locally strict Td-cone in (E, St).
Proof, (a)	(b): Let В be a ^-bounded subset of E. Then В is
^D-bounded by (a). Further, by the first part of proposition (4.1), G is a
locally strict Ad-cone in (A1, ^D). Hence, for any ^-neighbourhood F
of 0, there exist positive subsets B±, .B2 of E such that В c Bx — В % and
each of Въ B% is absorbed by the ^D-neighbourhood V C\ G—V A G
of 0, and in particular absorbed by V. Therefore (b) holds.
(b) => (a): Let В be a ^-bounded subset of E and let
V л G~V n C - U
be a ^D-neighbourhood of 0, where V is a ^-neighbourhood of 0.
We have to show that В is absorbed by U. By (b), there exist two
positive sets B19 B2 such that В £ B^B^ and each of B1; B2 is
absorbed by V and hence by V Л C (since Вг, B2 are positive) and a
fortiori by U. This implies that В is ^bounded.
(4.4)	Proposition. Let (E, G, At) be an ordered convex space and
Ad the family of all St-bounded sets. Then the following statements are
equivalent:
(a)	G is a strict Ad-cone in (E, At);
(b)	for each At-bounded subset В of E there exists a circled convex,
decomposable, and At-bounded subset A of E such that В £i A;
(c)	for each ^-bounded subset В of E there exists a circled convex,
absolute-dominated, and At-bounded subset A of E such that В £ A.
(a), (b), (c) are also equivalent if В £ A is replaced by В £ A in (b)
and (c) and in (a) C is a AS-cone.
Proof. Recall that if A is a circled convex subset of E then the
decomposable kernel D(H) is absolutely dominated and
1(_4 n C-A n C) £ £>(J.) £ A n C-A n C.
Hence (a) <=>(b) and (b) => (c). On the other hand, if A is an absolutely
dominated, circled convex set then A must be positively generated;
hence (c) => (a).
46
^-CONES AND LOCAL ^-CONES
(4.5)	Proposition. Let (E, C, ^) be an ordered convex space. Then
the following statements are equivalent:
(a)	G is a locally strict 3d-cone in (E,
(b)	for each .^-bounded subset В of E and each &-neighbourhood V
of 0 there exists a circled convex and decomposable subset A of E such
that A is absorbed by V and В A;
(c)	for each ^-bounded subset В of E and each ^-neighbourhood V
of 0 there exist a circled convex and absolute-dominated subset A of E such
that A is absorbed by V and В <= A.
(a), (b), (c) are also equivalent if В <= A is replaced by В A in (b)
and (c) and in (a) G is a local ^-cone.
Proof. Similar to that given for proposition (4.4).
(4.6)	Proposition. For any ordered normed space {E,G, ||.||), the
following statements are equivalent:
(a)	G gives an open decomposition in {E, ||.||);
(b)	G is a locally strict 3d-cone in (E, ||. |j);
(с)	C is a strict 3d-cone in (E, ||.||);
(d)	there exists a positive real number a such that G is ^.-generating
in (E, ||.||).
The following statements are also equivalent:
(a)' C gives a nearly-open decomposition in {E, ||.||);
(b)' C is a local 3d-cone in {E, ||. || );
(с)' C is a 3d-cone in {E, |[.||);
(d)' there exists a positive real number a such that G is nearly
a-generating in (E, ||. |]).
Proof. The equivalence (a) о (d) was established in corollary
(3.2). That (a) <=> (b) was proved in proposition (4.1). Similarly, it is easy
to verify that (b)o(c). The equivalence of (a)', (b)', (c)', (d)' follows
similarly.
If {E, |j.||) is of the second category (e.g. if E is a Banach space)
we can improve the implications (d)	(a) and (d)' => (a)' as follows.
(4.7)	Proposition. Let {E, G, ||.||) be an ordered normed space which
is of the second category. If for each x in E there exist two norm-bounded
sequences {yn}, {zn} in C such that {yn-~zn} converges to x in {E, ||.||) {in
particular, if E — C—C), then C is a 3d-cone in {E, ||.||) and hence is
nearly ^.-generating for some a > 0. If, in addition, G is ^.\\-complete,
then G is {aFef generating for all e > 0.
^-CONES AND LOCAL ^-CONES
47
Proof. Let S be the closed unit ball in (E, ||.||), and let Z>(S) be the
decomposable kernel of S. Then the stated decomposition assumption
oo ___________________________________________
of the proposition implies that E — (J %D(S). Since E is of the second
n=i	_____
category, there must exist some positive integer N such that 2Vj9(S)
has a non-empty interior. Since N Е(Т>) is circled convex, its interior
must contain the origin. In other words, Z)(S) is a neighbourhood of
0, and so is D(S). This shows that G gives a nearly-open decomposition
in (E, ||.||) and hence, G is nearly а-generating for some a > 0. Finally,
if C is also |j.||-complete, then it follows from theorem (3.9) that G is
/^-generating for each /3 > a.
Theorem (3.9) implies that if a cone G in a normed space (F, ||.||) is
||. ||-complete then G is a ^-cone if and only if it is a strict ^-cone. The
following proposition demonstrates another class of vector spaces with
such a property.
(4.8)	Proposition. Let (E, G, £P) be an ordered convex space and let
the topological dual E' be ordered by the dual cone G'. Suppose that every
a(E', Efbounded subset of G' is equicontinuous. Then G' is a strict
A§-cone in (E', o(E', E)) if (and only if) it is a Ad-cone in (Er, o(E'} E)).
Proof. Suppose G' is a ^-cone in (E', o(E', E)). Let В be a o(E', E)-
bounded subset of E'. Then there exists a a(E', B)-bounded subset of
E' such that В s A C\ & —A C\ G'. Clearly we can further assume,
without loss of generality, that A is o(Er, j&)-closed. Then А П G' is
<r(B',-Enclosed and bounded; also, by assumption, A C\ C' is equi-
continuous and hence o(E', B)-compact in view of the Alaoglu-
Bourbaki theorem. Consequently A C\ G'-—A П G' must be o(E', E)-
closed, hence В c A n O' —A r\ Gr = A n O' — A C\ G'. This shows
that G is a strict ^-cone in (Er, a(E’, E)).
It is clear that if an ordered, convex space (E, C, tP) is barrelled then
every positive o(E', _®)-bounded subset of E' must be equicontinuous,
and that the strong dual of a semi-reflexive space is barrelled. Thus we
have the following corollary immediately.
(4.9)	Corollary. Let (E, G, AP) be an ordered convex space. If E
is semi-reflexive, and if C is tP-closed, then C is a strict Ad-cone in (E, A0)
if (and only if) it is a Ad-cone in (E, tP).
5
LOCALLY O-CONVEX SPACES
Let (E, C) be an ordered vector space. Recall that a subset A of E is
order-convex (full) if [a1; a2] s A whenever a15 a2 g A and ar < a2. A
vector topology A in (E, C) is said to be locally order-convex (or locally
full) if it admits a neighbourhood-base at 0 consisting of order-convex
sets. In this case, we shall say that (E, C, A) is a locally order-convex
space (locally full space) and the cone C is called a normal cone in (E, A).
Assuming A to be Hausdorff (as we always do), a normal cone C must
be a proper cone. In fact, supposes e О П — C. Let tyl be a ^-neighbour-
hood-base at 0 consisting of order-convex sets. Then x e [0, 0] £ U for
each U g At. Since A is Hausdorff, {0} =	g hence x — 0.
(5.1)	Theorem. Let (E, O, A) be an ordered topological vector space.
Then the following statements are equivalent:
(a)	A is locally order-convex;
(b)	if {xn, n g D, <} and {yn, n g D, <} are two nets such that
0 < xn < yn for each n g D and if {yn} converges to 0 in (E, A) then so
does {.rj;
(c)	for any A-neighbourhood W at 0, there exists a A-neighbourhood
V of 0 such that (7-0) n О £= W;
(d)	for any A-neighbourhood W at 0 there exists a circled ^-neighbour-
hood U of 0 such that [О] <= J7;
(e)	A admits a neighbourhood-base at 0 consisting of circled and
order-convex sets.
Proof, (a) => (b): Straightforward.
(b)	=> (c): Suppose on the contrary that there exists a A-
neighbourhood W of 0 such that
(7—0) n О ф W
whenever V g At, where At denotes the family of all ^-neighbourhoods
of 0. Lor each 7 e At, there exists xr e ( 7—О) A О but xv f W. Let
yv g 7 be such that 0 < xv < уг. Notice that {yv, 7 e 2 } is a net
in (E, A) converging to 0; but {жг} does not converge to 0.
LOCALLY O-CONVEX SPACES
49
(c)	=> (d): Take a circled ^-neighbourhood Л] of 0 such that
+	By (c), there exists a ^-neighbourhood W2 of 0 such
that C) C\ G £ W±. Then take a circled ^-neighbourhood JFj of
0 such that W3+W3 <= W2. Let U = TFX nWs. Then U is a circled
^-neighbourhood of 0 with the property that [17] c jy.
Finally, the implications (d) => (e)	(a) are trivial, and the proof of
the theorem is complete.
(5.2)	Proposition. Let (E, C, .P) be an ordered topological vector
space. If 0 is locally ordered-convex then the order-convex hull of each
IP-bounded subset of E is P-bounded. The converse also holds in the case
when IP is metrizable.
Proof. Let В be a ^-bounded subset of E, and V a ^-neighbour-
hood of 0. If .P is locally order-convex then there exists an. order-
convex ^-neighbourhood U of 0 such, that U s V. Иепсс В is absorbed
by U and so [77] is absorbed by [?7] = in particular [B] is absorbed
by V. This shows that [B] is ^-bounded.
Conversely, suppose IP is a metrizable vector topology in (E, C) such
that the order-convex hull of each ^-bounded subset of E is IP-
bounded. We have to show that IP is locally order-convex. If not then,
in view of the preceding theorem, there exists a ^-neighbourhood W
of 0 such that	(F-C) n С ф W,
whenever V is a ^-neighbourhood of 0. Now take a countable neigh-
bourhood-base {Vn:n = 1, 2,...} at 0 such that Vn+1lWn+1 cz for
each n. Then
n С ф W ln = 1,2,...).
\n /
/1	\	.1
Hence, for each n there exist xn in / - Vn — С | ГЛ C and yn in - Vn such
\n	I	n
that	i
0 < xn < yn g - Vn and xn ф W.
Since each nyn e Vn, it is clear that {nyn} is a sequence in (E, IP)
converging to 0; hence that the set
A = {0} kJ {nyn:n = I, 2,...}
is -bounded. Notice that nxn e [0, nyj £ [Л]. Since nxnfn'W, it
follows that [Л] is not contained in any nW (n = 1, 2,...). This implies
50
LOCALLY O-CJONVEX SPACES
that the order-convex hull [yl] of the ^-bounded set is not ^-bounded,
which is a contradiction.
(5.3)	Corollary. Let (E, G, tP) be a metrizable ordered topological
vector space. Then is locally order-convex if and only if there exists a
locally order-convex topology in E for which 0 and <3? have the same,
topologically bounded sets.
Next let us consider, in ordered vector spaces, vector topologies which
are both locally convex and locally order-convex; such spaces will be
called locally o-convex spaces, and the topologies are called locally
o-convex topologies. Since a locally order-convex topology is not
necessarily locally convex, the concepts of locally order-convex
topologies and of locally o-convex topologies are distinct.
Recall that by an o-convex set we mean a set which is both order-
convex and convex. For example, the order-convex hull of a convex
(circled) set is an o-convex (circled) set. Hence a locally convex topology
0 in an ordered vector space (E, C) is locally o-convex if and only if IP
admits a neighbourhood-base at 0 consisting of o-convex (or circled
o-convex) sets.
Recall also that by an absolute-o-convex set we mean a set which is
both convex and absolute-order-convex, and by a positive-o-convex
set we mean a set which is both convex and positive-order-convex.
(5.4)	Theorem. Let (E, G, tP) be an ordered convex space. Then the
following statements are equivalent:
(a)	is locally o-convex;
(b)	admits a neighbourhood-base at 0 consisting of circled o-convex
sets;
(c)	S? admits a neighbourhood-base at 0 consisting of circled absolute-o-
convex sets;
(d)	admits a neighbourhood-base al 0 consisting of circled positive-
o-convex sets;
(e)	the family of all ^-continuous absolute-monotone semi-norms
determines the topology &;
(f)	the family of all tP-continuoiis monotone semi-norms determines
the topology &.
Proof. The equivalence (a) о (b) was noted by the remark preceding
to the theorem. It is trivial that (b) => (c) => (d). In view of the implica-
tion (c) => (a) of theorem (5.1), we also have (d) => (a). Therefore the
LOCALLY O-CONVEX SPACES
51
statements (a), (b), (c), and (d) are mutually equivalent. Furthermore,
by proposition (1.6), (с) о (e) and (d) о (f). The proof of theorem (5.4)
is thus completed.
(5.5)	Coeollaby. Let G be a cone in a locally convex space (E, IP),
and let G be the ^-closure of G. Then (E, C, IP) is locally o-convex if and
only if [E, G, IP) is locally o-convex.
Let a be a positive constant. A cone G in a semi-normed space
(E, ||. ||) is said to be a-normal if the following implication holds:
x < у < z in (E, G) => \\y\\ < а.тах{||ж||, ||z||}
(thus a must be larger than, or equal to, 1); in this case, we also say
that the semi-norm ||.|| is а-normal in (E, C). If S denotes the closed
unit ball in (E, ||.||) and F(S) denotes the order-convex hull of S in
(E, G) then G is а-normal if and only if F(S) £ aS. In this case the
family of the sets of the form {F(rS):r > 0} = {r#(S):r > 0} is a
neighbourhood-base at 0 in E with respect to the vector topology
induced by ||.||. Thus the first statement in the following proposition
is clear.
(5.6)	Proposition. Let \\.\\ be a semi-norm in an ordered vector space
(E, G). If there exists a> 1 such that G is а-normal in (E, II-II) the
||. ||-topology is locally o-convex. Conversely, if & is a semi-normable and
locally o-convex topology in (E, G) then there exists a semi-norm p on E
which is 1 -normal and is such that & is precisely the vector topology
induced by p.
Proof. It remains to show the second assertion of the proposition.
If is locally o-convex and semi-normable, then, in view of theorem
(5.4), there exists a circled and o-convex ^-neighbourhood V of 0 in E
such that {rV:r > 0} is a neighbourhood-base at 0 in [IL &) Then the
Minkowski functional of V is 1-normal (cf. proposition (1.6)) and induces
the topology IP. This completes the proof.
Clearly C is а-normal if and only if it is (a + e)-normal for each e > 0.
This remark makes the following result clear.
(5.7)	Corollary. Let (E, G) be an ordered vector space. Then a semi-
norm on E is a-normal in (2*7, G)for some a > 1 if and only if there exists
a semi-norm p equivalent to the given semi-norm such that p is 1-normal.
, fHEMATISCH INSTHTU
( SjKSUNlWRSrraT ТЕ	
BIBUOTHEEK
52
LOCALLY O-CONVEX SPACES
Wo now turn our attention to a study of convergence in locally
o-convex spaces. The following proposition is due to Bonsall (1955)
and the proof given here is due to Weston (1957a).
(5.8)	Proposition. Let (E, G, SP) be a locally o-convex space such
that G is (P-closed, and let {xT‘.r e D} be an increasing net in E. Then
xT converges to x with respect to & if and only if xT converges to x with
respect to a(E, E').
Proof. The condition is clearly necessary. To prove its sufficiency
we first recall, from proposition (2.1), that x = sup x?. Suppose that
ur = x— xr feD). Then 0 < адД and uT converges to 0 with respect to
a(E, E'f By applying the strong separation theorem, 0 is contained in
the ^-closed convex hull of {ut;t g I)}; thus, for any o-convex. and
circled ^-neighbourhood V of 0, there exists a finite subset
{rpi — 1, 2,..., n}
n	n
of D and real numbers Д > 0 with У h - 1 such that V LuT g G;
i=l	«‘=1
since .D is a direct set, there exists t0 g P such that тг- < т0 for all
n	n
i — 1, 2,..., n. We have, therefore, 0 < иг = У. кгит < when-
i=i	i=t	1
ever -г > r0; it follows from, the order-convexity of V that uT g V
whenever т > r0, and hence that xr converges to x with respect to AG
We recall that if (E, tP) is a locally convex space and if Ad is a family
of ^-bounded sets in E, we say that Ad is saturated if
(а)	Ad contains arbitrary subsets of each of its members;
(b)	kA g Ad for all к g R and A g Ad\
(c)	for each finite subfamily {Api = 1, 2,...,n}, say, of Ad the
n
closed circled convex hull of (J is in Ad.
<=1
The following theorem, due to Schaefer (.1966), may be regarded as
one of the central results in the theory of ordered topological vector
spaces.
(5.9)	Theorem. Let (E, G, kP} be an ordered convex space with the
topological dual E', and let Tl be a saturated family consisting of a{E', Ey
bounded subsets of E’ for which the linear hull of U {В: В e is o‘(E', E)-
dense in E'. Suppose that Ad is the ^-topology on E.
(a)	If G' is an ^A-cone then A? is a locally o-convex topology.
LOCALLY O-CONVEX SPACES
53
(b)	Suppose that is a locally o-convex topology. If & is consistent
with the duality {E, E'}, then O' is a strict ^-cone.
Proof, (a) Since the linear hull of U {В: В & is a{Ef, J57)-dense in
E', the topology is a (Hausdorff) locally convex topology in E. Let
F be a ^-neighbourhood of 0 in E. Then there exists a circled convex
set A in such that A® £ V, where the polar is taken in E. Since O'
is a i^-cone, A is contained in the a(E', Enclosure JD(B) of the de-
composable kernel of В for some circled convex В in . Then
(Р(Б))° = (2)(B))° is a ^-neighbourhood of 0. Clearly (j9(B))° is
order-convex (cf. the proof of theorem (2.11)) and contained in. A0 and
hence in F. This shows that is locally o-convex.
(b) Conversely, if & is locally o-convex, then each ^-equicontinuous
subset of E* is contained in a decomposable equicontinuous set. If,
in addition, & is consistent with the duality (2?, E'}, then is a
fundamental family for the equicontinuous sets. It follows in particular
that each member of is contained in some decomposable member of
, i.e. Gr is a strict ^-cone. This completes the proof.
The preceding theorem can be restated in the following form (cf.
theorem (3.10)).
(5.10)	Theobem. Let (E, G, £P) be an ordered convex space with the
topological dual E', and let be a saturated family consisting of relatively
a(E', E)-compact subsets ofE'. If G' is an TGcone then it is a strict 'tf-cone.
In particular, if we take % to be a fundamental family of equicontinuous
subsets of E', then is a locally o-convex topology if and only if C' is
an $-cone, and this is the case if and only if G' is a strict %-cone.
It should be noted that if (E, SF) is barrelled then £ coincides with
the family of all a(E’, J?)-bounded subsets of E'. If (E, 3?) is infra-
barrelled then £ coincides with the family of all p(E', 2?)-bounded
subsets of E'. Therefore the following result is a consequence of the
preceding theorem.
(5.11)	Corollary. Let(E,C, dP) be an ordered convex space which is
barrelled. Then the following statements are equivalent:
(a)	(E, G, tP) is locally o-convex;
(b)	G' is a strict Aft-cone in (E', a(Er, E));
(c)	G' is a Aft-cone in (Ef, a(E', Ef).
5
54
LOCALLY O-CONVEX SPACES
If (E, G, is infrabarrelled then (a), (b), (c) are still equivalent but
with (E', a(E', E)) in (b) and (c) replaced by (E', fi(E’, E)).
The following result should be compared with corollary (3.14).
(5.12)	Coeollaby. Let (E, C, ^) be an ordered convex space. Then
E' O' —G' if and only if (E, C, a(E, E')) is locally o-convex. In
particular, if IP is locally o-convex then so is u(E, E').
(5.13)	Corollary. Let (E, G, IP) be a metrizable ordered convex
space for which E is of the second category, and let G be IP-complete. If
o(E', E) (or fi(E', E)) is a locally o-convex topology on E' then (E, G, IP)
is locally decomposable, and hence E must be ^-complete.
Proof. If p(E', E) is locally o-convex then the order-convex hull
in E' of each fi(E', ./^-bounded subset is f(E', £7)-bounded. Since IP is
metrizable, fi(E', ./^-bounded sets are ^-equicontinuous; it follows that
the order-convex hull in E' of each ^-equicontinuous subset is IP-
equicontinuous. By theorem (3.11), IP has the nearly-open decomposi-
tion property, and hence is locally decomposable since G is ^-complete
(cf. theorem (3.8)).
Similarly, if o(E', E) is locally o-convex then, in view of the pre-
ceding corollary, G must be generating in E. It follows from theorem
(3.8) that IP is locally decomposable.
(5.14)	Corollary. Let (E, G, IP) be an ordered convex space. If G
is an IP-cone in (E, IP) then (E!, G', fi(E', E)) is locally o-convex. The
condition is also necessary for the case when (E, IP) is semi-reflexive and
G is IP-closed.
Proof. The first assertion follows from (a) of theorem (5.9). Con-
versely, if (E, IP) is semi-reflexive and G is ^-closed, then E may be
regarded as the dual of (E1, G', fi(Er, E)) with the ‘dual cone’ G. If,
further, (E'} G', f(E', E)) is locally o-convex then it follows from
theorem (5.10) that the dual cone G is a strict ^-cone in (E, IP) and a
fortiori a ^Lcone.
Specializing in ordered normed spaces, theorem (5.9) can be strength-
ened as follows.
(5.15)	Theorem (Grosberg-Krein). Let (E, G, be an ordered
normed space and a a positive real number. Then (E’,G’, Щ) is
^.-generating if and only if (E, G, |]. ||) is ^-normal. This is the case if and
LOCALLY O-CONVEX SPACES	55
only if
||ж|| < а.зир{|/(ж)| :f e O', ||/|| < 1}, forallxsE.
Proof. Let S, S' denote the closed unit balls in E and E' respec-
tively. By the duality theorem (theorem (2.11)) we have
(Е(Я)У = L(^) = D(S').
Therefore (E, 0, ||.||) is a-normal
о E(S) <= aS
o(E(S))7r =2 (aS)”’ =-i S'
a
oa.D(S') 2 S'
о (E', O', ||. ||) is a-generating.
This proves the first assertion of the theorem; the second assertion
follows easily from, the first and the Hahn-Banach theorem.
The following theorem was proved by Ellis (1964), and in a some-
what weaker form by Ando (1962).
(6.16)	Theorem (Ando-Ellis). Let (E, C, ||.||) be an ordered normed
space and suppose that 0 is \\.\\-complete. Let a > 1. Then the following
statements are equivalent:
(a)	C is «-normal in (E', ||.[|);
(b)	C is nearly «-generating in (E, ||.||);
(с)	C is (&L^-generating in (E, ||. ][) for any e > 0.
If one (and hence all) of the equivalent properties is satisfied then
(E, ||.||) is complete.
Proof. In view of theorem (3.9), we have only to show that (a)o(b).
Since C is ||.||-closed, the polar (C")° of O', taken in E, is exactly --C
(cf. the proof of theorem (2.11)). Thus
(S'+C")° - (S')° r> (C")° = -(S Л 0).
By the Alaoglu-Bourbaki theorem, S' + <7' is o(E', E)-closed. In view
of the bipolar theorem, we have the following chain of equivalent
56
LOCALLY O-CONVEX SPACES
statements:
G' is a-normal
о(2ЧП n (S'-C") £ aS'
о {(S'+C") n (S'2 - S
a
о Бо((2'+С")° U (S'-0')°) 2 - S
a
oco( —(S n (7) u (S n C))	~ S
a
2)(S) 2 - S
a
о G is nearly a-generating,
proving that (a) o(b).
(5.17)	Corollary, Let {E, G, j|.||) be an ordered Banach space, and
suppose that G is j|. \\-closed. Then fi(E', E) is a locally o-convex topology
on E' if and only if a{E', E) is a locally o-convex topology on E'.
We conclude this section with a method constructing a locally full
topology from a vector topology.
Let (E, G, be an ordered topological vector space, and let be
a neighbourhood-base at 0 for £P. Suppose that
F(W) - {[F]:fe	- {f(F):Fe
It is easily seen that there exists a unique locally full (locally order
convex) topology SPY, say> such that FlfU) is a neighbourhood-base
at 0 for SP^; moreover, is the least upper bound of all locally full
(locally order-convex) topologies which are coarser than PF. This
topology SPV is referred to as the locally full {locally order-convex)
topology associated with SP. If £P is locally convex, then is a locally
o-convex topology; in this case, is called the locally o-convex topology
associated with SP. It should be noted that SP^ need not be Hausdorff.
Observe that the ^-closure of G coincides with the ^-closure of G;
it then follows that SP-% is Hausdorff if the ^-closure of G is a proper
cone. The following dual characterization of the topology should
be compared with theorem (3.10).
LOCALLY O-CONVEX SPACES
57
(5.18)	Theorem. Let (E, C, be an ordered convex space, and let
be the locally o-convex topology associated with 0й. Then is the
topology on E of uniform convergence on the decomposable ^-equicon-
tinuous subsets of E'. Consequently HP is locally o-convex (i.e. HHP = HHPf}
if and only if each HP-equicontinuous subset of E' is contained in a
decomposable HP-equicontinuous subset of E'.
Proof. Let % be a neighbourhood base at 0 in (E, HP) consisting of
circled convex sets. Let F(&) = {F( F): F e %}. Then HP is the topology
of uniform convergence on {F°: F g %} and tPF is that on
{IF0: W g #(Я) = {E( F)°: F g
Thus the theorem follows immediately from theorem (2.11).
(5.19)	Corollary. The topological dual (E, nPff of (E, ^r) is
equal to C —O', where O' denotes the dual cone in E' = (E, HP)'. In
particular, if HP is locally o-convex then E' is decomposable in E*.
Remark. С —O' is the decomposable kernel of (E, H?)r.
Finally we consider the permanence property of locally o-convex
spaces; the following result should be compared with theorem (3.16):
(5.20)	Theorem. Let {(Ea,Ca, <^a):aG Г} be a family of ordered
convex spaces, and let F = © Ел be the algebraic direct sum of EK)
аеГ
ordered by the product cone. Let tPa_v be the locally o-convex topology
associated with HP л and let (© ^a)i? be the locally o-convex topology
аеГ
associated with HHtPa. Then
ж;Г
(©A)	(5.1)
\аеГ / F аеГ
Consequently the following assertions hold:
(a)	if each HPa is locally o-convex then so is QHP.p
аеГ
(b)	the decomposable kernel in F* of the product JJ ($a, is
аеГ
precisely the product of the decomposable kernels in E* of (Ea, HPf)'.
Proof. For each a in Г, let Va be a circled convex neighbourhood
of 0 in (Ea, 0a). Let
Л = co(u Fa) and В = co(u F(Fa)V
\«еГ /	\аеГ /
58
LOCALLY O-CONVEX SPACES
To prove the formula (5.1), it is sufficient to show that В j?(J) с 3B.
It is straightforward to verify В s F{A)-} it then remains to demon-
strate that F(A) с 3B. Let x e F(A) and suppose that ax < x < a2,
where a15 a2 e A. Let
У =	and a ...
Then 0<«/<аеЛсБ. By the definition of В there exist a finite
number of indices a1; a3 ... an in Г such that
г-1
n
where each bi g jF(K.) and hi > 0,	=- 1. ’For each a g Г, let 77 a
denote the ath projection on the ath coordinate space Ea. Then, since
0 < у < a, it follows that
0 < 77,^) < 77,.(«) E ^F(Vai) cz %.в (i = 1, 2,... n)
and 77,(2/) = 0 for all a g Г\{а1, a2 ... an). Hence, since 2^ — 1,
У “ 2 e 21B4 22B + ...+2rrB — В.
г=1
Consequently x = %y-\-ax g %B \ В = 3B. This shows that F(A') с 3B
as required. Thus formula (5.1) is verified, and assertion (a) in the
theorem follows immediately. The assertion (b) follows from (a),
corollary (5.19), and the following well-known result
A © AV = П (A- A)'-
al a
The following result should be compared with theorem (3.19) and
may be regarded as a dual to the preceding theorem.
(5.21)	Theorem. Let {(Ea>Ga, ^):аеГ) be a family of ordered
convex spaces, and let E be the product space ordered by the product cone G.
Let П SPa be the product topology and	lhe locally o-convex
аеГ	аеГ
topology associated with Then
аеГ
(n^)	(6-2)
\ аеГ / К аеГ
Consequently the following assertions hold:
(a)	if each is locally o-convex then so is
аеГ
LOCALLY O-CONVEX SPACES	59
(b)	the decomposable kernel in E* of the direct sum ®(Ea} is
precisely the direct sum of decomposable kernels in E* of (Ea,
.Proof. For each a e Г, let Va be a circled convex neighbourhood of
0 in (Ea, 0a), and suppose all Va, except a finite number of a in Г,
are equal to Ea. Then formula (5.2) follows from an easily-verified fact:
4пк) - ПЖ)-
By methods similar to those used in the preceding theorem, we can
verify that assertions (a) and (b) are easy consequences of formula (5.2)
and corollary (5.19).
6
LOCALLY SOLID SPACES
Let (E, C) be an ordered vector space such that E = C — C and lot V
be a subset of E. Suppose that
$(7)	e F}.
We recall that V is absolute-order-convex if S(V) V, that V is
absolutely dominated if V &(F), and that V is solid if 67(L) = V.
Let us consider a vector topology SP in [E, C) with a neighbourhood-
base % at 0 consisting of circled sets. Let
S{^) ={^F):fe <%},
F(W) = {E(Vy. V eW},
and	n C-W n С - {V n C-V n С: V e W}.
Let ^s, SP-%, SPD respectively denote the vector topologies in E for
which $(^), Fy%y and	C\O — °U П C are respective neighbourhood-
bases at 0. The topologies and have been studied in detail (in
Chapters 3 and 5); in particular, we know that is the finest locally
order-convex topology in E coarser than hP, and that is the coarsest
topology in E with the open decomposition property and finer than
Similarly one can construct topologies
etc.
(6.1)	Theorem. Let (E, C, be an ordered topological vector space
such that E C—C. Then the following statements hold:
(a)	AC is locally order-convex and has the open decomposition property;
(b)
Proof. Note that if K+K <= V then	<= and
2&(L() iS(V) H C —-S(V) П C; hence has the open decomposition
property. Also, since each S( V) is absolute-order-convex, must be
locally order-convex by theorem (5.1). Thus (a) is proved. Next we
show that	(6.1)
LOCALLY SOLID SPACES
61
The first inequality is obvious since F(F) Э &(F), and the second
inequality is true since
F* nC-F* n C £ £(F)
whenever	T/rf. ,	„ Tr
V* . • F * V.
By inequality (6,1) and (a) of this theorem, we immediately have
s < ^DF-
It remains to show ^FI) > ^Djr. Take a ^D]rneighbourhood of 0 in
E, say F(V r\ G—V H (7), where V is a circled ^-neighbourhood of 0
and F(F nd-F nQ is the order-convex hull of V n C — V n G.
ПГИрп
inen F(V) n c^^(7) a(7c f(F cC-FnC), (6.2)
In fact, if x - y-z G F(F) n C — F(V) n G, where y, z e F(V) n G,
then there exist u, v g V such that 0 < у < и and 0 < z < v. In
particular, u, —v eV C—V И G (since 0 is in F И (7), and
X = у-2Е[-^и\ £ F(V c\G — V HO),
proving formula (6.2). This shows that F(V Г) (7—-F П (7) is a
neighbourhood. of 0, and hence that > ^df-
(6.2)	Corollary. Let (E, G, 3й) be an ordered topological vector
space. If HP is locally order-convex and G gives an open decomposition then
Sfl = HP^.
Proof. The assumptions on (E, G, 3й} imply that HP = HPT and
HP “ hence ^FD —	= HP. By theorem (6.1)(b) we conclude
that — ^g.
A locally convex topology on an ordered vector space (77, (7)
will be called a locally solid topology if 3й admits a neighbourhood-base
at 0 consisting of solid sets. This is the case if and only if 3й — <^s.
Further, since £(F) is circled and convex whenever F is circled and
convex, HP is locally solid if and only if it admits a neighbourhood-base
at 0 consisting of circled convex and solid sets. If HP is locally convex,
3% is called the locally solid topology associated with HP.
A semi-norm p on (E, G) is called a Riesz semi-norm if it satisfies the
following two conditions:
(a)	p is absolute-monotone: --и < x < ад in E =>p(x) < p(u);
(b)	for each x g E with p(x) < 1 there exists и gE with p(u) < 1
such that —ад < x < ад.
62
LOCALLY SOLID SPACES
Clearly the condition (a) is equivalent to the absolute-order-convexity
of the 'open ball’ U -= {x e E :p(x) < 1}, and condition (b) is equivalent
to the absolutely dominated property of U; thus p is a Riesz semi-norm
if and only if U is solid. This remark, together with theorem (6.1) and
corollary (6.2), makes the following theorem clear.
(6.3)	Theorem. Let (E, C, P) be an ordered convex space. Then
the following statements are equivalent:
(a)	(E, G, <P) is a locally solid space;
(b)	(E, C, is both locally o-convex and locally decomposable;
(с)	& is determined by a family of Riesz semi-norms on E;
(d)	^;
(e)	tP = SPjjP.
(6.4)	Corollary. Let (E, G, ^) be a bornological locally o-convex
space. Then (E, G, SP) is locally solid if and only if G is a locally strict
Pfi-cone in (E, IP}.
Proof. In view of proposition (4.1), C is a locally strict ^-cone in
(E, 0) if and only if tP is locally decomposable, i.e. if and only if
P = ^D. Since IP is locally o-convex, it follows that C is locally strict
.^-cone in (E, 1P) if and only if IP — ^FD = <^s, i.e. if and only if & is
locally solid.
(6.5)	Corollary. Let (E, C, IP) be a locally o-convex space and
suppose that E ~ G—G. Then IP3 = and the topological dual
(E, Y of (E, ^s) is the smallest solid subspace of E* generated by
E' — (E, IP)'. In particular, if IP is locally solid then (E, IP)' is a solid
subspace of E*.
Proof. Since IP is locally o-convex, it follows from theorem (6.1)
that	~ Also since IPS is locally o-convex and locally
decomposable, it follows from corollaries (3.12) and (5.19) that (E, LPff
is a solid set in E* containing E'. Furthermore, (AT, IPe)' — (E, IPD)'
is the order-convex hull in E* of E'. Consequently (E, 1Ра)' must be the
smallest solid set in E* containing E'.
Similarly we can prove the following corollary.
(6.6)	Corollary. Let (E, G, IP) be a locally decomposable space.
Then IP§ ~ IP$, and the topological dual (E, Ps)' of (E, IPQ) is the
LOCALLY SOLID SPACES
63
largest solid subspace G' —G' of E* contained in E' = (E, P)', where
G' - G* П E'.
(6.7)	Corollary. Let (E, C) be an ordered vector space such that
E G—C, and let F be a directed subspace of E* such that F is total
over E. Then there exists a locally solid topology IP on E such that (E, IP)'
is the solid subspace of E* generated by F. In particular, if F itself is solid
in E* then (E, IP}' F.
Proof. In view of corollary (5.12), (E, Gj equipped, with the weak
topology c>(E, F) is locally o-convex. Let IP = &s(E, F) be the locally
solid, topology associated with a(E, F); then the result follows im-
mediately from, corollary (6.5).
Similarly, on applying corollary (6.6), we have the following result.
(6.8)	Corollary. Let (E, Gj be an ordered vector space such that
(j and ieg p an order-convex subspace of E* such that F is
total over E. Then there exists a locally solid topology IP on E such that
(E, iPf is the largest solid subspace of E* contained in F. In particular,
if F itself is solid in E* then (E, IP}' ~ F.
Proof. In view of corollary (3.14), (E, G}, equipped with the Mackey
topology t(E, F}, is locally decomposable. Let IP ra(E, F} be the
locally solid topology associated with r(E, F}- then the result follows
immediately from corollary (6.6).
We record another consequence of theorem (6.3).
(6.9)	Corollary. Let (E, G, IP} be a locally o-convex space with a
generating cone G, and let P& be the locally solid topology on E associated
zvith IP. Then the following assertions hold:
(a)	a positive subset of E is P-bounded if and only if it is P ^bounded;
(b)	if G is a locally strict P-cone, then a subset of E is P-bounded if
and only if it is Pa-bounded;
(c)	if G is IP-closed, then a monotone increasing net {жт} converges to x
in (E, P} if and only if it does in (E, Pf).
Proof. Since P is locally o-convex, Pa = P Thus the corollary
is a restatement of propositions (3.5) and (4.3).
64
LOCALLY SOLID SPACES
(6.10)	Corollary. Let (E, G, ^) be a locally o-convex space
satisfying the property that any o-convex circled subset of E which absorbs
all positive SP-bounded subsets of E is a ^-neighbourhood of 0. Then
(E, G, ^) is locally solid if (and only if)E = G—C.
Proof. It is not hard to see that no strictly finer locally o-convex
topology on E has the same positive ^-bounded subsets of E. The
result then follows from theorem (6.3) and corollary (6.9).
(6.11)	Corollary. Let (E,C, ||.||) be an ordered Banach space for
which G is ||.||-closed. Let £P be the vector topology in E induced by ||.||.
Then the following statements are equivalent:
(a)	(E, G, hP) is locally solid;
(b)	G is ^-normal and ^-generating in (E, ||. ||) for some a > 1, /3 > 1;
(с)	G and G' are generating cones in E and E’ respectively;
(d)	G is [3-generating in E and O' is ^.-generating in E' for some
a > 1, /3 > 1;
(e)	O' is у-normal and ^.-generating in (E'} ||.||), for some у > 1,
a > 1;
(f)	(E', G') is locally solid with respect to the vector topology induced
by the norm ||.|| on E'.
Proof. By theorem (6.3), (a)o(b) and (e)o(f). By theorem (3.8)
and proposition (4.6), (c)o(d). Finally, by the theorems of Krein-
Grosberg and Ando-Ellis (theorems (5.15) and (5.16)), (b)o(d) and
(b)o(e).
In Banach spaces (considering the metric aspect as well as the
topological aspect) we have the following more satisfactory result.
(6.12)	Theorem (Davies). Let (E, C, ]|.||) be an ordered normed space,
and let U be the open unit ball in (E, ||. ||). Let (E', G', Ц.Ц) be the Banach
dual space with the dual cone G', and let U', S' respectively denote the open
and closed unit balls in (E'} ||.||). We consider the following statements:
(a)	][. || is a Biesz norm in (E, (7);
(b)	U is solid in (E, C);
(c)	||.|| is a Biesz norm in (E', O');
(d)	U' is solid in (E', G');
(e)	S' is solid in (E', G').
LOCALLY SOLID SPACES
65
Then (a)<i>(b) ~>(c) <=> (d) <=>(e). Furthermore, if E and G are ||.||-
complete then (c)	(a) and hence all the statements (a)-(e) are mutually
equivalent.
Proof. It is easy to see that (a) o(b), (c) o(d), and (e) => (c). We
next show that (c) => (e). Since ||. || is absolute-monotone on (E', G'), S'
is certainly absolute-order-convex. To see that S' is absolutely
dominated let f g S', f 7^ 0. Then, for each positive integer n, there
exist gn e E' with < 1 such that
+______L____< q
ii/ll +(!/»)
(by the definition of Riesz norms). Let hn == {||/1Н-(1/^)}рп- Then
||AB|j < 1 + (l/n) for each n. By the Alaoglu theorem, {hn} has a a(E', E}-
cluster point, say h. Then ||A|| < 1. and ±/< h. This shows that S'
is solid. Thus (c) => (e). Therefore statements (c), (d), (e) are equivalent.
(b)	(e): Since U is solid, U — S(U). By theorem (1.17), we then
have	ТГ = RL)f =S(U”),
i.e. S' = $(S'). This shows that (e) holds.
We have shown that (a) o(b) (c) o(d) o(e), and it remains to
show (c)	(a) under the additional assumption that E and G are
||. ||-complete. Accordingly we suppose that (c) holds. Then, by the
established implication (b) => (c), the norm. ||. || on the second dual space
E" must be a Riesz norm. In particular, ||.|| is absolute-monotone on
E" so the norm in E must also be absolute-monotone since (E, |j . || ) is
isometrically isomorphic to a subspace of E". To verify that ||.|| is a
Riesz norm, we have only to show that U is absolutely dominated. To
this end, let us consider the vector topology HP on E induced by
||. ||. By (c), it follows from theorem (6.3) and corollary (6.11) that HP is
locally solid. Hence the family {SlfXJyA > 0} = {/h?(t7): 2 > 0} is a
neighbourhood-base at 0 in (E, HP}. Notice then that a$(£7) c (3S(U}
whenever 0 < a < Д. On the other hand, since &(S') -= S' (by (e)), we
have (cf. theorem (1.17))
(S(U}y = A((W) = £(S') - S',
so that (£(?7))’r’r = S. It follows from the bipolar theorem that
S s S(U}. Hence S c iS(U} c (]. -pe)S(U) for each e > 0. Conse-
quently U c S(U}, i.e. U is absolutely dominated.
66
LOCALLY SOLID SPACES
Finally we record the following permanence properties for locally
solid topologies for future references.
(6.13)	Theorem. The following assertions hold:
(a) the product of locally solid spaces is locally solid;
(h) the locally convex direct sum of locally solid spaces is locally solid.
Proof. By locally convex sum of {Ea: a g Г} we mean the algebraic
direct sum of {Pa} equipped with the locally convex direct sum topology.
Thus (b) follows immediately from theorems (3.16) and (5.20). Like-
wise, (a) follows from theorems (3.19) and (5.21).
7
THE ORDER-BOUND TOPOLOGY
Suppose that (E, 0} is an ordered vector space and that is the
family consisting of all the circled convex subsets of E each of which
absorbs all order-bounded subsets of E. It is easily seen that de-
termines a locally convex topology (not necessarily Hausdorff), denoted
by ^b on E', and this topology ^b is referred to as the order-bound
topology (or order topology} on E, It is also obvious that the order-bound
topology on E is the finest locally convex topology on E for which
every order-bounded subset of E is topologically bounded and that
(E, С, <^ъ)' = E\ consequently ^b is Hausdorff if and only if Eb is
total over E. From now on, when we consider the order-bound topology
<^b, we always assume that is Hausdorff. It should be noted that
Еъ is always an order-convex subspace of E*.
We state some elementary properties of as follows.
(7.1)	Proposition, Let (E, G} be an ordered vector space, and let
be the order-bound topology on E. Then -PYj is a bornological topology in
E and hence is the Mackey topology r(E, Eb). Moreover, the following
statements are equivalent:
(a)	G is a generating cone in E ;
(b)	G is a locally strict Sd-cone in (E,
(с)	(E, G, k^b) is a locally decomposable space.
Proof. It is easy to verify that (E, ^b) is a bornological space with
the topological dual Eb; hence ^b = r(E, Eb}. The equivalence between
(a), (b), and (c) follows from corollary (3.14) and proposition (4.1).
A sequence {жи} in an ordered vector space (E, G} is called a relative
uniform null-sequence if there exists a sequence of positive numbers a„
with -> co, such that {апжп} is an order-bounded subset of E. It is
clear that {xn} is a relative uniform null-sequence if and only if there
exist positive numbers en, with en -> 0, and an order-bounded subset
В of E such that xn s snB for all n.
68
THE ORDER-BOUND TOPOLOGY
(7.2)	Lemma. Let V be a circled convex subset of (E, C). Then V
absorbs all order-bounded subsets of E if and only if it absorbs all relative
uniform null-sequences in E.
Proof. Let be a relative uniform null-sequence in E. There
exist positive numbers sn, with en —> 0, and an order-bounded subset
В of E such that xn e snB for all n; since V absorbs В and since en 0,
it follows that V must absorb {#„}. Therefore the condition is necessary.
Conversely, if there exists an order-bounded subset В of E such that
the assertion В <= n2V is false for all n, then we have xn e В such that
xnfn2V. Therefore V does not absorb the relative uniform null-
sequence {xn[n}. This completes the proof.
Let (F, be a locally convex space, and let T be a linear mapping
of (E, C) into F. Let us say that T is order-bounded if it maps each
order-bounded subset of E into a -bounded subset of F. It is clear
that each order-bounded linear mapping of (E, 0} into R is an order-
bounded linear functional on E. If К is a cone in F, T is said to be
positive if T(C) с K. It should be noted that if each order-bounded
subset of F is SB-bounded then positive linear mappings from (E, C,
into (F, K, are order-bounded.
The following result was proved by Wong (1972a).
(7.3)	Theorem. Let (E, C, tP) be an ordered topological vector space,
and let be the order-bound topology on E. Then the following statements
are equivalent:
(a)	& is finer than
(b)	each circled convex subset of E which absorbs all relative uniform
mdl-sequences in E is a -neighbourhood of 0;
(c)	each order-bounded linear mapping of (E, C, tP) into any locally
convex space (F, .F') is continuous.
Proof, (a)	(b): Let V be a circled convex subset of E which
absorbs all relative uniform null-sequences in E. According to lemma
(7.2), V absorbs all order-bounded subsets of E, so V is a ^-neighbour-
hood of 0; consequently V is a ^-neighbourhood of 0, since is
coarser than
(b)	=> (c): Let T be an order-bounded linear mapping of E into F
and let U be a circled convex PF-neighbourhood of 0 in F. Then T'”1((7)
is a circled convex subset of E which absorbs all order-bounded subsets
THE ORDER-BOUND TOPOLOGY	69
of E, and so by lemma (7.2) T-^fU) absorbs all relative uniform null-
sequences in E. Therefore T-fiU) is a ^-neighbourhood of 0 in E, this
implies that T is continuous.
(c)	(a): Suppose that i is the identity mapping of (E, G, into
(E, ^b). Since each order-bounded subset of E is ^-bounded, it
follows that i is order-bounded, and hence that i is continuous. There-
fore is coarser than
(7.4)	Corollary. Let (E, G, ^) be an ordered convex space. Then the
following statements are equivalent:
(a)	SP is the order-bound topology 0йb;
(b)	each circled convex subset of E which absorbs all relative uniform
null-sequences in E is a 0-neighbourhood of 0, and each order-bounded
subset of E is ^-bounded;
(c)	every order-bounded linear mapping of (E, G, &) into any locally
convex space (F, $~) is continuous, and each order-bounded subset of E is
&-bounded;
(d)	S? is the Mackey topology r(E, Ef and E' = Еъ.
Proof. It is clear that (a), (b), and (c) are equivalent by theorem
(7.3). The equivalence of (a) and (d) was established in (a) of proposition
(7.1).
(7.5)	Corollary. The order-bound topology on (E, C) is the finest
locally convex topology for which every relative uniform null-sequence in E
is convergent to 0.
An ordered topological vector space (E, C, is said to be funda-
mentally a-order-complete if each increasing ^-Cauchy sequence in E
has a supremum in E. It is clear that if G is sequentially ^-complete
then (E, C, &) must be fundamentally cr-order-complete. In the next
chapter, we shall study the fundamentally cr-order-completeness in
detail.
(7.6)	Corollary. For any bornological ordered convex space
(E, G, &), if G is a strict &-cone in (E, 0й) and if (E, G, 0й) is funda-
mentally a-order-complete, then is coarser than & (and hence each
order-bounded linear mapping of E into any locally convex space (E,
is continuous); in particular, all order-bounded linear functionals (and
6
70
THE ORDER-BOUND TOPOLOGY
certainly all positive linear functionals) on E are IP-continuous. If, in
addition, each order-bounded subset of E is IP-bounded then IP is ^b.
Proof. Let V be a circled convex ^-neighbourhood of 0 in E.
Then V absorbs all order-bounded subsets of E. We wish to show that
V is a ^-neighbourhood of 0. Since IP is bornological, it suffices to
show that V absorbs every ^-bounded subset of E. Suppose, on the
contrary, that there exists a ^-bounded subset В of E which
is not absorbed by V. Since G is a strict .^-cone, for this В there
exists a circled convex ^-bounded subset A of E such that
В £ А П G—A C\ C. Then the set J n G must not be absorbed by V.
Hence, for each positive integer n there exists xn e А G G such that
n
xn f22nV. The sequence 2 2гкхк (n > 1) is an increasing ^-Cauchy
fc=i
sequence in E, so	n
У = sup J 2“~kxk
» Л=1
exists in E. Notice that 0 < 2~nxn < у and 2гпхп ф 2nV; therefore the
order-interval [0, y\ is not absorbed by the ^-neighbourhood У of 0,
contrary to the construction of ^-topology. This shows that V must
be a ^-neighbourhood of 0 and hence that < IP. If, in addition,
each order-bounded set in E is ^-bounded then 1РЪ = IP "by corollary
(7.4).
(7.7)	Cobollary. Let (E, G, IP) be a metrizable ordered topological
vector space, and let (F, ST) be any locally convex space. Then each of the
following conditions implies that each order-bounded linear mapping of E
into F is continuous (in particular, all positive linear functionals on E are
IP-continuous) and hence is certainly coarser than IP.
(a)	G gives an open decomposition in (E, IP), and (E, G, IP) is funda-
mentally а-order complete.
(b)	G is IP-complete and generating, and E is of the second category.
Proof. It is enough to verify (a) since (b) follows from (a) by
theorem (3.8). Let {Fn:n > 1} be a neighbourhood-base at 0 for IP
consisting of ^-closed, circled sets such that K+i+K+i c Vn for all
n > 1. Then {Vn П G—Vn n C:n > 1} is a neighbourhood-base at 0
for IP since G gives an open decomposition in (E, IP). Let V be a circled
convex set in E which absorbs all relative uniform null-sequences in E.
THE ORDER-BOUND TOPOLOGY
71
If V is not a ^-neighbourhood of 0, then the assertion
Vn n C—Vn n C <= n47
is false for each n > 1. For any n, let xn, yn in Vn n О be such that
(n	\	/ n	\
2 and I п~^Ук) are increasing
*=i	/	\fc=i	/
n
^-Cauchy sequences in C. By the hypothesis, sup	= x and
n	n Jc—'L
sup 2	" У exist in G. We conclude from
n k-.l
У < n~z(xn-yn) < X
that {n~z(xn —yn)} is a relative uniform null-sequence for which it is not
absorbed by V, contradicting our assumption on F. Therefore V is a
^-neighbourhood of 0, and so is coarser than & by theorem (7.3),
It will be seen that (FZ, ^) is complete provided that each order -
bounded set in E is ^-bounded (corollary (8.11)).
(7.8)	Corollary. For any ordered topological vector space (E,C, tP),
if G has an interior point e then £Pb is coarser than (and hence each
order-bounded linear mapping of E into any locally convex space (F,
is continuous); in particular, all positive linear functionals on E are
^-continuous.
Proof. The topology induced by the gauge of [ —e, e] is the order-
bound topology ^b. Since e is an interior point of O, there exists a
circled ^-neighbourhood V of 0 such that e + V c 0, and hence
V £ [—e, e]; this implies that ^b is coarser than &. This completes the
proof.
Let (F, be a topological vector space. A family % of ^-bounded
sets in E is called a ^-determined family if each circled convex
absorbing subset of E which absorbs all members of is a ^-neighbour-
hood of 0. A subset В of the cone G is said to be ^-bounded if for each
(n	1
ffLpx^.n > 1
7c=l	J
is order-bounded.
(7.9)	Corollary. Let (E, G, УР) be an ordered topological vector
space. If there exists a S?-determined family of tP-bounded subsets of E
such that each member % is contained in the difference of two ^-bounded
subsets of G, then is coarser than S? (and hence each order-bounded
72
THE ORDER-BOUND TOPOLOGY
linear mapping of E into any locally convex space (F,	) is continuous);
in particular, all positive linear functionals on E are &-continuous.
Proof. Let F be any circled convex set in E which, absorbs
all relative uniform null-sequences in E. If there exists A etfl
such that the assertion A c n4F is false for all n > 1, then the assertion
Ах— A2 s n4F is false for all n > 1, where A± and A2 are F-bounded
subsets of C such that А с A1—A2. For any n, there exist xn e Alt
yn e A2 such that n~s(xn—yn) fnV. It is easily seen that
{n^{xn-yn):n > 1}
is a relative uniform null-sequence for which it is not absorbed by V,
(П	'l	in	A
k~2xk\ and j	are order-bounded. This contradiction
7c=l	/	Vc==l	>
shows that V must be a ^-neighbourhood of 0, and hence the result
follows from theorem (7.3).
Let (F, ^F) be a locally convex space. A sequence {yn} in F is called
a local null-sequence if there exist positive numbers an, with an -> co
such that {awa?„} is a 2F-bounded subset of F. It is clear that {yn} is a
local null-sequence if and only if there exist positive numbers en, with
en -> 0 and a «F'-bounded subset В of F such that yn e enB for all n.
(7.10)	Theorem. Let (F, PF) be a locally convex space, and let T be
a linear mapping of (E, C) into F. Then the following statements are
equivalent:
(a)	T is order-bounded;
(b)	T maps every relative uniform null-sequence in E into a local
null-sequence in F;
(с)	T maps every relative uniform null-sequence in E into a sequence
in F which converges to 0 for SF;
(d)	T maps every relative uniform null-sequence in E into a FF bounded
subset of F.
Proof. Let {&„} be a relative uniform null-sequence in E. There
exist positive numbers аи, with co, and an order-bounded subset
В of E such that anxn e В for all n. Let T be order-bounded. Then
T(B) is a ^-bounded subset of F, and so {T(xn)} is a local null-
sequence in F. This proves the implication (a) => (b). The implications
(b) => (c) => (d) are obvious. It remains to show (d) => (a). Suppose
that T is not order-bounded. Then there exists an order-bounded subset
THE ORDER-BOUND TOPOLOGY	73
В of E such that T(B) is not «У-bounded. Let V be a circled convex
«У-neighbourhood of 0 in F such that the assertion T(B) c: n2V is
false for all n. For each n, let xn in В be such that T(xn) ф n2V. Then
we have found a relative uniform null-sequence {xn/n} in E such that
{T(xn!n}} is not «У-bounded. Therefore (d) must imply (a), and the
proof is complete.
The following result is an immediate consequence of theorems (7.3)
and (7.10).
(7.11)	Corollary. Let РРЪ be the order-bound topology on (E, C),
and let (F, be a locally convex space. A linear mapping T of (E, C, <£/b)
into (F, is continuous if and only if it is sequentially continuous, and
if and only if it satisfies one of the conditions (a)-(d) of theorem (7.10).
Let (E, G, PL) be an ordered convex space such that each order-
bounded subset of E is ^-bounded. Then each relative uniform
null-sequence {ж„} in E is ^-bounded, and so the polar, taken in E', of
the set {xn} is absorbing; we denote the topology of uniform convergence
on all relative uniform null-sequences in E by «/'o, and denote the
topology of uniform convergence on all local null-sequences in E by
We now present a dual characterization of the order-bound topology
as follows (cf. Wong (1972a)).
(7.12)	Theorem. Let (E, C, 0s) be an ordered convex space for which
C is PF-closed and generating. Then (P is the order-bound topology PPb if
and only if PF satisfies the following conditions:
(а)	«У is the Mackey topology т(Е, E');
(b)	each order-bounded subset of E is ^-bounded;
(с)	E' is ./'^-complete.
Proof, (i) Necessity. It is clear that (a) and (b) are satisfied by
PL. It remains to verify that E' is «/^-complete. Let {fT} be а «/У
Cauchy net in E'. Then {fT} is a a(E', L’)-Cauchy net since {x[n} is a
relative uniform null-sequence in E for any x g E, and so there exists
f e E* such that fr converges to f pointwise on E. Let {x^} be any
relative uniform null-sequence in E, and let A be the set consisting of
{£„}. Then/ converges to/uniformly on A since / is a «/^-Cauchy net.
It is clear from theorem (7.10) that each / is bounded on A, so that/is
bounded on A. Thus, by theorem (7.10), / g Еь ~ E'. Since {xn} was
arbitrary, then / converges to / for «/'o, and so E' is «//-complete.
74
THE ORDER-BOUND TOPOLOGY
(ii) Sufficiency. The condition (b) implies that E’ £ Еъ. We now show
that Eb S E'. Suppose that {жи} is a relative uniform null-sequence in
E, and that V is the circled convex hull of {xn}. Since E' is J^'o-complete,
by Grothendieck’s completeness theorem, it is sufficient to show that the
restriction of each f e Еъ to the set V is o(E, ^/)'continuous- I*1 facti
since C is generating there exist positive numbers Ли, with Лп —> 0, and
e g C such that xn e Лп[--e, e], and so xn c y[ — e, e] for some у > 0,
because Лп —>0; consequently V £ e, e] since C is ^-closed. On
the other hand, suppose that Ee = (J n\—e, e], and that jjx|| is the
n
gauge of [-—e, e] in Ee. Then )|. || is a norm on Ee, and [—e, e] is the
closed unit ball of (Ee, ||. ||), i.e. [—e, e] {x g Ee: ||ж|| < 1}. Observe
that V is a precompact subset of (Ec, (|. ||). There exists a finite subset
n
{yy 1 < i < n} of V such that 7 c U {Уг\~[~~С e]); since [—-e, e] is
a(E, _Zi7')-closed., it follows from 1=1
r — U (гЛ + [-е.е]) = U (»< + [-«> e])
that V is also a precompact subset of (Ee> ]|. ||). By making use of Kothe
(1969, § 28,5(2)), the norm topology ||. and the topology cffiE, E')
coincide on V because [—e, e] is a(E, ^'J-closed. Finally, since each
f e Еъ is bounded on [ —e, e], then the restriction of / to Ee is a contin-
uous linear functional on (Ee, ||. ||), and so the restriction of / to V is ||. || -
continuous and certainly a(E, .^'/continuous. This completes the proof.
(7.13)	Corollary. Let (E, C, be an ordered convex space for
which C is ^-closed and generating. Suppose that (E, ^) is bornological
and that each order-bounded subset of E is .^-bounded. If each J?' -
Cauchy net in E' is an J?rc -Cauchy net then & is the order-bound topology
Proof. By making use of theorem (7.12), we only have to verify that
E' is J^-complete. Suppose that {fr} is an «/"^-Cauchy net in E', then it
is an -Cauchy net. Since (E, tP) is bornological, it follows from
Kothe (1969, § 28,5(1)) that Ef is jC^-complete; and hence there
exists / in E' such that fr converges to / with respect to On the
other hand, since each order-bounded subset of E is ^-bounded, then
У'о is coarser than and so fT converges to / for This completes
the proof.
THE ORDER-BOUND TOPOLOGY
75
Let (E, C) be an ordered vector space. We recall that E has the
Riesz decomposition property if [0, u-j-w] = [0, u] + [0, w] whenever и
and w are in G. E is called a weakly Riesz space (or weakly vector lattice')
if E has the Riesz decomposition property and E = G—G-, and E is
called a Riesz space (or vector lattice) if each pair of elements x, у of E has
a least upper bound, written x v y, in E. It is clear that if E is a Riesz
space then O' is a generating cone and E has the Riesz decomposition
property; but the converse is not true.
Namioka (1967) has given an example to show that the order-bound,
topology on an ordered vector space need not be locally o-convex;
but Schaefer (1966) has shown that if E has the Riesz decomposition
property then the order-bound topology <^b is the finest locally o-
convex topology. Therefore, by making use of corollary (6.2), proposi-
tion (7.1) (c), and Schaefer’s result (mentioned in the above), the order-
bound topology .^b on a weakly Riesz space (E, G) is locally solid. We
now give a more direct and elementary proof of this result.
(7.14)	Theorem. Let (E, G) be a weakly Riesz space, then the order-
bound topology <^b on E is the finest locally solid topology.
Proof. We first show that is locally solid. Suppose that У is a
circled convex ^-neighbourhood of 0, we wish to find a solid convex
^-neighbourhood U of 0 such that U £ F. To do this, let
$ = и {[—w, u]:u g V and [0, и] £ F}. It is clear that 8 is solid. If
x, у g 8 and if 0 < Я < 1, there exist u,weG, with [0, u] £ F and
[0, w] c: F, such that — и < x < и and —w < у < w; and thus
— (Яад + (1—A)w) < Яж4(1 — Я)у < Xu4-(l — X)w. By the Riesz decom-
position property and the convexity of F, we have that
[0, Xu-b(l —A)w] = [0, Xu] -f-[0, (1— X)w~]
= Я[0, и] + (I — Я)[0, w] s XVX (1 -A)F
- V,
and thus Яж + (1 —X)y g 8. Therefore 8 is convex. Furthermore,
8 c 2 F; for if x g 8 then there exists и G G, with [0, u] £ F, such that
x-—a	.	u--x
—u < x < u. Since 0 < ------ < и and since 0 <----- < u, it follows
.	2	2
ж+w u—x
from x = — ——— that x g 2F, and hence that 8	2F. On the
2	2
other hand, we also have that & П С — {и e C: [0, u] <=: F}. Indeed,
obviously {и eC:[0,w] £ F} £ /S n G. Suppose that x g 8 n G. Then
76
THE ORDER-BOUND TOPOLOGY
there exists и g C with [0, и] V} such that —u < ж < u, and so
[0, ж] £ [0, и] c V, therefore x g {u g G. [0, u\ <= У}; this shows that
$ П G = {u g O’. [0, u] F}. Finally we show that 8 absorbs all
elements in G; from this 8 absorbs all order-bounded subsets of E
because G is generating. Suppose not; then there exists x g G such that
x ф n8 for all n, it then follows from {u g C: [0, u] £ V} = 8 C\ G that
[0, ж] £ nV is false for all n; this gives a contradiction because V
absorbs all order-bounded subsets of E. Therefore & is a solid convex
^-neighbourhood of 0 for which 8 Q 2V, and so U — $8 has the
desired property.
Since each order-bounded subset of E is bounded with respect to any
locally solid topology on E, it follows that is the finest locally solid
topology. This completes the proof.
It is known from proposition (7.1) that (E, G, 0%) is bornological,
where is the order-bound topology on£. We conclude this chapter
with an example which will show that the topology on a bornological
ordered convex space (E, G, ^), for which every order-bounded subset
of E is ^-bounded, need not be the order-bound topology.
(7.15) Example. Let E be the vector subspace of consisting of
all elements x — (xn: n g N) satisfying the condition xn = 0 for all but
a finite number of indices n, equipped with the norm
||ж)| = тах{|жп| :n g N}.
E has a natural cone G defined by
G = (x = (xn\x eN):zn> 0 for all n in N).
Let	,	,	,
I	1	I
V = ж = (xn;x g N): \xn\ < - for all n in N .
(	n	)
Then У is a convex solid absorbing subset of E which is not a ||. )|-
neighbourhood of 0. Therefore the norm topology j|. || is not the
order-bound topology.
8
METRIZABLE ORDERED
TOPOLOGICAL VECTOR SPACES
This chapter is devoted to a study of relations between order complete-
ness and topological completeness in metrizable ordered topological
vector spaces. We shall see that there are useful and elegant results that
do not hold in non-metrizable spaces.
Throughout this book zf1 denotes the ordered Banach space con-
sisting of all absolutely summable sequences of real numbers, equipped
with its usual norm and ordering; therefore (2„) in rf1 is positive if and
only if each is non-negative.
(8.1)	Definitions. Suppose that (В, C, is an ordered topological
vector space.
(1)	(E, is said to be boundedly о-order-complete if each sequence
in E which is increasing and Abounded has a supremum in E.
(E, is said to be boundedly order-complete if each net in E which is
increasing and Abounded has a supremum in E.
(2)	(E, is said to be fundamentally a-order-complete if each in-
creasing ACauchy sequence has a supremum in E.
(3)	(E, ^) is said to be monotonically sequentially complete if each
increasing ^-Cauchy sequence is convergent in E.
(4)	(E, is said to be fl-order-summable if for each positive SP-
bounded sequence {un} in E and any positive element (Я„) in fl, the
sequences of partial sums of {Anun:n >1} have a supremum in E,
i.e. supA^A:n > 1 exists in E\ the supremum will be con-
(fc=i	J
veniently denoted by (0)— 2
It is clear that an ordered topological vector space is fundamentally
cr-order-complete if and only if each positive, increasing ACauchy
sequence has a supremum, and that if E and C are ^-complete then
(B, .^) is fundamentally cr-order-complete.
In metrizable and fundamentally cr-order-complete spaces, we have
the following useful result characterizing the local order-convexity.
78
METRIZABLE ORDERED
(8.2)	Theorem. For any metrizable ordered topological vector space
{E, C, tP), if {E, tP) is fundamentally a-order-complete {in particular, E
and, C are ^-complete) then {E, C, is locally full if and only if each
order-bounded subset of E is -bounded.
Proof. The necessity is obvious. To prove the sufficiency, Jet
{Vn:n > 1} be a neighbourhood-base at 0 for consisting of circled
sets such that К-ы+K+i c: K- Suppose, on the contrary, that
{E, C, tP) is not locally full; then there exists a ^-neighbourhood W of
0 such that (7^(7) n С Ф 2nW for all n > 1.
For each n > 1, there exist xn e E, yn e Vn such that
0 < xn < yn and xn f 2nW.
It is clear that the sequence of the partial sums of {yn} is an increasing
^-Cauchy sequence. By the fundamental cr-order-completeness,
yk: m > 1J exists in E, and so
fe=l	J
o < Xn < yn < y.
Since xn$2nW, it follows that the order-interval [0, y] is not &-
bounded. This completes the proof.
(8.3)	Corollary. Let {E, C, be a metrizable, locally decomposable
space, and let {E, tP) be fundamentally a-order-complete. Then the
following statements are equivalent:
(a)	(E, C, is a locally o-convex space;
(b)	& is the order-bound topology
(c)	each order-bounded subset of E is tP-bounded.
Furthermore, if {E, C,	satisfies one of (a), (b), and (c) then {E, C,
is a locally solid space; if, in addition, {E, C) has the Riesz decomposition
property, then E' — Eb = E# and E' is a vector lattice.
Proof. The implication (b) => (c) is clear, and the implication
(c) => (a) follows from theorem (8.2). It remains to verify that (a)
implies (b). Let {JQ'.n > 1} be a neighbourhood-base at 0 for &
consisting of circled convex sets such that K+x+Kii-i c K- Since
(E, C, is locally solid, for each n there exists a solid convex
neighbourhood Un of 0 such that
17 cf.
'-z n fn'
TOPOLOGICAL VECTOR SPACES
79
If is not there exists a circled convex ^-neighbourhood Ж of 0
such that	ф 2„ w foj; aH n>y
For each n > 1, there exist xn g E and yn g Un such that
- yn < xn < yn and xn ф 2nW.
Since yn g Vn, it follows from the fundamental o'-order-completeness
(ТП	\
2 yk: m > 11 exists in E; hence
a;=i	J
У Уп <	< Уп < У'
Therefore W does not absorb the order-interval [~~y,y] which gives a
contradiction. The final conclusion that E' is a vector lattice follows
from theorem (1.10).
(8.4)	Corollary. For any ordered Frechet space (E, C, £P), if C
is .^-closed, then (E, C, ^) is locally o-convex if and only if E' ,£lb.
In this case, SP is the order-bound topology ^b.
Proof. Application of theorem (8.2) and corollary (8.3).
(8.5)	Proposition. Let (E, G, IP) be a metrizable ordered topological
vector space, and let C give an open decomposition in (E, ,^). If (E, C) is
Archimedean and if (E, tP) is fundamentally a-order-complete, then (J is
^-closed.
Proof. Let {Vn:n > 1} be a neighbourhood-base at 0 for SP con-
sisting of circled sets such that Vn+1 +Vn+1 <= Vn, and let x be in the
^-closure of G. For each n, there exists xn g G such, that
г.-А(Г. пС-Г.пС),
n
and so there exists un g Vn ПС such that n(xn-—x) < un. It is clear
!n	'I
is an increasing ^-Cauchy sequence. By the
)
fundamental cr-order-completeness. и = sup IS uk:n > 1 exists in E.
J
We conclude from the Archimedean property of E and from
1
zi t zyi	ri /
д-j vL	Cv *
n
that — x	0 and hence that x g C. This shows that C is ^-closed.
80
METRIZABLE ORDERED
(8.6)	Lemma. For any ordered topological vector space (E, C,
consider the following statements:
(a)	(E, is sequentially complete;
(b)	(E, SF) is monotonically sequentially complete;
(с)	(E, ^) is &-order-summable;
(d)	(E, kP) is fundamentally a-order-complete;
(e)	(E, is boundedly a-order-complete.
Then the following statements hold:
(1)	(a) • (b) and (e) => (d);
(2)	(b)	(c) provided that (E, is locally convex and C is ^-closed;
(3)	(b)	(d) provided that C is ^-closed;
(4)	(d)	(c) provided that (E, is locally convex;
(5)	(c) => (d) provided that (E, is metrizable—consequently, for a
metrizable ordered convex space (c) and (d) are equivalent.
Proof. Wo have only to verify assertion (5) since other assertions
are obvious. Let {Vn:n ~ 1, 2,...} be a neighbourhood-base at 0 for SP
consisting of circled sets such that K+i+K+i К f°r n > 1, and
let {xn} be an increasing ^-Cauchy sequence. There exists a subsequence
{xnJ of {xn} such that
жи, g 2~Wfc (k > 1),
then {T{(xn^-— xnf): к > 1} is a '^-bounded sequence in C, and so, by
the ^-order-summability,
x > 1]
\7c=^l	J
J
exists in E. Since У (ж„ — x„ ) = an -x„ , it follows that
' n*+l nTi!	ni+l nl’
k=l
у = гЦ(] - supK34i:J > 1}
exists in E. We now claim that у = sup{«m:w > 1}; it is equivalent to
verify that	r „
Xm < У *or all
In fact, for any m there exists a positive integer q such that nk > m
whenever к > q\ since {xn} is increasing, it follows that
< xnk < У-
This completes the proof.
There are ordered Banach spaces which are fundamentally u-order-
complete but not boundedly u-order-complete as shown by the following
example.
TOPOLOGICAL VECTOR SPACES
81
(8.7) Example. Consider the ordered Banach space 0(0, 1] con-
sisting of all real-valued continuous functions on [0, 1], equipped with
its usual norm and ordering. Suppose that
(0 if 0 < t < |
if | < t < l-H
n
1 if < I < 1.
" n
Then {ж„} is an increasing norm-bounded sequence in C[0, 1] but has no
supremum in CJO, 1]. Thus C[0, 1] is not boundedly u-order-complete.
We observe that the cone
О - {ж еф, 1]:ж(/) > 0 for а1П е [0, 1]}
is (norm) closed. Therefore (7[0, 1] is fundamentally u-order-complete.
The study of the relationship between order-completeness and
topological completeness can be broken down into two stages: in the
first stage we establish the fact that monotonically sequential complete-
ness, under certain conditions, implies completeness; and in the second
stage we establish some sufficient conditions to ensure that order-
completeness implies monotonically sequential completeness. First, we
prove the following theorem due to Jameson (1970).
(8.8) Theorem, For any metrizable ordered topological vector space
(E, C, 0), if (E, 0°) is monotonically sequentially complete and if C gives
an open decomposition in (E, ^), then (E, ^) is complete. The converse is
also true provided that C is ^-complete and generating.
Proof. Let {Vn :n > 1} be a neighbourhood-base at 0 for con-
sisting of circled ^-closed subsets of E for which K+i+K+r К f°r
all n > 1, and let Bn = Vn П C—Vn ("} C (n > 1). Then Bn is a
^-neighbourhood of 0. Any ^-Cauchy sequence has a subsequence
{zw} such that
^n+y.^n^Bn (n > 1).
There exist un, wn in П О such that
= un-~wn (n > 1).
n	n
Since zn+1 —z1 = 2 uk — 2 wJc> then the convergence of {zn} is equivalent
ы	&=i
82
METRIZABLE ORDERED
(W	I	/ n	\
2 : n 1 and 2 w7c: n > 1L
J	u=i	)
n+<I	n
2	«„ eVn+1 n C+...+V„+, n С s И„ n C s К
7c-l	й=-1
(n	\
2 uk: n > 1 is an increasing ^-Cauchy sequence, and
fc=i	J
(n	1
У wk: n > 11
fc=i	J
converges for SP. Therefore E is ^-complete. The converse follows from
theorem (3.8). This completes the proof.
As the second stage in establishing the relationship between order
completeness and topological completeness, we prove the following
result of Wong (19726) which should be compared with the equivalence
of (a) and (c) in theorem (3.8).
(8.9)	Theorem. For any metrizable ordered topological vector space
(E, C, fP), if (F < ^b, C is generating, and if (E, is fundamentally
о-order-complete then it is monotonically sequentially complete and locally
full. If, in addition, E is of the second category, then (E, is complete
and C gives an open decomposition in (E, &).
Proof. The locally full property of (E, C, 0} is an immediate
consequence of theorem (8.2). Let {Vn;n > 1} be a neighbourhood-base
at 0 for & consisting of circled ^-closed sets such that J^+1	£ Vn
(n > 1). Then {Vn П C— Vn П C:n > 1} forms a neighbourhood-base
at 0 for the metrizable topology ^D. Notice that tF <	and
each increasing ^-Cauchy sequence in E is also <^D-Cauchy. Hence, to
show that (E, 0s} is monotonically sequentially complete it is sufficient
to show that (E, is complete. Any -Cauchy sequence in E has a
subsequence {xn} such that
xn+i-^n en~2(Vn r\C—Vn n Cf
and so there exist yn, zn in Vn C\ C such that
n2(xn+1-xn) = y^-zn.
(m	a	/ m	\
X h~xyk'.m > 1 ,	2 h^z^-.m > 1
fc=i	J	U=i	J
(m	\	f m	\
^^~2d/c'm > 1 > and { У k~~2zlt.:m > 1} are increasing ^-Cauchy
fo=l	)	J
TOPOLOGICAL VECTOR SPACES	83
sequences. Since (F/, ^) is fundamentally o'-order-complete, it follows
that У о = (°)	= (°) ““2
A	к
ft	к
(m	\
2 к^уь 'm > 17 converges to у with respect
J
to For each m, we have
m	co	/	00	1	1
° < y-2 1<^Ук = (°)- 2 к~2Ук < w-1 (0)- J —г-; ym+1\ <
£	7c^ m-i-1	Ы^+j )
(m	A
2 k^2yk'.m > 1> con-
&=i	j
2 h~~2zk:m > 1}
fc=l	J
converges to z with respect to ^*D. Therefore, the sequence {xm}
converges to y— ziz1 with respect to <^D, and hence (7?, <^D) is com-
plete.
Finally, since & is coarser than ^D, then, the identity mapping from
(E, ^p) onto (.S', ^) is continuous, and hence coincides with ^p
by the open mapping theorem provided that (E, УР) is of the second
category; consequently (E, tP) is complete, and 0 gives an open
decomposition in (E, ^). This completes the proof.
Remark. It is worth to note that under the assumption of theorem
(8.9), the metrizable topological vector space (E, tPjf) has the following
properties:
(a)	each order-bounded subset of E is ^p-bounded;
(b)	C gives an open decomposition in (E,	;
(c)	a monotone sequence in E is ^D-Cauchy if and only if it is
^-Cauchy;
(d)	(E, ^p) is complete.
(8.10)	Corollary, Let (E, C, 0s} be a metrizable ordered convex space
such that C is tP-closed and generating, and let УР be coarser than £^b.
If (E, УР) is fundamentally a-order-complete then is the locally
decomposable topology on E associated with and {E, C, ^b) is
a complete, metrizable locally solid space.
Proof. In view of the remark of theorem (8.9), (E, C, ^p) is a
complete, metrizable locally decomposable space and is coarser than
84
METRIZABLE ORDERED
^b; hence, by making use of corollary (8.3),	= <^b and (E, G, IPD)
is also locally o-convex; consequently (.£/, C, <^b) is a locally solid space.
This completes the proof.
(8.11)	Corollary. For any metrizable ordered topological vector
space (E, G, IP), if SP is coarser than I?b, then each of the following
conditions implies the completeness of (E, IP):
(a)	G gives an open decomposition in (А\ ^) and (E, C, IP) is funda-
mentally a-order-complete;
(b)	G is IP-complete, generating, and E is of the second category.
If, in addition, IP is locally convex then IP — lPb and (E, 0, IP) is a
complete metrizable locally solid space.
Proof. Since, by making use of theorem (3.8), (b) is an immediate
consequence of (a), therefore we have only to show (a). According to
theorem (8.9), (E, G, IP) is monotonically sequentially complete, and
hence, from theorem (8.8), {E, IP) is complete since G gives an open
decomposition in (E, IP). The final assertion follows from corollary (7.7).
(8.12)	Corollary. Let (E, G, IP) be a metrizable ordered convex
space for which G is a locally strict Id-cone in (E, IP), and let IP be
coarser than lPb. If (E, SP) is fundamentally a-order-complete, then
IP ~ IPb, and (E, C, IP) is a complete metrizable locally solid space.
Proof. Since metrizable convex spaces are bornological, it follows
from proposition (4.1) that (E, G, SP) is a locally decomposable space,
and hence the result follows from corollary (8.11).
9
ORDERED NORMED VECTOR SPACES
By an ordered normed space we mean a normed space (E, |j. ||) equipped
with a partial ordering < induced by a cone G. Recall that (E, G, ||. [() is
an approximate order-unit normed space if there exists an approximate
order-unit {ел, A e A, <} in G such that the given norm |j. || is the gauge
of the circled convex set
q]:2eA).
Since $л is solid, the induced norm must be a Riesz norm and hence
(E, G, ||. ||) must be locally solid under the vector topology induced by
||. |]. The following result characterizes approximate order-unit normed
spaces among Riesz normed spaces.
(9.1)	Proposition. Let(E,C} ||.||) be an ordered normed space. Then
the following statements are equivalent:
(a)	E is an approximate order-unit normed space;
(b)	G is 1-normal in (E, ||. j|) and the open unit ball
U - {хеЕ: И < 1}
is directed upwards;
(c)	U is order-convex and directed upwards;
(d)	U is absolute-order-convex and directed upwards;
(e)	U is solid and directed upwards;
(f)	||. || is a Riesz norm on (E, G) and U is directed upwards.
Proof. In view of proposition (1.6) and theorem (6.3), (b) о (c) and
(e)o(f). If U is directed upwards then a fortiori it is absolutely
dominated; hence (d)o(e). Further, it is trivial that (c)	(d). Thus
to complete the proof we have only to show (a)	(b) and (e)	(a).
(a) => (b): Let ||. || be the gauge of SA =	e2]:Z eA), and
suppose that x < у < z in E. Let M = тах{||ж||, j|z||}. Let e, with
e > 0, be given. Then there exist hx, in Л such that
and
-(^ + Флж < я < (М + е)еЛх
-(^ + е)Ч < z < (TH + £)ev
7
86
ORDERED NORMED VECTOR SPACES
Let у > Лж, Xz in A. Then we have that
(Jf + e)eM < x and z < (M-\-e}e^.
Hence, since x < у < z,
•—(Jf-j-s)^ < x < у < z < (Hf-fie)^.
It follows that ||?/|| < M-\-s. Since e is arbitrary,
\\УII < > -= max{|H, I)z|j}.
This shows that G is 1-normal in (157, ||. |[). To show that U is directed
upwards, let ur, u2 e U and a a positive real number such that
maxfUuJI, ||u2||} < 1—a. Then there exist 2X, Л2 in Л such that
-{l-a)eA. < Ui < (1-а)ел. (i - 1, 2).
Let Л e A be such that Л > 7Lj, Я2 and и (1 — a)eA. Then и e V and
26	26 j ? *^2*
(c) -> (a): Suppose the open unit ball U in (E, G, ||. ||) is solid and
directed upwards. Then U c\ C is also directed. Lor each и e U П C,
let eu = u. Then {eu, и e U A (7, <} is a net in G. Let
A = u{[—eu> eu\:u e U г\ G}.
Since U is solid, it is not difficult to show that S = U. Hence the given
norm ||. |] is precisely the gauge of S; that is, |[. |j is an approximate
order-unit norm with the approximate order-unit {eM}.
In what follows the open unit ball in a normed space (E, |]. ||) is
always denoted by U, i.e. U = {x eE: ||z|| < 1}; if (E', ||. ||) is the
Banach dual space of (E, ||. ||), then U' will denote the open unit ball in
(E'} |j. ||). The closed unit ball in (E, ||. ||) is denoted by S, i.e.
2 = {x e E\ ||ж || < 1}; S' will denote the closed unit ball in (E', ||. )|).
An ordered normed space (E, C, |). ||) is called an order-unit normed
space if there exists an order-unit e such that |]. || is the gauge of [—e, e].
Thus order-unit normed spaces form a special class of approximate
order-unit normed spaces.
(9.2)	Proposition. Let(E,G, ||.||) be an ordered normed space. Then
the following statements are equivalent:
(a)	E is an order-unit normed space;
(b)	G is 1-normal in (E, ||. ||) and there exists e in E with ||e|| < 1 such
that e > и for all и e 17;
(c)	U is order-convex and all elements of U are dominated by some
e in E with || e || < 1;
ORDERED NORMED VECTOR SPACES	87
(d)	U is absolute-order-convex and all elements of U are dominated
by some e in E with |j e |] < 1;
(e)	U is solid and all elements of U are dominated by some e in E with
IH < I;
(f)	||. || is a Riesz norm on (E, 0) and all elements of U are dominated
by some e in E with ]|e][ < 1.
Proof, (a) => (b): If ||. || is an order-unit norm on E with order-
unit e, then e dominates all elements of the open unit ball in E', also G
is 1-normal by proposition (9.1). As in proposition (9.1), it is easy to
verify that (b) о (о) => (d) and (e) => (f). Next we show that (d)	(e).
Letw e U. Then there exists a such that ||u|| < a < l.By(d), -[zufcc < e
for some e G E with ||e|) < 1. Then ae e U and dominates и absolutely.
This shows that 17 is absolutely dominated. Thus the implication
(d) => (e) is clear. The implication (e) => (d) is trivial. Therefore
(d)o(e). To complete the proof it remains to show that (f) => (a). If
(f) holds, then the order-interval [—e, e] contains U and is contained
in the closed unit ball in (A1, j|. ||), hence j|. || is the gauge of [—e, e], and
thus an order-unit norm with order-unit e.
Example. The sequence spaces , c, function space , and the
space of all bounded real-valued continuous functions defined on a
topological space are examples of order-unit normed spaces. The space
c0 of all null sequences is an approximate order-unit normed space with
no order-unit.
The concept dual to (approximate) order-unit normed spaces is that
of base normed spaces. (See Edwards (1964) and Ellis (1.964),) A positive
subset В of an ordered vector space (E, C) is called a base of 0 if В is
convex and has the property that every non-zero element c in 0 has a
unique representation of the form a6 with a > 0 and Ъ e B. If, for each
c in С, Д(с) denotes the uniquely determined positive number such that
c _ h(c}b where b g B, then h is obviously a strictly positive, additive,
and positively homogeneous functional on C (h is said to be strictly
positive if h(c) > 0 for all non-zero positive elements c). Let F = C — G.
Tor each x = c1—c2 in F, where cls c2 g Gt we define
h(x) = Л(с2) -- Л(с2).
Then h is a well-defined linear functional on E such that h\C — h.
Notice also that В = {c g G:h(c) = 1}. By the Hahn-Banach theorem
h can be extended to define on the whole of E. Let/ be a linear extension
88
ORDERED .NORMED VECTOR SPACES
of h, then f is a strictly positive linear functional on E such that
В = G	={ce G\f{c) = 1}. This remark makes the following
result clear.
(9.3)	Lemma. Let В be a positive set in an ordered vector space (E, G}.
Then В is a base of G if and only if there exists a strictly positive linear
functional f on E such that В = G x(l).
If В is a base of G and if G—G = E, then the circled convex hull
Г(В) of В is absorbing in E. The gauge of Г(В) is called a base semi-norm
on E (defined by the base B). If the semi-norm is in fact a norm it will be
referred to as a base norm. An ordered normed space (E, O', ]] - ) is called
a base normed space if E — G—G and if there exists a base В of G such
that the given norm ||. || is the gauge of Г(В). If В is a base of G,
E G—G and if Г(В) — co(B U — B) is the circled convex hull of B,
then. Г(В) is the smallest solid set containing B. In fact, it is easily
seen that Г(В) is absolutely dominated; to see that Г(В) is absolute-
order-convex let
±Ж < 2/ = ЯЛ-ЯЛ e Г(Б),
where b1} b2 e В and Я15 Я2 > 0 with Ях + Яа 1. Since > 0, there
exist b,V eB and a, a' > 0 such that у-\~% = <xb emdy—x = a'b'. Then
2x — ab — ab'. To show that x e Г(В), we have to show that
|(a + a') < 1.
By lemma (9.3), there exists a strictly positive linear functional / on E
such that В — C ^/^(l). Notice that
2(ЯЛ — Я262) = 2y = ab-dab',
hence, on applying / on both sides, we have
I > Ях-Я2 = |(a + a'),
as required. This shows that x e Г(.В), and hence Г(В) is absolute-order-
convex and consequently Г(В) is solid. Finally, if S is a solid set
containing В and if z — p2z2 e Г(В), where z1; z2 <= В and
y1, 14 > 0 with ^4-^2 = 1, then let z' =	g В c S. Since
< z' it follows that z g 8, showing that Г(В) cz S. Therefore we
have shown that Г(В) is the smallest solid set in E containing B.
Consequently the gauge of Г(.В) is always a Riesz semi-norm on E.
ORDERED NORMED VECTOR SPACES
89
(9.4)	Proposition. Let (E, C, ||. ||) be a base-normed space with base
B. Then ||. || is a Riesz norm on В and the following assertions hold:
(а)	В = {ж g С: ||ж|| = 1};
(b)	||. || is additive on C, i.e, h+2/ll = hll +h|| for all x, у in C;
(с)	G is (1^-generating for each s > 0;
(d)	C is 2-normal.
(Remark. Clearly the additivity of ||. || on C implies that ||. [| is
monotone; thus the condition is a normality condition. Similarly that U
be directed upwards is a а-generating condition for appropriate a.)
Proof. That j|. || is a Riesz norm, has been noted before. To show (a),
let b g B. Then ||6|| < 1. Further, suppose a is a positive real number
such that b g aP(R). Then 6/a = a'bf-ai’b" for some b‘, b" in В and
a', a" > 0 with a'4-a" = 1. Let/be a strictly positive linear functional
on E such that В = О	Then
a' = /(aW) =	+	=- + a",
\a / a
hence I/a — a' —a" < a' + a" = 1, so a > 1. This implies that ||5|| > 1;
consequently ||&]| = 1 is valid for all b g B. On the other hand, if c g C
and |jc|| — 1, then c ~ yb* for some у > 0 and b* eB; further
1 = ||c|| = \\yb* || = y\\b* || = у
by what we have previously proved. Therefore c “ yb* — 6* g B. Thus
(a) is verified. To verify (b), let x, у g C. We can further suppose that
x 0, у L 0. Then, by (а), ж/||ж||, 2//Ц2/Ц e B. Since В is convex it
follows that
ж+у hll / a? \	|M|/ ?/ \ B
hll + hll hll + hll\hll/ hll + hllwll/
Applying (a) again, we have
as required in (b).
%+y
hll + hll
= 1, i.e. h+2/ll = hll + hll
Let e > 0 and let x E E. Since ||. || is a Riesz norm, there exists у e E
with ||«|(]| < (1+e) Ill’ll such that Lx < y. Let xx = l(y-\-x) and
=	Then хъ x2 e О, x — xz, and
hill + k2|| = hi+^all = \\v\\ < (i-H) h)|.
Thus (c) is proved.
90
ORDERED NORMED VECTOR SPACES
Finally, we verify (d). Suppose x < у < z in E. Then there exist
bx, b2 G В and Лх, Я2 > 0 such that y—x = Q^z—y --= Л2&2. By (b),
we have
hll + hll > h-^ll = HlM-ЛМ = Ц1М + Н2&2И
- h-M + h“«/|| > (blHhll)+(ll«/l|-hll),
it follows that hll + hll > ||y||. Consequently G must be 2-normal.
(9.5)	Proposition. Let (E, G, ||. ||) be an ordered normed space. Then
the following statements are equivalent:
(a)	E	is a base normed space;
(b)	||.	j| is additive on G and G is (1 + e)-generating for each	e	>	0;
(c)	||.	|| is additive on G and the open unit ball U in (E, |].	||) is	de-
composable;
(d)	I).	|| is additive on G and U is solid;
(e)	||.	j| is a Riesz norm on (E, G) and |]. || is additive on G.
.Proof. In view of proposition (1.7) and. theorem (6.3), it is clear that
(b) о (c) and (d)	(c). That (a)	(b) and (a)	(e) were proved in the
preceding proposition. Therefore to complete the proof we have only to
verify that (b)	(a) and (e)	(b).
(b)	=> (a): Let В = {x e G: h|| = .1.}. Since ||. || is additive on G,
В is convex. .It is then easily verified that В is a base of G. Let
Г(В) = co (В U — B). Then Г(В) is contained in the closed unit ball of
(E, ||. ||). On the other hand, it contains the open unit ball U; in fact, if
|| ж || < 1 then there exist хг, x2gG with ||aq|| -(- hall < 1 such that
x = xv-x2. Assuming that хг 0, ж2 0, we have —— , —— e В and
hili hall
ж = hill I у™? I + hall (vA?) e Г(В).
vl^alr
This shows that U с Г(В). Consequently the given norm )|. || is
precisely the gauge of Г(В), and (E, G, ||. ||) is a base normed space.
(e)	(b): Let x g E and s > 0. Since ||. || is a Riesz norm., there
exists у in E with ||y|| < (1-f-e) hi! such that < y.Letaq = i(y~px)
and x2 — i(y—x). Then x Xy — x2, x1} x2 e G, and
hill + hall hi+^ll = hll < (i+e) hll>
showing that G is (1 -|-£)-generating. Therefore (e) => (b).
ORDERED NORMED VECTOR SPACES
91
In what follows we shall show that the concepts of base norms and
approximate order-unit norms are dual to each other. To this end, we
shall first establish the fact that the additivity of the norm is dual to the
directedness of the open unit ball. More precisely, we have the following
theorem (cf. Ng (1969a)):
(9.6)	Theobem, Let (E,G, ||.||) be an ordered normed space and
(E', ||. ||) the Banach dual space with the dual cone C. Then the following
statements are equivalent:
(a)	|i. || is additive on G;
(b)	the open unit ball U' in E' is directed upwards;
(c)	the closed unit ball S' in E' is directed upivards.
Proof > (a)	(b): Suppose that/15/2 are elements of U'. Let 5 be
such that max{||/1||, ||/2||} < 6 < 1. We define
p(x) = d ||x ||	(x e E)
and
q(x) == sup{/](a;1) +/а(ж2) ’-%i>	= x} (x e (7).
Thenp is sublinear and q is superlinear on where they are defined. Also,
since ||. || is additive, it is easily seen that q(x] < jp(a;) for each x in G.
By Bonsall’s generalization of the Hahn-Banach theorem (1.15), there
exists a linear functional f on E such that q(x) < f(x) and/(?/) < p(y)
for all x e G and all у e E. Then/13 /2 < f and \\f\\ < <5 < 1 (so/ e U').
(b)	=> (c): Let f, g be an arbitrary pair of elements in S'. Then, by
(b), for each positive integer n, there exists hn in U' such that
n+1
f,g < —
n
By the Alaoglu theorem, S' is a{E'E)-compact, so {hn} has a a(E'E}-
cluster point Ao, say. Then hQ e S' and/, g < hQ. This shows that S' is
directed upwards.
(c)	(a): Let x, у be in G. By the Hahn-Banach theorem, there
exist /, g e S' such that f(x) ~ ||ж|| and g(y) = |j?/||. For this pair of
functions/, g, there exists h e S' such that/, g < h by (c). Then
kll + hll =/(^)+0r(y) < h{x)ph{y) < h+y||,
consequently, ||ж|| Т1Ы| = ||ж+у|| by virtue of the triangle inequality
for ||. ||.
The following result is dual to the preceding theorem and is parallel
to theorems (5.16) and (6.12):
92
ORDERED NORMED VECTOR SPACES
(9.7)	Theorem. Let C be a cone in a normed space (E, ||. ||) and let
(E', ||. ]|) be the Banach dual space with the dual cone Gr. И+ consider the
following statements:
(a)	||. || is additive on G';
(b)	the open unit ball U in (E, ||. ||) is directed upwards.
Then (b)	(a). If, in addition, C is assumed to be ||. \\-complete, then
(a)	(b), hence (a) and (b) are equivalent. Furthermore, if C is ||. ||-com-
plete and if (a) or (b) holds, then (E, [|. ||) must be complete.
(Remark. It is easy to see that if the closed unit ball is directed
upwards then so is the open unit ball. The converse is incorrect: for a
counterexample see Asimow (1968). In the case where X and G are
assumed to be ||. || -complete, the theorem was proved independently
by Asimow (1968) and Ng (1969a). A more general theorem than the
present form was announced by Ng and Duhoux (1973).)
Proof. If (a) or (b) holds, then we established the fact in theorem
(5.16), that (E, ||. ||) is complete whenever G is ||. ||-complete.
(b)	(a): Let f,geG' and let x,y e U. By (b), there exists z g U
such that x, у < z. Then
/(^)+^) </(ФЫ/> < \\f+g\\ hll < llf+^l-
Since x, у are arbitrary in U, it follows that ||/|[ + 1Ы1 < II/ bf/ll 5 conse-
quently II/H +ll^ll = ll/+<7ll by virtue of the triangle inequality.
Finally we show that (a) => (b) under the assumption that C is ||. ||-
complete (hence E is also ||. ||-complete). Let xf, x" be in U. Let 5 be
such that тах{||ж'||, ||+'||} < <5 < I. We shall find an x g U such that
x > x’, x". To do this, let
= (/еЛ
and
W) = suptf'M +/"(«") eC'.f =/'+/"}	(/eC').
Then P is sublinear and Q is superlinear in the domains where they are
defined; also, by (a), Q(f) < P(f) for each/in Gr. Further, P is lower
g(E', E)-semi-continuous on E. Next we show that Q is upper semi-
continuous on S' C\ O' under the relative o(E', E)-topology. If not,
then there exist a real number Я and a net {/a} in S' C\ G' convergent
to/0 such that Q(fa) > A > Q(/o) for each a. Then, for each a, there exist
/«, f" in °' with A = +/« such that
ORDERED NORMED VECTOR SPACES
93
Notice that 0 < /a < fa, so ||/a|| < |(/J| < 1 by (a). This shows that each
/a is in S' И O'. Similarly/" e S' И O'. Since S' П & is a(E’, .^-com-
pact, {/„}, {/"} have convergent subnets with limits, say Д and f£, in
S' n C' respectively. Since fa = ff-lff we must have /0 = /o+/o-
Passing to the limit in the last displayed inequality, we have
Wo)
contrary to the fact that Я > Q(f0). The contradiction shows that Q is
upper semi-continuous on S' nd'. By the separation theorem, given
in corollary (2.19), for 0 < e < 1—-<5, there exists x in E such that
and
W) <»
» < W)+s \\f\\
tfzO')
By the last inequality, it follows from the Hahn-Banach theorem that
И = sup{/(a:):/ e E’, |)/|| < 1} < <5 + e < 1.
Also, for each / in 6", f(x'} < $(/) < /(x). Since G is complete (hence
closed), it follows from the Hahn-Banach separation theorem (cf.
theorem (2.16)) that x' < x. Similarly x" < x. Thus x is an element of U
which dominates x', x".
The following theorem is due to Krein and Ellis (cf. Ellis (1964-)).
(9.8)	Theorem. Let (E, G, ||.||) be an ordered normed space and let
(Ef, ||. |j) be the Banach dual space with the, dual cone O'. We consider the
following statements:
(a)	(E, C, ||. ||) is a base normed space;
(b)	(£?', C", ||. ||) is an approximate order-unit normed space;
(c)	(E', O', ||. ||) is an order-unit normed space.
Then (a) => (b) o(c). If C is assumed to be |). || complete, then (c) => (a)
and hence (a), (b), and (c) are mutually equivalent. Further, if C is
||. ||-complete and if one of the statements (a), (b), and (c) holds, then
(E, ||. ||) must be complete.
Proof, (a)	(b): If (E, G, ||. ||) is a base normed space, then ||. || is
a Riesz norm on (E, G) and is additive on G by proposition (9.4). It
follows from theorem (6.12) and theorem (9.6) that the norm on the
Banach dual space (E', O') is also a Riesz norm and the open unit ball
U' is directed upwards; hence (E', O', ||. ||) is an approximate order-unit
normed space by proposition (9.1).
94	ORDERED NORMED VECTOR SPACES
(b)	(c): If (b) holds, then the open unit ball U' in (£?', ||.||) is
directed upwards. For each и e U', let eM = u. Then {eu:u e U'} is an
increasing net in the closed unit ball X' which is o{E', 2?)-compact by
the Alaoglu theorem. Let e be a cluster point of {eM}. Then e > eu for all
u. In view of proposition (9.2), we conclude that (E',C, ||.||) is an
order-unit normed space.
Finally we show that if G is ||. ||-complete and if (c) holds then
(E, ||. ||) is complete and (a) holds. By (c), O' is 1-normal in {E', ||. ||).
Hence, since C is ||. ||-complete, it follows from theorem (5.16) that
(E, ||. ||) must be complete and G is (1+r)-generating for each s > 0.
Furthermore, the open unit ball in the order-unit normed space
(Er, G', j|. ||) is directed upwards, and it follows from theorem (9.6) that
the norm ||. || on E is additive on G. This, together with the (1 + e)-
generating property of G, implies that (E, G, ||. ||) is a base normed
space by proposition (9.5), i.e. (a) holds.
The following theorem is dual to theorem (9.8) and is due to
Ng (1969).
(9.9)	Theorem. Let (E, C, ||. ||) be an ordered normed space and let
(E'} C', ||. ||) be the Banach dual space with the dual cone G'. We consider
the following statements:
(a)	(E, C, ||. ||) is an approximate order-unit normed space;
(b)	(E’, C', ||. ||) is a base normed space.
Then (a) => (b). If G is assumed to be |]. ^-complete then (b) ==> (a) and
hence (a) and (b) are equivalent. Further, if C is \\. \\-complete and if
either (a) or (b) holds then (E, ||. ||) must be complete.
Proof. If C is |]. ||-complete, then we have the following chain of
equivalences (cf. proposition (9.1), proposition (9.5), theorem (6.12), and
theorem (9.7)):
(a)	о the open unit ball is directed upwards and ||. || is a Riesz norm
on E
о the norm on E’ is additive on G' and is a Riesz norm on E' <=> (b).
The other assertions can also be easily verified by virtue of the proposi-
tions quoted.
In general, that E' be a base normed space does not imply that E is
an order-unit normed space, as the following example shows.
Example. Let c0 be the Banach lattice of all null sequences of real
numbers (with the natural ordering and the supremum-norm). Then the
Banach dual space of c0 is the space G of all summable sequences of real
numbers. The norm on c0 is a Riesz norm and the open unit ball is
ORDERED .NORMED VECTOR SPACES
95
directed upwards; so c0 is an approximate order-unit normed space and
G is a base normed space. Further c0 has no order-unit. In fact, consider
an element x — (ж15 ж2,...) of c0. Let у = (v%.>	Then у e c0,
and lim Mfixn = 0 for any constant M > 0. Hence there does not
exist a constant M. with the property that
(V^i, \/x2...) = у < Их — (>жп Mx2,...).
This shows that the order-interval [ — ж, ж] does not absorb y, so x is not
an order-unit. Thus c0 has no order-unit at all.
In the situation of the preceding theorem, it is easily seen that the
set В — {/ e C': ||/|| = 1} is a base of O' and the defining base norm is
the same as the given norm. It turns out that the existence of an order-
unit in E is related to a topological property of B.
(9.10)	Theorem (Edwards). Let (E, G, ||.||) be an ordered normed
space, and let (E’, O', ||. ||) be the Banach dual space with dual cone C.
We consider the following statements:
(a)	(E, G, ||. ||) is an order-unit normed space;
(b)	(E!, &, |j. ||) is a base normed space with a o{E', Efcompact base
В defining the norm ||. ||.
Then (a) => (b). If C is assumed to be ||. \\-complete, then (b)	(a); hence
the statements (a) and (b) are equivalent. Moreover, if G is ||. \\-complete
and if either (a) or (b) holds, then (E, ||. ||) must be complete.
Proof. The final assertion can be proved easily, as before. Next we
show that (a)	(b). Suppose (E, C, ||. ||) has an order-unit e defining the
order-unit norm ||. || on E. By the preceding theorem, (E', Gf, ||. ||) is a
base normed space; hence the set
в ^{fEC-Wfw 1}
is a base of G' and the corresponding base norm is precisely the norm
||. || on Ef. It is not difficult to see that
В =	< 1 and /(e) = 1}.
By the Alaoglu theorem, В is o[E',Ef compact, proving (b).
Conversely, suppose (b) holds and that G is ||. |[-complete. Then, by
theorem (9.9), (E, ||. ||) is also complete and is an approximate order-
unit normed space. By (b), co(B U — B) contains the open unit ball U'
and is contained in the closed unit ball S' in {E', ||. ||); further, since В
is o(E', ^-compact, it is easily seen that co(B U — B) must be equal
96
ORDERED NORMED VECTOR SPACES
to S'. Next, we note that each / e E' can be expressed as/ f1—f2 for
some /1? /2 e C'; define
M) = IIAII - 11ЛИ-
Since ||. || is additive on O', p(f) does not depend on the particular
choice of the components /15 /2 of/. Hence /-> ^(/) is a well-defined
linear functional on E'} and its restriction to O' is identical to that of
||. ||. Notice that
IB-АВ = 2' n
(9.1)
In fact, the set on the left-hand side is certainly contained in that on the
right-hand side; on the other hand, let / e S' П iu“i(0) and write
/ = 'Мь. — 'У)2> where b1} Ъ2е В and 2г, Я2 > 0 with ^+^2 = 1. Then
О = /<(/) = IIVillHIAsM = А-Л.
it follows that | — Я2. Therefore / = l/i —iA e i-S—|-B. Thus
formula (9.1) is proved. Since В is o(E', j®)-compact so is the set —
Therefore the kernel ;u“1(0) of p intersects S' in a a(E', ^)-compact set.
By the Krein-Smulian theorem (cf. e.g. Schaefer (1966, p. 152)), д-ЦО)
is o(E', A?)-closed and p is a(E', .^-continuous. Consequently, there
exists e in E such that p(fj = f(e) for all feE'. As noted before,
(E, C, ||. ||) is, at least, an approximate order-unit normed space: thus,
in view of proposition (9.2), to complete the proof we have only to show
that ||e|| < 1 and e > x whenever ||ж|| < 1. To verify this, let
/ —	—Я2&2 g S' = co(B U —B) for appropriate Ьг, b2 and 213 Я2.
Then /(в) = Л(/) = и лап - плац = ях-лг < 1.
By the Hahn-Banach theorem, it follows that [|e|| < I. Moreover if
x eE and || ж || < 1 then
Ж) < i = VII = ХЛ =/(«)
for all / g B; and so /(ж) < /(e) for all / g O'. Since C is ||. ||-complete
(hence closed), it follows that ж < e, whenever ||ж|| < 1. This completes
the proof of the theorem.
Let К be a compact convex subset of a locally convex space. A
real-valued function a on К is said to be affine if
a(AiA?i + .Я2&2) = Л,]/й(&1) +Aft(^2)
whenever Jc1} k2 e К and Я15 Я2 > 0 with = 1. Let C’(K) denote
the ordered Banach space of all real-valued continuous functions on E,
and let A (A) be the closed subspace of C(.K) consisting of all affine
functions. Then the constant function 1 is an order-unit in C(K) and
defines the order-unit norm which is precisely the usual supremum in
ORDERED NORMED VECTOR SPACES
97
C(K). Further, since 1 g A(K), A(K) is also an order-unit normed space
in its own right. Thus the implication (b) => (a) in the following theorem
is clear.
(9.11)	Theorem. Let (E, G, ||. ||) be an ordered Banach space with a
closed cone G. Then the following statements are equivalent:
(a)	(E, G> ||. ||) is an order-unit normed space;
(b)	there exists a compact convex set К in a locally convex space such
that (E, G, ||. ||) is isometrically order-isomorphic to A{K).
Proof. It remains to show (a)	(b). Let
В =^= {f e E': \\f\\ = l,/eQ
In the proof of the preceding theorem, we noted that В is a o[E', Ef
compact base of G', and Г(В) = S'. Let H(_B) be the space of all affine
g(E', E)-continuous real-valued functions on JB. We shall represent
(E,G, ||. ||) as J.(2?).
For each x in E, define x е Л (.13) by the rule
</)=>)	(f£B).
Since F(jB) — S', it follows from the Hahn-Banach .theorem that
||ж|| = ||x||. Further, since G is closed, it is easily seen that x > 0 if and
only if x > 0. It is now clear that the map ж —> ж is isometrically order-
isomorphic from E into A(B). To complete the proof we have to show
that the map is onto. Let a e A(B). Since В is a base of G', a can be
uniquely extended to become an affine function on C', and consequently
a can be uniquely extended to become a, linear functional on G' —G' = E'.
Notice that if f e S' = Г(В) and f — Lb—pc, where b,c g В and
p, 2 > 0 with 2 + p = 1, then
a(/) = ha(l) — pa(c).
Since S' = Г(В) and a is continuous on the g(E* , j^-compact set B,
it follows easily that a is continuous on S'. In particular, the inter-
section u~1(0) П S' of the kernel with the closed unit ball is o(E', E)-
closed; consequently, it follows from the Krein-Smulian theorem that
cW’-fO) is g(E', £i)-closed, and hence a is a g(E', E)-continuous linear
functional on E'. Therefore there exists x in E such that £ — a; this
shows that the map ж -> ж is onto J(B), and completes the proof of the
theorem.
98
ORDERED NORMED VECTOR SPACES
In order to represent approximate order-unit normed spaces in the
spirit of the preceding theorem we need to study a special type of
compact convex sets. Let {E, C, be an ordered convex space. A non-
empty .^-compact convex subset К of C is called a cap of C if C\K is
also convex. Notice that any cap necessarily contains the origin. К
is said to be universal if 0 — pos K.; i.e. if C — (J {AK: A > 0}. If C has
a compact base B, for instance, then the set {Ab: 0 < A < 1, b e B} is a
universal cap of C. Another example of a universal cap is the positive
part of the unit ball in the Banach dual space of an approximate order-
unit normed space, as the following lemma shows.
(9.12)	Lemma. Let (E, C, ||. ||) be an approximate order-unit normed
space, and let (E', C', ||. ||) be the Banach dual space with the dual cone C'
and with the closed unit ball S'. Let К ~ S' П C'. Then К is a universal
cap of C' in the locally convex space (E', a(E', E)).
Proof. By the Alaoglu theorem, К is a compact convex subset
of E' with respect to the o-(E', A)-topology. Also, by theorem (9.9),
(E'} C, ||. II) a base normed space, so ||. || is additive on C". Conse-
quently C'\K is convex. Therefore К is a cap of C. Finally it is easily
seen that К must be universal in C.
If A is a cap, A0(A) denotes the subspace of J. (A) consisting of all
functions vanishing at the origin 0. Then A0(A) is an ordered Banach
space in its own right.
(9.13)	Proposition. Let (E,C, ||.||) be an approximate order-unit
normed space, and suppose that (£, ||. ||) and C are complete. Then there
exists a universal cap К such that (E, C, ||. ||) is isometrically order -
isomorphic to A0(A), where К may be taken to be the positive part of the
unit ball in the Banach dual space E'.
Proof. Let A — S' П C, as ia the preceding lemma. Then A is a
universal cap of C in (E', a{E', A)). For each x in E define x e A0(A) by
ЭД-Ж (/SA).
Then, as in the proof of theorem (9.1.1), x x is an isometrically order-
isomorphic map of E onto A0(A).
In what follows we shall show that the converse of proposition (9.13)
holds. To this end, we first prove the converse of lemma (9.12).
ORDERED NORMED VECTOR SPACES
99
(9.14)	Proposition. Let (E, G, be an ordered convex space such
that G is &-closed and E — G—G. Let К be a universal cap of G, and let
j|. || be the gauge of co(.7f U — K). Then (E, C, ||. ||) is a base normed
space, and there exists an approximate order-unit normed space (V, W, ||. [|)
such that (E, G, ||. ||) is isometrically order-isomorphic to the ordered
Banach dual space (V', W, ||. ||) of (F, W, ||. ||). Furthermore, if
S = [% g El ||ж|| < 1} then
К - S П C.	(9.2)
Proof. For each x e C, we define
p(x) inf{2 > 0:ж g Ж).
Since К is ^-compact convex and since C\K is convex, it is not difficult
to verify that p is a positively homogeneous, additive functional on G
and К = {xe Gip(x) < 1}. Moreover, since К is ^-compact, у»(ж) Ф 0
whenever x e G and x 0. Let В ~ {xeC ip(x) = 1}. Then it is easily
seen that В is a base of G and co(A U —K) = co(B U — B). Hence
the gauge ||. || of co(l£ U —K) is precisely the base semi-norm defined
by the base B. Furthermore, since К is ^-compact, co(A U —K) is
^-compact, and hence ||. || is in fact a norm and
со(Б U — B) = co(X U — K) = {x eEi ||ж|| < 1},
that is, co(B U — B) ~ S. To verify formula (9.2), let x e S n G, and
suppose that x = 2,1b1— d2&2, where &3, Z>2 g В and Я2 > 0 with
Лх-|-Я2 = 1. Then
^(«) + Я2 - У>(жЦ-А262) ^plf^bj =
so p(x) =	-- Л3 < 1 and x g K. This shows that S П G c X; conse-
quently S П С = К since it is obvious that S n G 2 K. Thus formula
(9.2) is proved. We have shown that (E, G, ||. ||) is a base normed space
with the ^-compact closed unit ball S, and that К is the positive part
S П G of the unit ball S. Finally we show that (E, ||. ||) may be iden-
tified with a Banach dual space. Accordingly, let (E, &)' and (E, j|. ||)'
denote the dual spaces of E und . and ||. || respectively. Let V be the
space of all linear functionals f on E such that f is ^-continuous on
S. Then
(E,	£ V с (E, ||.||)'.	(9.3)
The first inequality is obvious. To see the second inequality let f g V,
then/(S) is the continuous image of the ^-compact set S, so is compact
and hence bounded in R. Therefore f is continuous on (E, ||. ||), and
100
ORDERED NORMED VECTOR SPACES
formula (9.3) is proved. Now it is easily seen that У is a closed subspace
of the Banach space (E, ||. ||)'. Thus V may be regarded as a Banach
space in its own right. Let W be the cone in V consisting of all positive
linear functionals on E. Then (V, W, ||. ||) is an ordered Banach space
and W is ||. |j-closed. Let (У', W', ||.||) denote the Banach dual of
(V, ||. ||) with the dual cone W'. Lor each x in E, define iplx) by the rule
(^(ж))(г>) = v(x) (v e V).
Then it is easy to see that ip is a 1-1 continuous (in fact norm-reducing)
map from E into the Banach dual space V' of V. Since the cone G in E
is ^-closed, it is easy to see that x > 0 in E if and only if ip(x) > 0 in
V. Also, since each v in У is ^-continuous on 2, the restriction y| S of
to 2 is continuous with respect to the relative ^-topology in 2 and
the y)-topology in У'. Since 2 is ^-compact it follows that ^(2)
is <т(У', y)-compact. Also the set ^(2) is convex. By the bipolar
theorem, it is precisely its bipolar (?/;(2))’гя’ with respect to the duality
(У', У). Note that
(^(2))”' = {w e У:(^(я))(г>) < 1 V# s 2}
= {v e У:г(ж) < 1 V# e 2},
which is the unit ball in У, and hence (^(2))’ггг (that is, y(2)) is the
unit ball in V1. In other words, ip maps 2 onto the unit ball in V'.
Therefore ip is an isometric order-isomorphism from (E, G, |[. ||) onto
the Banach dual space (У', Wf, ||. ||). Since (E, C, ||. ||) is a base normed
space, so is (У', W', ||. ||) and it follows from theorem (9.9) that
(У, W, ||. ||) is an approximate order-unit normed space.
Remark. The condition that G is ^-closed is in fact automatically
satisfied in view of formula (9.2) and the Krein-Smulian theorem.
(9.15)	Theorem. Let (E, G, ||.||) be an ordered Banach space with a
closed cone, C. Then the following statements are equivalent:
(a)	(E, G, ||. ||) is .an approximate order-unit normed space;
(b)	there exists a universal cap К of a cone P such that (E, C, ||. ||) is
isometrically order-isomorphic to A,(A).
Proof. The implication (a) => (b) was established in proposition
(9.13). Conversely, suppose A is a universal cap of a cone P, and let
P — P — F. Let ||. || be the gauge of co(A U —A). Then, by the
preceding proposition, there exists an approximate order-unit normed
space (У, W, ||.||) such that (A, P, ||. ||) may be identified with the
ORDERED NORMED VECTOR SPACES	101
ordered Banach dual space (F\ W', ||. ||), and К is identified with the
positive part of the unit ball in (V, W', |] -1|) (cf. formula (9.2) in
proposition (9.14)). By proposition (9.13), (7, W, ||. ||) is isometrically
order-isomorphic to J.0(X). Thus A0(.A) is also an. approximate order-
unit normed space. This proves that (b)	(a) and completes the proof
of the theorem.
We recall that a semi-norm p on an ordered vector space (.£’, G) is
called a Riesz semi-norm if it satisfies the following two conditions:
(i)	absolute-monotonicity—if — у < x < у then p(x) < p(y);
(ii	) for any x e E with $>(ж) < 1 there exists у eG with p(y) < 1 such
that ~~y < x < y.
Now suppose that (£?, G) is a Riesz space (i.e. vector lattice), and
that p is a Riesz semi-norm. Then, in view of the absolute-monotonicity
property of p and — |ж| < x < |ж|, we have p(x) < 7>(|ж|). If there is x
in E such that p{x) < jp(|^|), and we can assume without loss of
generality thatjp(^) < 1 < 79(|ж|), then there exists у e C with p(y) < 1
such that —y < x < y, and thus —у < |ж| < у; using the absolute-
monotone property of p again, we obtain ^()ж|) < p(y) < 1, contrary
to the fact that 1 < ^(|ж|). This contradiction shows that p(ж) = ^(|ж|)
for all x eE. This remark makes the following result clear.
(9.16)	Lemma. Let (E, O') be a Riesz space and p a semi-norm on E.
Then the following statements are equivalent:
(a)	p is a Riesz semi-norm;
(b)	p is monotone and p{\x|) = p(x) for all x e E;
(c)	if |ж| < |y| then we have p(x) < p(yf
Let fE} C) be a vector lattice. A Riesz semi-norm on E is called a
Riesz norm if it is, in fact, a norm. A vector lattice equipped with a
Riesz norm is called a normed lattice (or a normed Riesz space). If the
norm in a normed lattice is complete then the normed lattice is called
a Banach lattice (or briefly В-lattice). It is elementary and well known
(cf. proposition (10.3), in the next chapter) that — y^\ < \x—y\
where x, у are in E; hence, if ||. || is a Riesz norm on E, then
k+-2/+|l <
This implies that the map x —> is continuous in a normed lattice
(E, G, ||. |j). Similarly we can show that the lattice operations x —> x~
and x — > |ж| are continuous. Consequently the positive cone in a normed
8
1.02
ORDERED NORMED VECTOR SPACES
vector lattice must be closed (and hence the positive cone in a В -lattice
must be complete). In Chapter 11 we shall generalize the theory of
normed vector lattices to a more general case of so-called locally convex
Riesz spaces. Nevertheless, in the remainder of this chapter wo study
some results peculiar to the normable case.
A Banach lattice (X, G} ||. ||) is called:
(i)	an AM-space (or ALm-space) if ||oq V x2|| = ||жг|| V ||ж2|| for each
pair of elements x1} ж2 in (7;
(ii	) an AL-space (or AlA-space) if ||aq-f;ra)| = ll^ill + ll^all f°r each
pair of elements aq, x2in G;
(ii	i) an AL®-space (1 < p < oo) if
Hi v«2|p < IKF+W3’ < ki+aM®
for each pair of elements x2, x2 in G.
The following result characterizes AAf-spaces.
(9.17)	Proposition. Let ||. || be a norm on a vector lattice (X, G)
(Biesz space) and suppose that (A, ||. ||) is complete. Then the following
statements are equivalent:
(a)	(A, (7, ||. ||) is an AM-space;
(b)	||. || is an approximate order-unit norm on (A, G);
(c)	(A, G, |j. ||) is a Banach lattice and the closed unit ball S in (A, ||. ||)
is directed upwards;
(d)	(A, G, ||. j|) is a Banach lattice and the open unit ball U in (A, ||. ||)
is directed upwards.
Proof. It is easy to see that (a) => (c)	(d). In view of lemma (9.16)
and proposition (9.1) it is clear that (b) o(d). Thus it remains to show
(d) ^=> (a). Let aq, x2 e G. Then 0 < xx, x2 < xt V x2; by (d), it follows
that ||aq||, ||ж2|| < Jl^ V x21|; hence ||oq|| V ||ж2|| < Цж-l V ж3[[. We show
further that the strict inequality cannot hold. Otherwise there exists a
real number M such that H^JI V ||ж2|| < M < ||aq V x21|. Since U is
directed upwards, there exists x e A with ||ж|| < M such that x > ж15 x2.
Then 0 < xL V x2 < x and ||oq V ж2|| < ||ж)| < M, a contradiction.
Therefore we must have ||aq V ж2|| = ||aq|| V ||ж2|| whenever aq, x2 e G.
This implies that A is an AM-space.
The following result characterizes AL-spaces, and is dual to the
preceding proposition.
ORDERED NORMED VECTOR SPACES
103
(9.18)	Proposition. Let ||. || be a norm on a vector lattice (X, C) and
suppose that {X, ||.||) is complete. Then the following statements are
equivalent:
(a)	(X, (7, ||. |[) is an AL-space;
(b)	||. || is a Riesz norm on (X, G) and is additive on G;
(c)	||. || is a base norm on (X, G).
Proof. By lemma (9.16), (a)<=>(b). By proposition (9.5), (b)o(c).
Let (E, G, ||. ||) be an ordered Banach space for which G is closed,
and we suppose further that ||. || is a Riesz norm on E. If, in addition,
(E, C) is assumed to have the Riesz decomposition property, then
Eb = E* ~E' andL" is a vector lattice by corollary (8.3). Consequently
E' is a Banach lattice. In what follows we shall show the converse: if E'
is a Banach, lattice then E must have the Riesz decomposition property.
In order to prove this result let us return to examine the proof of the
Riesz theorem, once again, where (E, G) is an ordered vector space with
the Riesz decomposition. The principal construction in the proof is to
define, for f1}f2 g Eb, that
(A v A)(^) = sup^i^j) f>2):x - xt +«2, x1; x2 g С} (xg C)
and that
(А л /3)(ж) - inf{A(«i) +A(^) ‘x =xr +ж2, x2 g 0} (xg C).
Then, because E has the Riesz decomposition property, ft V f2 and
A A A are positively homogeneous and additive (i.e. affine) on G.
Consequently, A v A (or> mor© precisely, the linear extension of
A V A) is the supremum of A and A, and А Л А the infimum of A, A-
The construction can be adopted in an ordered vector space even
without the Riesz decomposition property.
Let us consider an arbitrary ordered vector space (E, C). Let pt, p2
be two sublinear functionals on G and suppose they are bounded on
order-intervals in G. Define
(Pi A pffx) = inApi^J +р2(ж2): x = xr -рхх, жг, e 0} (x g Cf
Then Pi A p2 is an ‘infimum’ of p15 p2; that is, px A p2 is the greatest
sublinear and real-valued functional on G which is smaller than (or
equal to) both p1} p2. Similarly, if q1> q2 are two superlinear functionals
on G and bounded on each order-interval in C, then the functional
104
ORDERED NORMED VECTOR SPACES
Q.1 v 7г defined by
(7i v 7г)И ; J 8ир{7г(ж1)+7йЮ:ж = М~Я2> x2 £ 0} (x g C)
is a ‘supremum’ of g15 ^2; that is, qA V q2 is the smallest superlinear
functional on G which is larger than дд, q%. We remark that, even if
7i> 7г < Pi> is not necessarily true that gq V q2 < Pt. A p2. Of
course, this will be the case if pi} qt arc all affine (that is, both sublinear
and superlinear).
The following result sharpens corollary (2.19) in the presence of the
Reisz decomposition property.
(9.19)	Theorem. Let (E,G, ||.||) be an ordered Banach space with
closed cone G, and suppose that G is normal and generating in (E, ||. ||).
Let jp, q be linear functionals on the dual cone G' with the following
properties:
(a)	q < p on G', i.e. q(f) < p(f)for all f e O';
(b)	q is upper semi-continuous on С' n S' with respect to the a(E', E)~
topology, where = {f g E': \\f{\ < 1};
(c)	p is lower semi-continuous and bounded on G' П S'.
If the ordered Banach dual space (E'} G') has the Riesz decomposition
property, then there exists x in E such that
1(f) < /И < p(f), for all feG'.
(Remark. In view of the Krein-Smulian theorem, (b) implies that
q is, in fact, upper semi-continuous on the whole of G' with respect
to the a(E', E)-topology. Similarly (c) implies that p is lower semi-
continuous on C". Let Ё denote the canonical image in the second dual
E" of E. Then the conclusion of the theorem states that there is a
a(E', E)-continuous linear functional x lying between the semi-con-
tinuous functionals p and q.)
Proof. By corollary (3.2) and proposition (5.6), there exist positive
real numbers a, /3 such that G is а-normal and /З-generating in (E, ||. ||).
Hence, by corollary (6.11), there exists a Riesz norm j|. ||' on (E, G)
equivalent to the given norm ||. ||. Therefore we can assume without
loss of generality that ||. || is a Riesz norm on E (otherwise, consider
|[. ||' instead of ||. ||); hence the norm |j. || on the dual space E' is also a
Riesz norm.
ORDERED NORMED VECTOR SPACES
105
By corollary (2.19), there exists £0 in E such that
<?(A <Ж) < WH
(9.4)
Let »0 denote the canonical image in E" of ж0. Let pt = p A x0 be the
‘infimum’ of the sublinear functionals p and ж0 on G' i.e.,
Mfl -inlWi)+A(A)7=A+/2>/i>AeC"} (feC).
Similarly let qt — q V
be the ‘supremum’ of the superlinear
functionals q and S$o--
i.e.,
?i(A = sup ^(Д) -И0(А)	:/ = Л d f2, fv A e (/ e O').
Then pt is lower semi-continuous and qx is upper semi-continuous on
С' О S' (and hence on G' by the Krein-Smulian theorem) with respect
to the i?)-topology. Furthermore, qx < pz on G'. In fact, let
f = fi+fz = 91+02 g O', where Д, A, g15 g2 are in G'. Since (E’, G!) has
the Riesz decomposition property, there exist Лп, A12, A31, Л22 in G'
such that
fl — Л1Т^2О A ^12 d ^22
01	^11 d ^12?	02 — ^Й1“Ь^22-
Since ||. II i-s a Riesz norm, ||Д12И < IIAII- linearity of p, q, it
follows from inequality (9.4) that
9'(A)d- ^o(A)“”^N^ < 0(^11) + #(^2i)d' ^0(^12) d-£0(A2)~~“y^
< у>(Л1) + Л(М +7?(Л2) d- А(АЙ)
= 7?(A11d-Ai2)d-^0(Aid-^22)
By the definition of and^q, it follows that q^f) < p-ff) for all / e G'.
Applying corollary (2.19) again, there is xx in E such that
	0i(f) <fM < 3h(/H		2a	(f^C)
Then	0(f) < f(%i)	< p(f) +	ll/ll 22	(fcC')
and	Ш f, .	-/(^o) <	ll/ll 22	(feGf
106	ORDERED NORMED VECTOR SPACES
Inductively, we can construct a sequence {zm} in E such that
?(Л <Ж) <?(Л+“ГЙ <feG">
cl'flCl	It p 11	M rt ||
< f(xn+1 -~xn) <	(f e O')
for each positive integer n. Since the norm ||. || on E' is a Riesz norm, i(
follows easily from the Hahn-Banach theorem and the last displayed,
inequalities that {xn} is a Cauchy sequence in the Banach space (£?, ||. ||),
and hence converges, say, to x in E. For this x, we clearly have
M < № < P(f)
as required.
The preceding separation theorem enables us to establish the following
important duality theorem.
(9.20)	Theorem (Riesz-Ando). Let (E, G, ||. ||) be an ordered Banach
space with closed cone and suppose that G is normal and generating in
(E, ||. ||). Let (E', O', ||. ||) be the Banach dual space with the dual cone C'.
Then the following statements are equivalent:
(a)	(E, G) has the Riesz decomposition property;
(b)	(E’, G') is a vector lattice;
(c)	(Er, G') has the Riesz decomposition property.
Proof. By corollary (8.3), (a)	(b), and the implication (b) => (c) is
elementary and well known. Thus it remains to show (c) => (a). Suppose
(c) holds and that x1} x2 < yx, y2 in E. Let
7 = V x2 and p - y± A y2,
where and y4 (i = 1, 2) are the canonical images in E" of and yt
respectively. Then p, q satisfy the conditions of the preceding theorem,
so there exists x in X such that
M < /И < p(f) (f e G').
Then/(aq) < qff) < f(x) for all / e C. Since G is closed, it follows that
aq < x. Similarly we can show x2 < x and x < уъ у2. This shows that
(a) holds.
(9.21)	Corollary. Let (E}G, ||.||) and (E',G', ||.||) be as in the
preceding theorem. Then the following statements are equivalent:
(a)	(15", C', ||. ||) is a Banach lattice;
ORDERED NORMED VECTOR SPACES
107
(b)	(E, G) has the Riesz decomposition property and the norm j]. || on E
is a Riesz norm.
In particular, the Banach dual of a Banach lattice is a Banach lattice.
Proof. Follows from theorems (6.12) and (9.20).
The following result was proved independently by Davies (1967) and
Ng (unpublished) at about the same time.
(9.22)	Corollary. Let (E,G, ||.||) and (E’,C, ||.||) be as in the
preceding theorem. Then the following statements are equivalent:
(a)	(E', G’, у. ||) is an AL-space;
(b)	(E, G, ||. ||) is an approximate order-unit normed space and has the
Riesz decomposition property.
In particular, if E is an AM-space then E’ must be an AL-space.
Proof. By proposition (9.18), (a) holds if and only if (E', C') is a
vector lattice and the norm on E' is a base norm. Thus the equivalence
of (a) and (b) follows immediately from theorems (9.20) and (9.9).
Remark. It may happen that (E, G) is not a vector lattice even
though {E',C, ||. ||) is an AL-space. For a counterexample, see
Lindenstrass (1964). The following result of Ellis (1964) is dual to
corollary (9.22).
(9.23)	Corollary. Let (E, G, ||. ||) be an ordered Banach space with
closed cone G, and suppose that G is normal and generating in (E, ||. ||).
Let (E', O', ||. ||) be the Banach dual space with the dual cone G'. Then the
following statements are equivalent:
(a)	(E, G, ||. ||) is an AL-space;
(b)	(E', G', ||. ||) is an AM-space;
(с)	(E, G) has the Riesz decomposition property, and the norm (|. ||
on E is a Riesz norm and additive on G;
(d)	(E, G) has the Riesz decomposition property, and (E, G, ||.||) is
a base normed space.
Proof. By proposition (9.5), (c)o(d). By proposition (9.17), (b)
holds if and only if E' is a vector lattice and the norm on E' is an
approximate order-unit norm. Thus the equivalence (b) and (d) follows
from theorems (9.20) and (9.8). Therefore statements (b), (c), and (d)
are mutually equivalent. Further, it is trivial that (a)	(c). Thus to
complete the proof, we have only to show (b) => (a). Accordingly,
108
ORDERED NqRMED VECTOR SPACES
suppose (b) holds. Then (c) holds and the second dual space (E", C", ||. ||)
is an HL-space by corollary (9.22). By virtue of (c) and proposition
(9.18), to show that E is an HL-spacc it is sufficient to show that it is a
vector lattice. Now, let x g E. By (c), for each positive integer n, there
exists xn in E with ||жи|| < ||ж|| Tl/n such that < xn. Let
Уп =	and zn  	-ж).
Then yn> zn > 0, xn = yn+zn and x ~ yn—zn, Let yni zn, and x respec-
tively denote the canonical images of yn, zni and x in the HL-space E".
Let x+ — x V 0 and x~ = — x V 0 in the vector lattice E". Then
yn > x+ and zn > x~. Hence, since E" is an HL-space, we have
II Ml = \\Уп\\	114-MI + llMl
and	IM = 1141! = II4-MI-HIMI-
Consequently
||ж||4-1 > H^II = IK+M = IIMI + IIM = II4-MI
'Tb
+ II4-MI+ IIMI+ IIMI = II4-MIH4-MI+ IM+MI
= \\yn-^+\\ +114-MI + ||ж|| -= \\yn-& |j и 114-Ml + 1141-
Passing to the limit, as n co we see that yn x+ and zn —> x~. Since
(E, ||. ||) is complete, its image Ё in E" must be closed; hence x+, x~ g Ё.
This shows that E must be a vector sublattice of E”; hence E is a vector
lattice in its own right. This completes the proof of corollary (9.23).
Recall that a Banach, lattice (E, G, |]. ||) is an AL- or (HL1-) space
if the norm is additive on. E, and that it is an AM- (or H.L00-) space if
the open unit ball is directed upwards. Thus, in a general ordered
normed space, the additivity of the norm on the cone may be called an
HZd-condition and the directedness of the open unit ball may be called
an ALm-condition. In view of theorems (9.6) and (9.7), these two
conditions are dual. In the following we introduce what may be called
Lp-(1 < p < co) conditions and study some duality problems in-
volving such conditions. For simplicity we shall discuss the case when
(E, ||, ||) is a Banach space and G is a ||. \\-closed cone in (E, ||. ||). Except
where we state to the contrary, p and q will denote positive real numbers
such that l/p + l/(? = L In (E,G, ||.||), we consider the following
conditions linking the norm and the ordering.
Lv-condition (г): if x, у e G then ||ж +?/||® > 1М|да~НЫР-
JA-condition (ii): if x,y g E and e > 0 then there exists z e E with
||г||р < ||ж|р+ ||y F + s such that z > x, y.
ORDERED NORMED VECTOR SPACES	109
Let (E', C, ||. ||) denote the ordered Banach dual space with the dual
cone C. We shall show that the ZAcondition (i) and the TP-condition (w)
are dual conditions.
(9.24)	Lemma. For any ordered Banach space (E, C, ||. ||) with closed cone
С, E satisfies Lp-condition (ii) if and only if E' satisfies iF-condition (i).
Proof, (a) Necessity. Let/, g e Er be such that/, </ > 0. We have
to show that	[|/+<7Г> ll/H’+hll’-	(0-5)
If / = 0 or g = 0, then inequality (9.5) is trivial. We may therefore
suppose that /	0, p 0. Take a real number such that
0 < e < (I/Ц, |L<
Then there exist u, v in E with ||w|| < 1 and ||г> [| < 1 such that
ll/ll-e <f(u) and ||p||-£ < g(v).
x = (Ц/ll — s)qlvu and у — (||p|[ — s)'D,,1v.
Since E satisfies Lp-condition (ii), there exists zeE such that z > x, у
and hll’ < IH’+M’+e < (VII-£)’+(hll-s)’+a.
Since/, g are positive and q — 1 + q[p, we have
(\\f\\~-e)qF(\\g\\-sfi <f(x)+g(y) <f(z)-\g(z) < \\f+g\\ ||<
and it follows that
«ii/и-^+(hii-~e),+e}irt J y
Passing to the limit as e —> 0, we have
(ll/ll’+hll’)1'1 = (II/F+hll’)1-1'» < 117+9'11.
proving inequality (9.5).
(b) Sufficiency. Let x, у be in E and e > 0. Define
Q(h) = sup{/(z) Fg(y):/, g e O', h = f-\-g} (h e C')
and	+W = W (hh + ll’/ll’)1'1 CheE'Y
Then Q is an upper o(E', j^)-semi-continuous real-valued superlinear
functional such that Q(h) > h(x), h(y) for all h e C'; and P isa lower
a(E’, E)-semi-continuous real-valued sublinear functional. Also, since
по
ORDERED NORMED VECTOR SPACES
E' satisfies Lq-condition (if by Holder’s inequality we have
f(x)±g(u) < ll/ll 1М + 1Ы1 hl! < (II/F+
< Wf+gW (kF+h!l^ = m
wheneverf-\-g = h and/, g, h are in Cr. This shows that Qlfi) < P(A) for
all heC. Since E is complete, by a standard Hahn-Banach separation
argument (cf. corollary (2.19)), we can findzeE such that
Q(h) < h(z)	(h e G'f
ВД < P(A)/6 \\h\\	(heE'),
where <3 is a positive real number such that
{(lkir+hF)1/J’+^}1’ < hF+hF+e.
Then ||z||s’ < hF + e. Also, since C is closed, it follows from
h(x) < Q(h) < h(z) (h e C)
that x < z. Similarly у < z.
(9.25)	Lemma. Lei (E, C, ||.||) be as in the preceding lemma. Then
E satisfies Lv-condition (г) if and only if E' satisfies LP-condition (m),
i.e. if and only if E' satisfies the following condition:
(iii)	if f, g are in E', then there exists h e E' with ||Л||в < И/1М~1ЫР
such that h > fig.
Proof. We first remark that in the situation of Banach dual spaces,
the e that appeared in //-condition (и) can be dropped (by a compact-
ness argument).
If E’ satisfies //-condition (u), then it follows from, the necessity part
of lemma (9.24) that E” satisfies //-condition (г), so does E, since E is
isometrically isomorphic to a subspace of E" (under the canonical
embedding). Conversely, suppose that E satisfies //-condition (г), and
let /, g eE' . Define
Q(z) = sup{/F) Pg(y} :x, у eG,z xf-y} (z e (7)
and	A*) = Wl (11Л° + |1О1/г	(геЯ).
Then P and —Q are sublinear functionals on where they are defined.
Also, since E satisfies //-condition (г), by Holder’s inequality, we can
show that Q(s) < P{E) for all z eG. By Bonsall’s generalization of the
Hahn-Banach theorem (1.15), there exists a linear functional h on E
ORDERED NORMED VECTOR SPACES
Ш
Q(w) < A(w) and h(z) < P(z]
for all w e G and z e E. Then A e E' and satisfies the required property
in (iii), and a fortiori E' satisfies Z5-condition (и).
Remark. From our proofs it is clear that the preceding lemma and
the necessity part of lemma (9.24) are still valid even if E and G are not
complete.
An ordered Banach space (not necessarily a vector lattice) is said to
satisfy the IP-conditions if it satisfies both Z^-conditions (i) and (гг).
Combining lemmas (9.24) and (9.26) we arrive at the following theorem.
(9.26)	Theorem. Let p, q be real numbers such that Ifppi/q — 1,
and (E, G, ||. ||) an ordered Banach space with a closed cone G. Then E
satisfies Lv-conditions if and only if the ordered Banach dual space E'
satisfies Lq-conditions.
It should bo noted that an ordered Banach (E,G, ||.||) is an
AL^-space if it is a Banach lattice and satisfies the IP-conditions. Notice
that in an AZ^-space E, the following equalities hold:
IMP - II И IP - IMP+IMP (xeEf
where |ж| — x V —x, xь = x V 0 and x~~ = —x V 0.
(9.27)	Theorem. Letp, q be real numbers such that 1/ppi/q ~ 1, and
(E, G, ||. ||) an ordered Banach space with closed cone G. Then E is an
AlP-space if and only if E' is an AL^-space.
Proof. If E is an Л/Аярасе, then E' is a Banach lattice (by theorem
(6.12)), and satisfies A/Aconditions (by the preceding theorem); hence
E' is an AIAspace. Conversely, suppose that E' is an AZff-space. Then
E" is an AZp-space and E satisfies AZp-conditions by theorem (9.26).
Further, by corollary (9.21), (E, G) has the Riesz decomposition
property, and the norm ||. || on E is a Riesz norm. Thus, to show that E
is an AZAspace, it remains to show that E is a vector lattice. We prove
this by a similar argument as that given in corollary (9.23). Let x be in
E. For each positive integer n, there exists xn e E with ||жп|| < ||ж|| +-
n
such that < xn. Suppose yn = ^(xnpx) and zn = |(rrw— x). Then
112
ORDERED NORMED VECTOR SPACES
yn, zn> 0 and x yn—zn- Since E satisfies Lp~conditions, it follows that
ШР+1ЮР < ll^+^ll1’ = IKF < (|И+~) •	(9-6)
On the other hand, let </> denote the canonical embedding of E into E"
and let <У>(ж)+ — ф{х) V 0 and </>(#)~ = •— ф(х) V 0 in the vector lattice
E". Since E" is an ATA-space, we have
W)^+ll^rF - IIW - IFF-
Notice also that ф{уп} > </>F)+ and ф(%п) > ф(х)~~. Hence
Ш11 = WWII" = ШУп')--'Кх')+ +'Яж)+11‘’
Я Yi fl
IKII’ = \\ФЮ\\” -= |1ЖЬ<И®)“+ЖН’
so it follows from formula (9.6) that
(м+;У > llAll’+kV > Н(л)-^^)+11’+|1^Ю--^М-Г+||«!1’.
Passing to the limit as n -> go, we see that ф(уп) -> ф(х)+ and
f(zn) $(%)". Since E is complete, it follows that ф(х)+ e ф(Е) and
</>(ж)_ g ф(Е). Therefore ф(Е) is a vector sublattice of AF and so E is a
vector lattice. Consequently E is an HZAspace.
Remark. As in theorem (9.26), the proof of the necessity part does
not require the assumption of the completeness of E and G.
(9.28)	Theorem. Let (E,G} ||.||) be an ordered Banach space with
closed cone G. Suppose that the norm ||. || is a, Riesz norm on (E, G) and
that (E, G, jj. II) satisfies ALP-conditions (1 < p < go). Then E has the
Riesz decomposition property if and only if it is an ALp-space.
Proof. Let q be the (extended) real number such that lfp-\Alq = 1
(if p — 1 then q — co). We only have to show the necessity part.
Accordingly, suppose that E has the Riesz decomposition property and
satisfies the assumptions of the theorem. Then the ordered Banach dual
space E' is a Banach lattice (by corollary (9.21)) and satisfies the
ALq-conditions (by theorems (9.26) and (9.6)); hence E' is an AIAspace.
Brom the preceding theorem (and corollary (9.23)), we conclude that E
is an A-ZA-space.
10
ELEMENTARY THEORY OK RIESZ SPACES
We recall that an ordered vector space (X, O') with a proper cone G is
called a Riesz space (or vector lattice} if each pair of elements x, у of X
has a least upper bound, written x V у (or sup(x, y)), in X. Equivalently,
(X, (7) is a Riesz space if and only if each pair of elements x, у of X
has a greatest lower bound, written xKy (or inf(rr, y}), in X. Indeed,
x V у — —(—-ж Л —у} if one of them exists.
From now on (X, 0} (or simply X} will denote a Riesz space with the
positive cone C. It is clear that each Riesz space must be a weakly
Riesz space. The following example shows that there are weakly Riesz
spaces which are not Riesz spaces. Consider R2 with the cone defined by
Cw — {(aq, ж2) g R2:xr > 0, x2 > 0} U {(0, 0)}.
Then (R2, Gw) is a weakly Riesz space but it is not a Riesz space. For
further examples, see Fuchs (1966).
(10.1)	Proposition. Let (X, O') be a Riesz space. Then the following
statements hold:
(a)	(x V y)-fz ~ (x-j-z) V (yT-z) and (x A y)-fz — (ж+2) A (y-f-z);
(b)	if u, v, w are in G then A w < (wA w)-ffv A w);
(c)	if x A z == у A z = 0 then (x-f-y) Az = 0;
(d)	if и ~ x-f-y — z-\-w then и — (x V z}-f-(y A w);
(e)	x-f-y = x V y-f-x A y.
Proof. It is straightforward to verify (a); (c) follows easily from (b)
since x, y, z are certainly in (7; (e) follows from (d) (by putting z ~ у and
w = x). To prove (b), let z ~ (ufi«) A w. Then 0 < z < u-j-v, we have
by the Riesz decomposition property that z = x-f-y, where 0 < x < u,
0 < у < и. On the other hand, since x < z < w, у < z < w, it follows
that x < и A w and у < и A w; these imply that
(u 4~^) A w = z < и A w-\-v A w.
Finally, note that the statement (d) is equivalent to
y fy A w) ~ (ж V z) — x.	(10.1)
114
ELEMENTARY THEORY OF RIESZ SPACES
Notice also that
У~{У Л w) - y-\-( --y v -w) = 0 v {y-w},
x M z~~x = (ж—ж) V (z—x) — О V (z — x).
By hypothesis y—w = z — -x, thus equality (10,1) is clear.
(10.2)	Proposition. A Riesz space (X, (7) is distributive, that is:
(a)	if аир{жй: a g 1} exists in X then
у A sup{aq: a e 1} = sup{y A xa: a g I},
(b)	if inf{aq: a G 1} exists in X then
у V inf{xa: a g 1} = inf{y V xa: <x e I}.
Proof. Let x = sup{«a: a g I}. Then it is obvious that у A x > у A xa
for all a g I. Suppose that z > у A xa for all a ef. To prove (a), we have
to show that z у A x. In view of proposition (10.l)(e), it is equivalent
to verify that z > xAy~{y У x). Now, for each a g I, we have
%Х(У V x) > z + (y V xa) > у Л x^Py V x, ^ yPx,,
Consequently,	. . „ .	.
zpfy V ж) >
as required. The proof of (b) is similar and will be omitted.
If (X, (7) is a Riesz space and if x is in X, we define
x+ = x V 0 x~ — ( —x) V 0 and |a?| = x V (—ж).
x+ and x~~ are called the positive part and the negative part, respectively,
of the element x, while |ж| is referred to as the absolute value of ж. It is
easily seen from proposition (10.1)(e) that x = x+—x~. Two elements
x and у of X are said to be disjoint, written x I y, if |ж] A \y\ = 0. Bor
any subset В of X, we write
Ba = {ж g X:x _L 6 for all b in B}.
(10.3)	Proposition. Elements of a Riesz space X satisfy the following
properties:
(a)	x+ X x~;
(b)	|ж| = ж+Н-ж-" = ж+ V x~~;
ELEMENTARY THEORY OF RIESZ SPACES
115
(c)	x — x is the unique representation of x as a difference of two
disjoint positive elements;
(d)	(ж+у)'1' < x+Py+and (ж +«/)"” < x~Py~;
(e)	\xPy\ < И+Ы;
(f)	[ж—y\ = ж V у — x Л у = \х V z—y V z| +|ж Л z—y Л z|;
(g)	|ж+—у+\ < |ж — 2/| and	< 1Ж~“У\'>
(h)	х ± у if and only if |ж| + |?/J = | |ж| — \y| |;
(i)	|ж] < и if and only if —u < x < u.
Proof, (a) Observe that
X+ A %~—X~ — (ж+ — ж~) A 0 = X A 0 — —X~,
it then follows that ж+ J ж~.
(b)	In view of proposition (10.1)(a) and what we have just proved,
we have
ж+4-ж~' = x+ V x~ = x V 0 v ((—ж) V 0) ~ ж V (—ж) = |ж|.
(c)	Suppose that ж -= и— w, where и, w > 0 and и | w. It is
required to show that и x+ and го = ж~. Notice that
0 < u--x+ < u, 0 < w—xr < w,	и— ж'н — w—x~,
it follows from и | w that w —-x~ = u—x+ — (и— ж+) A (w~~x~) — 0,
and hence that и = ж+, w = ж~.
(d)	Clearly 0, xPy < ж+-р-у+; hence (ж-|-^)+ < x+PyP Similarly we
can show that (ж+«/)“’ < x~ d y~*
(e)	In view of (b) and (d) of this proposition, we have
|ж+?/| = (ж +«/)++(жРу)- < х+Ру+рх-ру- = |ж] + |«/|.
(f)	Observe, for any ж, у in X, that
[ж-2/l = (ж- 2/)++(ж-2/)“ = (p-У) V 0 — (х-~у) л 0
.-= Ж V у-у-(р А у--у) = Ж V у-х л у,
we then have, by proposition (10.I)(e), that
]ж V z—y v +|ж A z—y A z| = (ж V z) V (y V z) — (x V z) A (y V z)
4~(ж A z) V (y A z) -—(ж A z) A (ij A z)
= X V у V Z — (x Л у) V zp(x V y) \ Z—X \y A Z
— (Ж V yPz)-~(x A y pz) X У у —Ж Л у = |Ж— у\.
116 ELEMENTARY THEORY OF RIESZ SPACES
(g)	In view of (f) of this proposition, we have
|ж+—?/+| < |ж+—-?/+[+	л 0—y A 0| = \%—y\
\x~-yA < («+-y+\ +| -x~-Vy~\ = \x-y\.
(h)	By proposition (10.I)(c) and (f) of this proposition, we see that
kl+lz/l = и v |г/|+к| л |^| = |	|+2 И л [у\.
Recall that х _£ у if and only if	A |?/| — 0, thus this is the case if and
only lf	kl+l«d = | И-|^l I-
Finally, it is trivial to verify (i).
We recall that a set 8 in an ordered vector space E is solid if and only
if 8 = U{[~~u, u]:0 < ue8}. In terms of lattice structure, we are
able to give some characterization of solid sets as follows.
(10.4)	Proposition. A set 8 in a Riesz space X is solid if and only
if it satisfies the following property:
|ж| < \y\ with yeS=>xe/S.	(10.2)
Proof. If 8 is solid and if |ж| < \y\ with у e 8, then there exists
0 < и g8 such that —и < у < и and so, by proposition (10.3)(i),
|t/| < u. Observe that ]ж| < u's apply proposition (10.3)(i) again,
—u < x < u, consequently xeS. Conversely suppose statement (10.2)
holds. Then \x\ g 8 whenever x g 8; it follows from — |ж| < x < |a;] that
8 c U{[-и, ад]:0 < и eS}. On the other hand, if —и < x < и for
some 0 < и e 8 then, by proposition (10.3)(i), |ж| < и, and hence
x g 8. Therefore 8 ~ U{[—и, и]: 0 < и g $}, and the proof is complete.
It is easily seen that the intersection of a family of solid sets in a
Riesz space X is either empty or solid, and that the union of a family
of solid sets in X is solid. If В is a subset of X, the smallest solid set
containing B, written 8B, is called the solid hull of B. It is clear that
SB = и{[ЧЬИЬП:Ъ g B}.
The solid hull of an element x in X will be denoted by 8X; therefore
8X = [—\x\, |ж|]. If В is a subset of X, the set defined by
sk(B)	c B}
is called the solid kernel of B. It is clear that sk(B) is either empty or the
largest solid subset of X contained in B, and that
sk(B) = U{[ —u, u]: и, и] £ B}.
ELEMENTARY THEORY ON RIESZ SPACES
117
We recall that a set В in X is absolute-order-convex if and only if
8(B) В and absolutely dominated if В <= 8(B), where
$(B) — u{[—w, w'|:0 < и e B}.
Therefore, if В is absolute-order-convex then 8(B) is the solid kernel
of B, i.e. 8(B) = sk(B). If В is absolutely dominated then 8(B) is the
solid hull of B, i.e. 8(B) = 8B.
Some elementary, but useful, properties of solid sets are summarized
in the following proposition.
(10.5)	Proposition. Let (X, C) be a Riesz space, and let V be a
subset of X. Then the following statements hold:
(a)	the convex hull of each solid set in X is solid;
(b)	if V is convex then so is sk(F);
(c)	if sk(F) is non-empty then sk(F) is absorbing if and only ifsk( F)
absorbs every order-bounded subset of X;
(d)	F absorbs every order-bounded set in X if and only if sk(F)
absorbs all order-bounded subsets of X;
(e)	if A, В are solid subsets of X then A -\-B and AB are solid for any
real number X.
Proof. (&>) It is clear that 8kx ASX for any real number A and
x e X. We now claim that 8x+y SXA-8V. If z e 8x+y, then
Fl < k+g/i < ki+|y|
and so 0 < (г + |ж|+ |г/|)/2 < |ж| + Ы- By fb.e Riesz decomposition
property, there exist and w2 with 0 <	< |ж|, 0 < w2 < |?/| such
that	z + И + Ы = 2w1-]-2w2.
Take и = 2wx —|ж|, v = 2w2~-\y\; then и e 8x, v е8у and z = ири;
this implies that 8хЛ.у 8xp8u.
If A is a solid subset of X and if x e co A, there exist ai e A and
n	n
At e [0, 1] with 2	= I such that x = 2 ^iat- We now conclude, from
i=l
8X c V L8a. £ У A{A c co A,
2=1	i = l
that co A is solid.
(b)	Notice that sk(F) is the largest solid subset of X contained
in F, and that co(sk(F)) is solid. We conclude from co(sk(F)) cz F that
sk(F) — co(sk(F)), and hence that sk( F) is convex.
9
118
ELEMENTARY THEORY OF RIESZ SPACES
(c)	Since sk(F) is solid, sk(F) absorbs и e C if and only if it absorbs
[—и, и].
(d)	The condition is clearly sufficient. To prove its necessity, it is
sufficient to show that sk(F) absorbs each order-interval of the form
[—it, u] where и e G. Since V absorbs [—it, u], there exists A > 0 such
that [—и, u] 27, and so [—u, u] sk(AF) = Л sk(F).
(e)	It is clear that Л В is solid. It remains to verify that A /-В is solid.
Let x g A, у e B, and let z, in X, be such that |^| < |ж-Ь«/|. It is known
from the proof of (a) that 8XH, £	hence there exist a g 8x and
b e8v such that z = a 4-6. Since A and В are solid, it follows that
8X c: A, 8V B, and hence that z eA-\B. Therefore A-\-B is solid,
and the proof of this proposition is complete.
It should be noted that the solid hull of a convex set in a Riesz space
is, in general, not convex. By way of example, consider R2 with a
positive cone G defined by
О - {(ж, у):ж > 0, у > 0},
then (R3, G) is a Riesz space. Suppose that
В = {A(-2, 0)4(1 — A)(l, 3):A g[0, 1]}.
Then В is convex, but the solid hull 8B of В is not convex because
{A(—2, 0)4~(l —2)( —1, 3):2e [0, 1]} is not contained in. 8B.
A vector subspace 8 of a Riesz space (X, C) is called a Riesz subspace
if x+ e 8 whenever x e 8. Solid subspaces of X are referred to as lattice-
ideals (or simply ^-ideals). An /-ideal В of X is called a normal subspace
of X if it follows from xT 'f x in X with xT in В for all т that x belongs
to B. It is clear that the intersection of a family of normal subspaces
of X is a normal subspace. An /-ideal В of X is called a a-normal
subspace of X if it follows from xn 'f x in X with xn in В for all n that x
belongs to B. The intersection of a family of cr-normal subspaces of
X is a cr-normal subspace, of /-ideals in X is an /-ideal and of
Riesz subspaces of X is a Riesz subspace. If В is a subset of X,
the smallest /-ideal containing В is called the /-ideal generated by B.
If В is a subset of X, the smallest Riesz subspace containing В is called
the Riesz subspace generated by B. It is clear that if В is a Riesz subspace
of X, then the order-convex hull [ B] of В is the /-ideal in X generated
by B. If 8 is an /Ideal in X, the smallest normal subspace containing
B, written {B}, is called the normal subspace generated by 8. If 8 is an
/-ideal in X, the smallest ^-normal subspace containing 8, written
ELEMENTARY THEORY OF RIESZ SPACES 119
{$}ff, is called the o-normal subspace generated by 8. For instance, if A
is a subset of (X, G), then Ad is a normal subspace of X. If $ and В are
/-ideals in X, 8 is said to be order-dense in В if В c. {&}; 8 is o-order-
dense if В {$}a. In particular, 8 is said to be order-dense if X = {$}
and o-order-dense if X = {/?}„.
(10.6)	Proposition. Let (X, C) be a Riesz space, 8 an Aideal in X,
and suppose that и gC, Then the following statements hold:
(a)	и g {&} if and only if there exists a positive increasing net {ur}
in 8 such that uT f u.
(b)	и g {$}a if and only if there exists a positive increasing sequence
{un} in 8 such that un } u.
Proof. The proof of (b) is similar to that of (a) and will be omitted.
For the proof of (a), it is clear that if uT f и with ur e 8 then и e {8}.
Conversely, suppose that
В ~ {и e C: there exists an increasing net ur in 8 Ci G such that ur f u}
and that	g _ g _
Then 8	8 ez {&} and
В = {иeC: u sup {w e8: 0 < w < ад}} (10.3)
because 8 is an /-ideal in X. Notice that U{[0, u]:u е- В} с B; it then
follows that	n r 4..
В	g o}.
We complete the proof by showing that 8 is a normal subspace of X.
If щ , ад2 are in B, then u1pu2 is an upper bound of the set
{w g$:0 < w < ux-\-u2}.
On the other hand, if v is any upper bound of the set
{w g 8:0 < w < адх +ад2};
then v > Wjl+Wj, for any w2 e 8 with 0 < w± < ux, 0 < wz < ад2-
According to formula (10.3), there are
u± = sup{m g 8:0 < m < u±} and ада = sup{n c 8:0 < n < u2},
it follows that и^-щ < v, and hence that
^i+M2 = sup{w g£:0 < w < ад1Тад2}-
120 ELEMENTARY THEORY OF RIESZ SPACES
This shows that B-\-B <= B. Clearly IB £ В for each non-negative I
and it follows that S is a vector subspace of X. Furthermore, S is an
/-ideal. If у e 8 and if x g X is such that |ж| < \y\, there exist u15 u2 e В
such that у = щ — u2. Then иг-^и2 e B, and so x+ e В by the con-
struction of В and 0 < x+ < Mj+ug. Similarly, x~~ e B. Therefore
x	~ 8; this shows that 8 is an /-ideal. We now claim
that $ is a normal subspace of X. Suppose that 0 < uT f и in X, where
uT g B. Then
и = sup uT supr sup{«; g 8: 0 < v < uT}
= sup{r g 8: 0 < v < uT for some t).
Note that for a fixed r, the set {s g 8: 0 < s < uT} is directed upwards;
therefore и e В and a fortiori и g 8. This shows that 8 is a normal
subspace of X, and the proof is complete.
(10.7)	Proposition. For any set A in a Riesz space X, the set Ad
is always a normal subspace of X. Consequently {8} 8dd whenever
8 c X.
Proof. It is easy to verify the first assertion. To see the second, we
note that 8dd is a normal subspace containing 8; hence 8dd contains the
normal subspace {8} generated by 8.
It should, be noted that {8} and 8dd are, in general, not equal, as
shown by the following example: consider R2 equipped with the
lexicographic ordering. Then R2 is a Riesz space which is not
Archimedean since 0 < n(0, 1) < (1, 0) for all n; the only /-ideals in
R2 are {0}, the «/-axis, and R2 itself; all of these /-ideals are normal
subspaces of R2. Now if we take 8 to be the «/-axis, i.e.,
S = {(0,
then 8 = {8}, 8d = {0}, and 8dd = R2; it then follows that 8 8dd.
(10.8)	Lemma. Let Lr, L2 be subspaces of a Riesz space (X, C) such
that Lx Ci L2 = {0}, and let L be the algebraic direct sum of Lr and L2.
If Lx and L2 are Cideals in X, then L is the ordered direct sum of Lx and
L2 (i.e. if xx g Lx and ж2 g L2, then xl-f-x2 > 0 if and only if xx > 0 and
x2 > 0), denoted by L — Lt®L2.
ELEMENTARY THEORY OF RIESZ SPACES
121
Proof. Suppose that xi g Li (i = 1, 2). It is clear that	> 0
whenever xt > 0 (i == 1, 2). Conversely, if Px2 > 0 then
0 < Xi+xf <	xf, xf e Llt and x2, W g L2
because L± and L2 are /-ideals in X; in particular,
0 < xf < xfpxf.
By the Riesz decomposition property of X, there exist w1} w2 with.
0 <	< xf (i = I, 2) such that xf = w-P~w2; therefore wi gL{
(i — 1, 2) since are /-ideals in X. It now follows from
w2 = .'T —wt g L±
and from L{ Г\ L2 — {0} that w2 ~ 0, and hence
Xt = X'f--Xj' = Xi --W} > 0.
Similarly we can show that x2 > 0. Therefore L is the ordered direct
sum of and L2.
(10.9)	Proposition. Let (X, (7) be a Riesz space. Then the following
statements are equivalent:
(a)	(X, C) is Archimedean;
(b)	for any Aideal A in X, if Md = {0} then A is order-dense;
(c)	for any Aideal A in X, M©Md is order-dense;
(d)	for any normal subspace В of X, we have В — Bdd;
(e)	for any Aideal A in X, we have {Л} yldd.
Proof, (a) => (b): Let A be an /-ideal in X such that Ad — {0}. In
view of proposition (10.6), we have to show, for each и e C, that
и -= sup{t> e A: 0 < v < u}.
Let Au -~= {v g A: 0 < v < u}. Then и is certainly an upper bound of
Au. Let w be another upper bound of Au such that w X u. We have to
show that и < w. To do this, let w' = и A w. Then w' is again an upper
bound of Au. We claim that w' = и (so и < w). In fact, if w' p u, then
let z = u — w' > 0. Since AA — {0}, z f Ad so there exists a0 e A such
that z Л |a0| X 0- bet z' = z A |a0|. Since z' -= z A |a0| < ]a0| e A and
A is an /-ideal, we have z' e A; also z' < z < u, it follows that z' g Au.
Moreover, since z'Aw' < z-\-w' = и and w' is an upper bound of Au, we
have z' + c Au and a fortiori 2z' e Au. Inductively we have nz' e Au, i.e.
nzr < и for all n; but z 0, contrary to (a). This proves the implication
(a) => (b).
122
ELEMENTARY THEORY OF RIESZ SPACES
(b)	=> (c): Observe first that if A is an /-ideal then so is the direct
sum	Farther, (A®Ad)d = Ad П Add — {0}; hence A(i)Ad is
order-dense by (b).
(c)	=> (d): By proposition (10.7), В = {B} c Rdd. Since both sets are
positively generated, to see the opposite inclusion it is sufficient to show
that С H Bdd cr B. Let и g C C\ BdA, ^Nq have to show that и E B. By
(c), B@Bd is order-dense; hence there exists a net uT > 0 in B®Bd such
that ur | u. For each r, suppose uT — хт-\-Уг, where 0 < хт e В and
0 < yr g Bd. Notice that
0 < yT < uT < и e Bdd,
hence yT e Bdd. But we also have yT e Bd\ therefore yT = 0, valid for all
r. Thus xT — u7 f и and хт e В; it follows that и e В because В is a
normal subspace of X.
(d)	=> (e): We start by observing that Ad = {A}d. Next if В = {A}
then В is a normal subspace of X; hence, by (d), we have
{А} - В - Bdd = {A}dd - Add.
(e)	(a): Suppose that X is non-Archimedean. Then there exist
ад, v in C such that 0 nv < и for all n. Let Xv LH ~~v> Then Xv
is an /-ideal in X. We shall show that {Xv©Xd} is not (XiJ@Xd)dd. We
verify this by showing that и ф {ХуфХ1} because of
(X.@Xd)dd = (0/ - X.
Let	Bu = {w g X„®Xd:0 < w < u}.
Then (u— v) is an upper bound of Bu; in fact, let w e Bu. Then w has a
unique decomposition of the form w = wq-Hfq where g Xv and
w2 g Xd. Since Xv is an /-ideal, we have that wx-Av e X„, and hence that
w-Av = (Wi+v)+w2	Further, since Xv I Xd,
(Wi-Av) A w2 = 0
80	wAv = (w1-|-'y)-]-w2 = (w1X'v) V w2 < u.
This shows that (u—v) dominates every element w in Bu. Therefore
{u— v) is an upper bound of Bu and и is not the supremum of Bu. In
view of the proof of proposition (10.6), we conclude that и ф {X^©Xd}.
If X is an order-complete Riesz space and A. a subset of X, then Ad
and Add are disjoint normal subspaces of X, and hence the algebraic
direct sum of Ad and Add is the ordered direct sum, and Ad©Add is a
subspace of X. In fact, X = Ad©Add, as the following result, due to
Riesz (1940) shows.
ELEMENTARY THEORY OK RIESZ SPACES
123
(10.10)	Proposition. Let (X, (J) be an order-complete Riesz space,
and let A be a subset of X. Then X is the ordered direct sum of normal
subspaces Ad and Add of X, that is, X = Ad®Add.
Proof. Given a positive element и in X, we show that и — 7/t+<M.2,
where g Ad, u2 g Add, and иг > 0, u2 > 0. Define u± by
иг = snp{?z A |«| :x g Ad}.
Since и is an upper bound of the set {и А |ж|: x g Ad}, it follows from the
order completeness of X that ur exists, and hence that ut g Ad because
Ad is always a normal subspace of X. Notice that ut is the largest
element in Ad which is majorized by u. It is clear that 0 < иг < и. Let
u2 — u—u1; then u2 > 0. We further show that u2 e Add. To sec this, let
x e Ad, and let z = (u—uf) A |x|. Since Ad is an /-ideal, z eAd, thus
g Ad. Notice also that the element of Ad is dominated by u;
hence z-\-ux < and z < 0. Since it is obvious that z > 0, we must
have z — 0 and z u2 л [ж| = 0. This shows that u2 e Add, and hence
that и = utpu2 e _z4d©.4dd. Therefore X — zld®Add.
(10.11)	Corollary. Let (X, C) be an order-complete Riesz space. Tor
any normal subspace В of X, we have X. В ®Bd.
Proof. We remark that, since X is order-complete, the ordering in
X must be Archimedean. Thus the result follows from propositions
(10.9) and (10.10).
Let (X, C) be an order-complete Riesz space, and let В a normal
subspace of X. Define PB by
Р/?(ж) = sup{zH’ л |y| :y e B}--sup {x~ A |?/] :y e B}.
According to proposition (10.10), PB(x) e B, and so PB is a linear
transformation of X onto B. This Pf} is referred to as an ^-projection
on B.
(10,12)	Theorem. Let (E, CJ) be a weakly Riesz space. Then (Xb, (7*)
is an order-complete Riesz space, and the following statements hold:
(a)	for any и e C and f e Xb, we have.
f+(u) ~ sup{/(a?): 0 < x < u},
f'~\u) ~ sup{/(x): —u < x < 0},
|/|(w) = sup{/(«): — и < x < и} ~ sup{|/(ic)|: — и < x < u},
1Ж1 < 1/1 (1«И) whenever у gE,
124 ELEMENTARY THEORY OF RIESZ SPACES
(b)	if {fT:r e .D} is a majorized increasing net in Eb, then
h = sup{/r: r g 1)}
exists in Eb, where h(u) sup{/T(w): r g J)} for any и in C.
Proof, (a) We have shown in theorem (1.10) that (^ь, C*) is a Riesz
space and
/+(u) = g(u) sup{/(#):0 < x < u} (u g G).
Therefore, in view off~ = (—f)^, we obtain
f~fu) = (-f)+(u) = sup{ -f(y) ;0 < у < и}
sup{/( - y): -u < --y < 0}
= sup{/(x): — и < x < 0}.
Since \f\ — 2/+—/then
IЛИ = 2ЛИ"/И 2 sup{/(<r):0 < x < гОНН
= sup{/(2a?—u): 0 < x < w}
= sup{/(?/): -u < у < it}
= sup{|/(?/)|: ~~u < у < и},
in particular,
1/КЫ) = snp{|/(w)|: — |?И < w < Ы} > L/W
(b) Given и g C, we define
h(u) = sup{/r(w): r g D}.
Then h is positively homogeneous and additive on C because/rf. Since
G is generating, h can be extended uniquely to a linear functional h on
E. Notice that, since {/J is majorized in E\ h eEb. It is clear that
fT<h for all r g D. On the other hand, if g g Eb is such that/r < g for
all 7 g D, then h(u) — sup{/r(w): т g D} < g(u) for all и g C, and so
h = sup{fr: т g D}. Therefore (Eb, (7*) is an order-complete Riesz space.
Remark. If/, g are in Еъ we have, by making use of the preceding
result, that
(a)	(/v g){u) = sup{/(x) -\-g(u—x"): 0 < x < u}
=~ sup{/(v) -Gg(w) \v, w > 0 and и — vfi-w};
(b)	(/л g)(u) = inf{/(x)+(?(%— ж):0 < x < u}
= mi'{f(v)Pg(w') :v, w > 0 and и — vpw} (ueC);
ELEMENTARY THEORY OF RIESZ SPACES
125
(c)	for any и e C, [— u, nf	{f GXb:Jff(w) < 0, where [— u, uf is
the polar of the set u, u] taken in E\
Recall that a semi-norm p on an ordered vector space (E, 0} is a
Riesz semi-norm if it satisfies the following two conditions:
(i) p is absolute-monotone, i.e. —- и < x < и in E =-> p(x) < p(u);
(ii) for each x e E with p(x) < 1 there exists и e E with p(u) < 1
such that —u < x < u.
In view of lemma (9.16), if (X, G) is a Riesz space, then a semi-norm p
on X is a Riesz semi-norm if and only if it follows from |ж| < |y| with
x, у in X that p(x) < p(y).
(1.0.13) Theorem (Luxemburg-Zaanen). Let (X, G) be a Riesz space,
p a monotone semi-norm on X, and suppose that Y is a Riesz subspace of
X. Iff is a positive linear functional on У which is dominated by p on Y,
then there exists a positive and linear extension ф of f such that
|^(^)| < р(И) for any x g X. Furthermore, if p is a Riesz semi-norm
then ф is dominated by p.
Proof. Suppose that
q(x) = p(x]~) (zeX).
It is easily seen that g is sublinear on X and that
Ш < f(y+) < Р(У+) = Ш for all у g У.
In view of the Hahn-Banach extension theorem, there exists ф e X*
such that ф(у) = f(y) for all у g Y and ф(х} < q(x) for all x eX. For
any и g G, we have
ф(-и) < qf—u) ^pf-u)+) ^p(O)	0,
and so ф g G*. On the other hand, by theorem (10.12)(a), we have
1Ж)1 < \ф\(|ж|) - <£(|ж|) < q(|ж|) - ^(|ж|) for all х е X.
In particular, if p is a Riesz semi-norm then
|#r)| < p([x\) ^p(x),
and so ф is dominated by p. This completes the proof.
Before giving a dual result of theorem (10.12)(a), we need the follow-
ing lemma.
(10.14)	Lemma. Let (X, C) be a Riesz space, and let f g (7*. For any
и g 0, there exists g g C* such that 0 < g < f, g(u) = f(u), and g(x) 0,
whenever x и.
126
ELEMENTARY THEORY OE RIESZ SPACES
Proof. For any w e C, define
h(w) = sup/(w A nu).
n
Then h is positively homogeneous on C, 0 < h(w) < f(w) for all го g C,
and h(u) = /(u); further, h is additive on C. In fact, if w3 and w2 are in
C\ by proposition (10.1)(b), we have
(wxd-w2) A nu < WT A nu-PWz A nu
and so	7/ i \ у/ \ \ t./	\
On the other hand, it is clear that
го1 A nu-l-w2 A mu < (wi+w2) A (n-\-m)u,
and thus	7,	4,7/	x^.7./	।	\
H-/z(?u2) < h(w1-\~wz).
There exists a linear functional g on X such that g(w) ~ h(w) for all
w e C; in particular,
g(u) = h(u) = f(u),	0 < g(w) A(w) < f(w) for all w e C.
Finally, if |m| Л и ~ 0, then |«] A nu = 0 for all n, consequently
0 < |^(ж)| < £/(|ж|) — sup/d^j A nu) = 0.
Now we easily deduce a dual result of theorem (10.12)(a) as follows,
(10.15)	Proposition. Let (X, C) be a Riesz space. For any f gC*
and x e X, the following equalities hold:
(a)	f(x+) - sup{^(;r):0 < g < /};
(b)	f(x~) = sup{A(ti): -/ < h < 0};
(c)	f(\x\) - 8пр{0(ж): -f < g <f) = 8ир{|д(ж)|: -f < g < f}.
Proof. It is clear that (b) and (c) follow from (a), and hence we only
have to show the assertion (a). For any g e Eb with 0 < g < f, the
following inequalities hold:
g(x) < </(z+) < f(x+),
hence sup{</(a;):0 < g < f} < f(x+). Apply lemma (10.14) to obtain
h e C* such that 0 < h < f, h(x+) ^=f(x+), and h(y) = 0, whenever
У I ad-. Therefore
f(x+) = h(x+) = h(x+)—h(x^) = h(x),
ELEMENTARY THEORY OF RIESZ SPACES
127
consequently /(ж+) = supW): 0 < , < д
This completes the proof of this proposition.
Remark. For any f gO*, [—f,f]Q {x eX‘.f(\x\) < 1}, where, of
course, the polar is taken in X.
The following result is a consequence of theorem (10.12)(a) and
proposition (10.15).
(10.16)	Proposition. Let (X, C) be a Riesz space and let Y be a
Riesz subspace of (Xb, (7*). Then the following statements hold:
(a)	if A. is a solid subset of X then the polar Arr( Y) of A, taken in Y,
is a solid subset of Y;
(b)	if В is a solid subset of Xb then the polar Bn of B, taken in X, is a
solid subset of X.
Proof. Let g e Л1Г(У), and let f g Y be such that [f\ < |</|. For any
a g A, we have, by theorem (10.I2)(a), that
/(«) < l/KN) < |0|(И = sup{f/(a'): |m[ < |a|}.
Since A is a solid subset of X and since g eAk(Y), it follows that
|gr|(|a|) < 1, and hence that/(u) < 1. Therefore A”(Y) is a solid subset
of Y. This proves the assertion (a). The proof of (b) is similar, by making
use of proposition (10.15).
Let (X, C) be a Riesz space with the order-bound dual Xb, and let ф
be in Xb. ф is called a normal integral (or order-continuous) on X if for
any net {ur} with uT J, 0, then inf |<4(ur)| = 0; ф is called an integral (or
order- a-continuous) if, for any sequence {ади} with un | 0, then
inf = 0. The set of all normal integrals on X is denoted by Xb
and the set of all integrals on X is denoted by Xb. It is clear that
Xb c Xb
(10.17)	Proposition. Let (X, C) be a Riesz space. Then Xb and Zb
are normal subspaces of Xb.
Proof. It is clear that Xb is a vector subspace of and that if
0 < g < h g Xb then g g Xb. In order to show that Xb is an /-ideal, it
suffices to verify that if f g Xb then f+ g Xb. Suppose un J, 0. For any
w in X with 0 < w < zq, we have, by proposition (10.1 )(a), that
0 < w—w Л un < u1—un
and w A un | 0.
128
ELEMENTARY THEORY OF RIESZ SPACES
It follows from/(w — w A un) < f+(w—w A un) < f+tp^—uf) that
0 < f+M < /'(fo)	+/(w A un) for all n,
and hence that
0 < inf/1 (un) < f+(uf) —f(w) for all w c [0, w3].
By theorem (10.12)(a), inf J1-(mJ — 0, and so/+ g Xb.
Now suppose that {/J is a positive increasing net in Xb guch. that
fT t/ in Xb, and that un j, 0 in X. By theorem (I0.12)(b),
7>i) sup//mJ.
For any e > 0, there exists t0 such that 0 < f(ul)—fT (uf) < e. Since
un | 0 and since/—/Tq > 0, it follows that
0 < fM -fTo(un) < /(uj --ДЫ < e for all n,
andhencethatlim[/(Mj—/./mJ] < e. Notice that 0 < Д eXb and that
lim/jMj inf/jwj - 0.
n	n
Therefore m£f(un) = lim/(uj < e. Since e is arbitrary, this implies
n	n
that / g Xb, and so Xb is a normal subspace of Xb. This proves the
result for Xb, and the proof for Xb is similar.
Let (X, C) be a Riesz space. For any / e Xb, we define
^ = {«еХ:|/|(И)=0}.
It is easily seen that Nf is the largest /-ideal in X contained in the kernel
of/, and that if/is a normal integral thenN/ is a normal subspace of X.
Furthermore, we also have that N.f ~N,n — Ж+ П Nf-, and that
Nf = 3ktf-W
The following two results, which will be needed later, are of some
interest in themselves.
(10.18)	Proposition. Let (X, C) be a Riesz space, h e Xb, and
let феХьв. If h ±ф then N% <= Хф,
Proof. Without loss of generality one can assume that h and / are
positive. Notice that the statement that N& cz Хф is equivalent to the
statement that if 0 < w e then w e Хф because and Хф are
/-ideals in X. We now suppose 0 < w e N&. Since ф A h = 0 and since
(ф A A)(w) = inf{ф(х) + h(y):x, у e G, xXy = w},
ELEMENTARY THEORY OF RIESZ SPACES
129
there exist positive sequences {«„} and {yn} with xn +yn = w such that
ф(хп) < 1/2” for all n 1, 2,... .
For any e > 0, if we can find positive sequences {y’n}, {y"}, and {zn} with
the following properties:
(a)	0 < y'n < zn I 0, yn Уп+У'^
(b)	ф(Уп) < z,
then </>(w) — 0, and so w g A7'^: in fact, since xn ~~ w—yn = w— y'n —y"n,
we then have
ф(ги—у'п) = ф(хпХу'п) < 2~nj^£ for all n = 1, 2,... .
Note also that w > w— y'n > w— zn f w; it follows from 0 < ф e X
that ф^—у'п) -> </>(w) as n~>oo, and hence that ф(ш) < e; conse-
quently <£(w) = 0.
We complete the proof by showing the existence of positive sequences
{y^}, {y"}, and {уи} satisfying (a) and (b). For each n > 1, let
Уп.т = sup{yK, yn+1,..., ym} (m > n).
Then yn < уП1)П < w for all n, уП1П1 are in N& for all n, m, and
Ук,т > У1.т, Уп.к < Уп.з whenever к < j, so that ф{уп,т)	< <£(w). It
follows that lim ф(уп-т) exists, and hence that there exists a sequence
Wl—>CO
{mn} of natural numbers with mn f such that
ф{Уп.J ~Ф(Уп,mn) < for all к > mn.
Let	. r,	,
- - mf{yliWh, у2)й?а,..., yn>nin}.
Then zn[, zn < yWiWji, and so zn e ; furthermore zn | 0. In fact, since
< Уп-тп < Уп ХУп+1 H~ • • • ~\~Утп1
we have that
0 < k{zn) < 2-”-p2^(w+1) + ...-|-2^mm~> 0 (as n -> oo).
Hence if v e G is a lower bound of the set {sra} then h(v} — 0, which is to
say that v e Nn. Notice that 0 < v < zn e and that is an /-ideal,
hence v g Consequently v — 0.
We now define
Уn	Уп A and yn  Уп~~Уп-
Then 0 < y'n < zn | 0, y"n > 0, and yn — УпЛ~у"п> it remains to show
that ф(у'п) < £ For each n, let
(Уг,тп Уг.т^)-
1.30
ELEMENTARY THEORY OF RIESZ SPACES
Then
ФМ I ШУг.тп)~Ф(Уг.т^ < I	&Г all П.
j-=l	i~1
Since ykm > yjtm and?;w 7[. < yn>.} whenever к < j, it follows thatvm > 0
and
Уп.тп~~Ъп < У г.mt ^ОГ йТН = 1J 2,..., П,
and hence from the definition of zn that
Уп.тп	a^
We observe that w„ < y„ m .thus
tj it	M fl'i ‘tVyi 7
yn--vn <. zn for all n.
On the other hand, since
y'n = УиУ'п = yn ~Уп A zn sup{0, yn-zn} < vn,
since ф > 0, and since ф(рп) < e for all n, then
Ф(Уп) < ф(рп) < e f°r a^ n>
and so {y'n}, {y'n}> and {zw} are the required positive sequences.
A partial converse of the preceding result is given below.
(10.19)	Corollary. Let (X, C) be an Archimedean Riesz space, and
let h, ф be in X^. Then the following statements are equivalent:
(a)	h ± ф;
(b)	N£ C N„;
(c)	<= Nh;
(d)	Si ± Nj.
Proof. Since X is Archimedean, proposition (10.9) gives {A} = Add
for any «f-ideal A in X; in particular, Nh = {Nh} = (XJdd for every
h e X^. Therefore (b), (c), and (d) are equivalent. On the other hand, the
implication (a) => (b) follows from proposition (10.18). Conversely if
< с Хф, then \ф\ A |A| eXj and (\ф\ А |Л|)(ж) = 0 for all x ENh®Nf
Since X is Archimedean, it follows from proposition (10.9) that Nh®N^
is order-dense, and hence that \ф\ A |A| — 0 on X; i.e. ф | h.
(10.20)	Proposition. Let (X, C) be an Archimedean Riesz space,
0 < h g X£, and let и eC be such that h(u) > 0. Then there exists w g X
with 0 < w < u such that h(w) > 0, and
= 0 for all ф e X^ with ф X h.
ELEMENTARY THEORY OF RIESZ SPACES
131
Proof. Since (X, C) is Archimedean, it follows from proposition
(10.9) that Nh®N% is order-dense, and hence from proposition (10.6)
и — sup{s+w:0 < s e Nh, 0 < w eNfi, spiv < u}.
Since 0 < h e X^, we have
h(u) — sup{A(s) ph(w): 0 < s e Aft, 0 < w e Nf, s-j-w < u}
= sup{A(w):0 < w < u, w eN^},
and so, by making use of the fact that h(u) > 0, there exists w e Nf with
0 < w < и such that h(w) > 0. If ф e X° is such that ф | A, then, by
proposition (10.18),	<= Хф, and so w e Nф, because wsNf. We
conclude from |</>(w)| < \ф\ (|w|) that ф(ш) = 0. This completes the
proof.
(10.21)	Corollary. Let (X, C) be an Archimedean Biesz space, and
let В be a normal subspace of X^, If 0 f e Xf\B, then there exists
го e G such that
f(w) > 0 and w e B°,
where B° is the polar of В with respect to the duality (X, X^).
Proof. Set BA = {h e X^-h | B}. Since X^ is order-complete, it
follows from corollary (10.11) that X^ = В®ВЛ, and hence that/has
the unique decomposition / -- g-ph, where 0 < g e В and 0 < h e B&.
Note that h -=f=- 0 because f ф B; hence there exists и e C such that
h(u) > 0. Since h | B, and since Xj) с X1/ we conclude from proposi-
tion (10.20) that there exists w e X with 0 < w < и such that h(w) > 0
and w e B°, and hence that f(w) = g{w)-ph(w) > 0. This completes the
proof.
Let A be a subset of a Riesz space (X, C). denotes the polar of A
taken in X£, i.e. =^fE X^.f(a) < 1 V a e A}.
A^ denotes the polar of A taken in X^, i.e.
A” - {/eX^:/(a) < IVaed}.
If A is a subspace of X, then
^n	{/ e ^:/(a) = 0 У a e A}
and	Avg ={fEXb0:f(a) -OYaeA}.
132
ELEMENTARY THEORY OF RIESZ SPACES
(10.22)	Corollary. Let (X, C) be an Archimedean Riesz space. If
В is an /-ideal in then {B} = (B0)^, where
B° = {x e X :f(x) = 0 V f e B}.
Proof. We start with the observation that (B0)^ is a normal subspace
of X^ such that В cz (B0)^; it then follows that {B} <= (B0)*. Note that
B° = {B}°. If 0 < f e Х^\{В}, then, by corollary (10.21), there exists
w e C such that
flw) > 0 and w g {B}°;
hence f f (B0)", consequently {B} = (B0)(J. This completes the proof.
(10.23)	Corollary. Let (X, C) be an Archimedean Riesz space, and
let A, В be /-ideals in X^. Then the following statements hold:
(a)	{X.} cz {B} if and only if A° B°;
(b)	{J} = {B} if and only if A° — B°.
Proof. The conclusion (b) is an immediate consequence of the
assertion (a); and the assertion (a) follows from corollary (10.22). This
completes the proof.
Let X, Y be .Riesz spaces, and let T be a linear operator of X into Y.
T is called a lattice homomorphism (or briefly an /-homomorphism) if it
preserves the lattice operations; and the /-homomorphism is called an
/-isomorphism if it is injective. It is clear that each /-homomorphism is
positive.
We list some elementary properties of /-homomorphisms as follows.
(10.24)	Proposition. Let X, Y be Riesz spaces, and let T be an
/-homomorphism of X into Y. Then the following statements hold:
(a)	the kernel of T, written k(T), is an /-ideal in X;
(b)	Tlx) > 0 if and only if there exists и in С П k(T) such that
x/m > 0;
(c)	Tlx) < T(y) if and only if there exists z e X such that x V у < z
and T(y) Tlz);
(d)	|T(a?)| < |TQ/)| if and only if there exists w ek(T) such that
\w\ < |x| and |ж—-w\ < Jg/]/
(e)	if A is a solid set in X then T(A) is a solid set in T(X);
(f)	if В is a solid s J in T(X) then so is TpB) in X.
ELEMENTARY THEORY OF RIESZ SPACES
133
Proof, (a) Straightforward.
(b)	The sufficiency is obvious. To prove its necessity, we note that
0 < T(x) = (T(x))+ = T(x+)
and that x~ — x+—x eC nk(T). Then и = x~ is the required element.
(c)	The sufficiency follows from T(x) < T(z) = T(y). For the neces-
sity, we observe that T(y — x) > 0, there exists, by (b), и e C C\ k(T)
such that y-x-G и > O.Letg = u-h^/. Then rr V у < zandT(y) = T(z).
(d)	(i) 'Necessity. We start with the observation that there exists
v e G П k('71) such that
|ж| < 13/1+^ and 0 < v < |x|.
In fact, since Т()ж|) = |Т(ж)| < |T(y)|	T(|?/|), there exists, by (b),
и e С П k(T) such that — |ж| > 0, so |ж|— \y\ < и and hence
|ж| — [y| < и A |«|. Taking v = иЛ |ж|, then	|ж]-~|y|	<	v, 0	< v	<	|гг[,
and v e G C\ k(T) because 0 < v < u.
On the other hand, since 0 < v < |ж|	=	Ъу	the	Riesz
decomposition property of X there exist	v2	g X with 0	< vr	<	<r+,
0 < v2 < x~~ such that v — Obviously v± A v2 ~ 0. Set
w ~ —v2.
We obtain, by virtue of proposition (I0.3)(h),
|w| = ]^1—V2j = KI+KI =
so w e k(T), [w| < \xSimilarly, since (ж+—Л v%) = 0, we also
have
|ж—w| = lx ^-—х~-—у1-У-м21 =	-г>х)= |ж[— v,
which implies that \x—w\ < |?/|.
(ii) Sufficiency. Since ги ек(Т) and since
||ж— w\ — |ж[| < |ж— W—xl = |w|,
it follows that [x— w| — |ж| g k(T). The sufficiency now follows from the
following computation.
1ЖН T(\x\) = T(\x^w\) < T(\y\) = \T(y)\.
(e)	Let z e T(X) be such that \z\ < |T(a)| for some a e A. There
exists x e X such that z = T(x). By (d), there exists w such that
|w| < |x| and \x — w\ c |aSince A is solid, it follows that x—w e A and
io
.134
ELEMENTARY THEORY OF RIESZ SPACES
hence that T(x—w) e T(A). We conclude from T(x) = T(x —w) that
T(x) e T(A). Therefore TtA.') is a solid subset of T(X).
(f)	Let z be in T~BB)> and let x e X be such that |ж| < |z|. Then
|Т(ж)| = Т(|ж|) < T(|2|)	|T(z)J, so T(x) e В because В is a solid
subset of T(X), and thus x e	Therefore Т~Л(В) is a solid subset
of X. This completes the proof.
Let J be an /-ideal in a Riesz space (X, C), and let x [ж] be the
canonical mapping of X onto X/J defined by [ж] = ж 4 J. If
[O]t/ = {[ж] e Xf J: there exists j e J such that e C},
then (X/J, [CJj) is a Riesz space, and the canonical mapping ж [ж]
is an /-homomorphism of X onto X[J.
If {(Xa, Ca): a e Г} is a family of Riesz spaces, then the product space
X JJ Xa is a Riesz space under the ordering C = JJ Ca> and the
аеГ	аеГ
algebraic direct sum ф Xa is an /-ideal in {X, G). Moreover, each
аег
projection 7тй: П Xa —> Xx is an /-homomorphism; also each injection
ac-L’
>ФХа is an /-isomorphism. Observe that the properties of
аеГ
order-completeness as well as of u-order-completeness are preserved
in the formation of products and algebraic direct sums.
Let (X, C) and (У, K) be Riesz spaces. We say that (Y, K) is an
order-completion of (X, C) if:
(a)	(Y, K) is order-complete;
(b)	there exists an /-isomorphism, say ж ж, of (X, C) into (У, К);
(с)	for any у е Y, we have
у = sup{d; а е X, а < у} = inf{S: Ъ е X, у < 6}.
It should be noted that the condition (c) in the definition of order-
completion can be replaced by the following condition:
(d)	for any у e Y with у > 0, there exist a, b in X such that
0 < d < у < 6.
For a proof, see Luxemburg and Zaaren (1971).
For the sake of convenience, we shall identify X with the Riesz
subspace {d: a e X} of У. The following result is concerned with the
existence of the order-completion of a Riesz space. The construction of
an order-completion of an Archimedean Riesz space is a straightforward
generalization of the Dedekind procedure for completing the rational
number system, therefore we leave the proof which can be found in
Peressin (1967)) to the reader.
ELEMENTARY THEORY OF RIESZ SPACES 135
(10.25) Theorem (Nakano). A Riesz space (X, C) has an order-
completion if and only if X is Archimedean. Furthermore any two
order-completions of X. are ^-isomorphic.
As we shall see in Chapter 13, the order-completion of a locally
convex Riesz space may be identified with the topological dual of some
locally convex Riesz space under certain additional assumptions.
11
TOPOLOGICAL RIESZ SPACES
By a topological Riesz space (or topological vector lattice) we mean, a
Riesz space (X, C) equipped with a vector topology & which admits a
neighbourhood-base at 0 consisting of solid sets in X. A Riesz space
equipped with a locally solid topology is referred to as a locally convex
Riesz space (or locally convex vector lattice). Therefore, by making use of
theorem (6.3), (X, C, &) is a locally convex Riesz space if and only if
it is both locally o-convex and locally decomposable.
(11.1)	Proposition. Let (X, C) be a Riesz space, and let (X, tP) be
a topological vector space. Then the following statements are equivalent:
(a)	(X, G, IP) is a topological Riesz space;
(b)	the mapping (x, y) —> x V у is uniformly continuous on X xX;
(c)	the mapping x —>	is uniformly continuous on X;
(d)	(X, C, 3s) is locally full and G gives an open decomposition on
(X, 0>);
(e)	(X, G, &) is locally full and the mapping x -> x+ is continuous at 0;
(f)	for any two nets {жа:а g J)} and {ya:^ e &} in %, if < \yf
for all a g D and if converges to 0 with respect to then xa converges
to 0 with respect to SP.
Proof, (a) => (b): Let V be a solid ^-neighbourhood of 0, and let
W be a solid ^-neighbourhood of 0 such that W + W <= R For any
elements x, y, w1; and w2 in X,
wi A w2Tx л у < (ж+wj) V (y+w2) < V w2X-x V y,
— (IwJ+lwJ) < W-JL A W2 < wt V W2 < |wj +|w2|,
and so
-(И11+Н1) < (ж+Wj) V (y+w2)-x V у < |wj+|wa|.
Therefore, if w15 w2 g W then (xX-wf) V (y+w2)~- x V у g V; this
implies that the mapping (ж, у) -> x V у is uniformly continuous on
XxX.
(b)	(c): Obvious.
(c)	(d): Let IF be a ^-neighbourhood of 0. By the uniform con-
tinuity of the mapping x -> x+, there exists a ^-neighbourhood V of
TOPOLOGICAL RIESZ SPACES
137
0 such, that x+ — y+ g W whenever x~yeV. We now claim that
(F—- G) P\C W, and then it would follow from theorem (6.1) that
(X, C, A6) is locally full. Suppose that 0 < w < u, where и e V and
w e X. Then w —(w — и) и g V, and so w — w~r w"h — (w —«)+ e W.
For the proof of the open decomposition property, we first notice that
the mapping x —> x+ is continuous at 0. Let W be any ^-neighbourhood
of 0; then there exists a ^-neighbourhood V of 0 such that x+, x~ e W
whenever x e V, so V <= W n C — IF C\ G, and thus G gives an. open
decomposition on (X,
(d)	=> (e): Let W be a circled, order-convex ^-neighbourhood of 0,
and let V ~ W C\ G—W C\ C. Then V is a ^-neighbourhood. For any
x eV, there exist w1} w% in W C\ G such that x — w1-- w2, so
0 <	0 < x~ < w2; it then follows from the order-convexity of
W that x+ e W, and hence that the mapping x -> x+ is continuous at 0.
(e)	=> (a): For any ^-neighbourhood W of 0, there exists a circled
order-convex .^-neighbourhood U of 0 such that U a W. Since the
mapping x x+ is continuous at 0, then the mapping x -> |ж| is also
continuous at 0, so there exists a ^-neighbourhood V of 0 such that
|ж| g U whenever x e V, and so V sk(JF); this shows that the solid
kernel sk( W) of W is a ^-neighbourhood, consequently (X, C, is a
topological Riesz space.
Therefore statements (a) -(e) are mutually equivalent. Finally we
show that these statements are equivalent to (f). The implication
(a)	(f) is easy to verify. On the other hand, suppose (f) holds. Then,
in view of theorem (5.1), A6 must bo locally full; further, the map
x is clearly continuous. This shows that (f) => (a); consequently
statements (a)-(f) are equivalent.
If (X, G, 0s} is an order-complete topological Riesz space and if В is
a normal subspace of X then, by the preceding result and corollary
(10.11), X is the topological direct sum of В and B(\
(.11.2) Proposition. Let (X, G, SG) be a topological Riesz space.
Then the following statements hold:
(a)	G is &-closed;
(b)	(X, C) is Archimedean;
(c)	the solid hull of each ^-bounded set in X is ^-bounded, conse-
quently if В is &-bounded then so are B+, B~, and |B|, where
B+— fb+\Ъ e В}, B~ - {b--.be B}, and \B\	{\b\:b e B};
(d)	C is a strict ^d-cone in (X, ^);
(e)	the A6-closure of each Riesz subspace of X is also a Riesz subspace.
138
TOPOLOGICAL RIESZ SPACES
.Proof, (a) Since О = {u c X: и u+}, it follows from the con-
tinuity of the mapping x at 0 that C is ^-closed.
(b)	follows from (a) and proposition (2.1).
(c)	follows from the fact that IP admits a neighbourhood-base at 0
consisting of solid sets in X.
(d)	follows from (c).
(e)	Obvious.
(11.3)	Proposition. Let (X. C, IP) be a topological Riesz space.
Then the following statements hold:
(a)	the IP-closure H of each solid set II is solid;
(b)	the solid kernel sk(H) of each IP-closed set II is IP-closed;
(с)	H absorbs every ^-bounded set in X if and, only if sk(.H) absorbs
all IP-bounded subsets of X, where II cz X.
Proof, (a) Let be a neighbourhood-base at 0 for IP consisting of
solid sets in X. Since H гл{11 + V: V g %}, it follows from proposition
(10.5)(e) that Й is solid.
(b)	Since sk(H) is the largest solid subset of X contained in II, it
follows from (a) that sk(LZ) is ^-closed.
(c)	The sufficiency is obvious, and the necessity follows from
proposition (11.2)(c) and from the fact that sk(H) is the largest solid
subset of X contained in II. This completes the proof.
(11.4)	Corollary. Let {X, C, ^) be a topological Riesz space. Then
the IP-closure of each P-ideal in X is an P-ideal in X.
In view of proposition (11.3), for any topological Riesz space (X, C, IP)
there exists a neighbourhood-base at 0 for IP consisting of ^-closed
solid sets in X; in fact
% = {P: V is a solid ^-neighbourhood of 0}
is such a neighbourhood-base at 0 for IP, where V is the ^-closure of V.
(11.5)	Proposition. Let (X, С, IP) be a topological Riesz space.
Then each normal subspace of X is IP-closed.
Proof. Let В be any normal subspace of X. Since (X, C) is Archi-
medean, it follows from proposition (10.9) that В = _Bdd, The closedness
of any set of the form Ad is an immediate consequence of the continuity
of lattice operations, and hence В is ^-closed.
TOPOLOGICAL RIESZ SPACES
139
It should be noted that if В is an /-ideal in a locally convex Riesz
space (J, G, &), then, by proposition (11.6), В <= {B}. Therefore the
question arises whether В — {В}; we shall see (in theorem (13.1))
that this is the case if and only if all continuous linear functionals are
normal.
(11.6)	Proposition. Let (X, G, &} be a locally convex Riesz space.
The topological completion (X, .P) of (X, G, ^) is a locally convex Riesz
space ordered by the cone C, where 0 is the P-closure of G in (X, P).
Proof. Since the lattice operation (x, y) x V у is uniformly con-
tinuous from X xX into X, it follows from a well-known result that
this mapping has a unique uniformly continuous extension from.
XxX into X. It is easily verified that this extension makes (X, 6) into
a Riesz space. On the other hand, if the locally solid topology P is
determined by a family {pT: r e D} of Riesz semi-norms on (X, C), then,
for each т e I), pT can be uniquely extended to a continuous Riesz
semi-norm pT on (X, 0, P), and P is determined by the family
{pT: 7 e JD} of Riesz semi-norms on (X, 0); it then follows from theorem
(6.3) that (X, (f, P) is a locally convex Riesz space.
(11.7)	Proposition (Kawai). Let (X, C, P) be a locally convex
Riesz space, and let (F, K) be the order completion of (X, G). If (X, С, P)
satisfies the condition that {xy: у g 1} converges to 0 for P whenever
xy | 0, then there exists a locally solid topology P on (F, K) such that
P induces P on (X, G) and X is dense in (F, К, P).
Proof. Let {pa: a e D} be a family of Riesz semi-norms on X generat-
ing the locally solid topology P. In view of the definition of order-
completion, for each у e F there is an increasing net {aA} of X with
0 <	< \y\ such that |?/j supfipj, we now define
ffix) = sup{pa(aA)} (a e D).
Л
Since (X, G, P) satisfies the property that pa(xy) —> 0 as xy j, 0 (y e D),
then each pa is independent of the particular choice of an increasing
net {uA} with f |y|. It is easily seen that each pa is a Riesz semi-norm
on F for which
pa(d) — pfia) for all a e X.
140
TOPOLOGICAL RIESZ SPACES
Therefore the family (ррхЕ!)} determines a locally solid topology
0s on Y such that 0s is the relative topology on X induced by and
that X is dense in (У, 7^).
A Riesz semi-norm on X is called a Riesz norm if it is a norm, A
Riesz space equipped with a Riesz norm is called a normed Riesz space;
and a Riesz space with a Riesz semi-norm is called a semi-normed Riesz
space. Obviously, normed Riesz spaces are locally convex Riesz spaces.
A normed Riesz space is called a Banach lattice (or В-lattice) if it is
complete for the norm. A locally convex Riesz space (X, C, is
called a Frechet lattice (or F-latlice) if it is metrizable and complete
for the topology 3s. Similar to the case of locally convex spaces, we
shall see (below) that each locally convex Riesz space (X, (7, 72s) is
/-isomorphic and topologically isomorphic with a Riesz subspace Y
of the projective limit of 15-lattices {L^ \ a e Г}.
It is easily seen that every Riesz subspace of a locally convex Riesz
space, equipped with the relative topology, is a locally convex Riesz
spacej and that the Cartesian product of a family of locally convex
Riesz spaces, equipped with the product topology, is a locally convex
Riesz space. If J is a closed /-ideal in a locally convex Riesz space
(X, (7, then the quotient space (X/ Jt [(7]^), equipped with the
quotient topology, is also a locally convex Riesz space; for the sake of
convenience, it is referred to as the quotient Riesz space. We shall see
that the locally convex direct sum of a family of locally convex Riesz
spaces is also a locally convex Riesz space.
(11.8)	Proposition. Every locally convex Riesz space (X, (7, d^) is
^-isomorphic and topologically isomorphic to a dense Riesz subspace of the
projective limit of a family of Banach lattices; this family can be so chosen
that its cardinality equals the cardinality of a given neighbourhood-base at
0 for
Proof. Let {pa:a e Г} be a family of continuous Riesz semi-norms
on (X, C, ^) generating iP. Г is a directed set when we define a < f
if px(x) < PjfF) for all x g X. For each a e Г, let
Л - {жеХ:^(ж) = 0};
then is a ^-closed /’-ideal in X. Further, we define
[CL = M e X/J.: e J, such that z+j, e C},
II ИХ
TOPOLOGICAL RIESZ SPACES
141
then (X/Ja> [6% |]. ||a) is a normed Riesz space and the norm topology
||. [| a is coarser than the quotient topology on. X(Ja induced by
Moreover, a continuous /-homomorphism gaf} from (X/J^,	|(.||^)
onto (X/Ja, [GJa, ||. ||a) is defined by setting [ж]а = ^(Ид) whenever
a < (J. By proposition (11.6), the completion La of (X{Ja, [CJa, ||. ||J is
a Banach lattice (a e Г), and hence can be uniquely extended to a
continuous /-homomorphism. ga/j from Lp into La whenever a < /?. If
7тй denotes the projection of JJ La into ,La, then the projective limit
«бГ
lim of the family of Banach lattices {La: a g Г} with respect to the
/-homomorphisms ga(j(a, ft e Г, a < /5) is a closed/-ideal in the product
space JJ Ъл, because
a
1™ 9«pLp = {2 бП ^a'(^a-gap °	= 0, a < Д a, I? e Г}
4	a
and TTa“-gaj5 ° Ttj} is a continuous /-homomorphism from JJ La into La.
a
Notice also that the product space JJ La is complete with, respect to the
a
product topology J; it then follows that lim gapL^ is a complete
locally convex Riesz space with respect to the relative topology r
induced by J.
Now if we define a mapping у from X into JJ La by setting
a
^) = ([<:аеГ) (жеХ),
then tp is an /-isomorphism and a topological isomorphism from
(X, C, 3s) into (lim g^Lp, e/rf and y(X) and X[Ja are /-isomorphic;
a
this implies that y(X) is a dense Riesz subspace of (lim gajiLp,
We now turn our attention to seek some conditions ensuring that the
inductive topology with respect to a family of locally convex Riesz
spaces is locally solid.
(11.9)	Proposition. Let (X, C) be a Riesz space, and let & be the
inductive topology on X with respect to locally convex Riesz spaces
{(Xa, (7a, ^a):a g Г} and /-homomorphisms {TK:a g Г}. If X is the
linear hull of fj Ta(fXf) and if each TJfXf) is an I-ideal in X, then & is a
«еГ
locally solid topology and hence (X, C, is a locally convex Riesz space.
Proof. Let V be any convex ^-neighbourhood of 0 in X. Then each
27“ky'i is a ^-neighbourhood of 0 in Хж, and so there exists a solid
142
TOPOLOGICAL RIESZ SPACES
and convex ^-neighbourhood Wa of 0 in Xa such that Wa <= T/^T).
By proposition (10.24)(e), each Ta(Wa) is a solid subset of TJX^, and
hence Ta(Wf) is a solid subset of X because each Ta(Xa) is an /-ideal
in X. Observe that the union of a family of solid sets in X is a solid
subset of X. It follows that W = (J Ta(Wa) is a solid subset of X, and
схеГ
hence, from proposition (10.5)(a), the convex hull U of И7 is a solid
subset of X. Since X is the linear hull of (J Ta(Xa), U is absorbing,
аеГ
and so U is a convex, solid, and absorbing subset of X. It is clear that
U is a ^-neighbourhood of 0 since Wa 77a“1(B;r) cz Tf^U), and that
U qV since V is a convex set containing W. Therefore we have found
a convex, solid ^-neighbourhood U of 0 in X such that U <= J7;
consequently is a locally solid topology, and this completes the
proof.
In view of the definition of a strict inductive limit and the preceding
result, it is known that if {Xw} is a sequence of /-ideals in X, and if
(Xn, Gn, are locally convex Riesz spaces, then the strict inductive
limit of {Xn} is a locally convex Riesz space.
(11.10)	Coboll ary. The locally convex direct sum of a family of locally
convex Riesz spaces {(Xa, Сж, G Г), denoted by © (Xa, Ca, tPf), is
a
a locally convex Riesz space. Furthermore, © (Xa, Ca, tPf) is complete if
a
and only if each Xa is complete for &a.
Proof. If X denotes the algebraic direct sum of {Xa; a g Г], ja
denotes the injection map of Xa into X, and if О = X Л JJ Oa, then
аеГ
the Riesz space (X, C) is the linear hull of (J ja(Xa), and each©(X J is an
аеГ
/-ideal in X. The result now follows from proposition (11.9) and the
definition of locally convex sum topology on X.
Let (E, C) be an ordered vector space, and let e be an order-unit in
E. Then the gauge ||. ||e of [—e, e], defined by
||ж[|е = inf{2 > 0:# g Я[--e, e]},
is a semi-norm on E, and it is referred to as the order-unit semi-norm
corresponding to e. It is clear that ||. ||e is a norm if and only if (E, O') is
almost-Archimedean, and that the order-unit semi-norms corre-
sponding to two different order-units are equivalent; so the topology
given by order-unit semi-norms can be regarded as a topology determined
TOPOLOGICAL RIESZ SPACES
143
by the ordering. An order-unit semi-norm, which is also a norm, is
called an order-unit norm.
A normed Riesz space (X, C, ||. |j) is called a unital normed Riesz
space if the norm ||. || is an order-unit norm. It is clear that if (X, C, ||. ||)
is a unital normed Riesz space, then the norm topology is the order-
bound topology. This leads to the following result.
(11.11)	Proposition. Let (X, C, be a locally convex Riesz space.
Then & is the order-bound topology ^b if and only if there exists a
family of unital normed Riesz spaces {(XM, Cu, ||. ||M): и e Г} and a
family of Ghomomorphisms {Tu:u e Г} with the following properties:
(a)	X is the linear hull of U Tu(Xf);
меГ
(b)	each TU(XU) is an Gideal in X;
(c)	& is the inductive topology of {(XM, Cu, ||. e Г} with respect
to {Tu:u e Г}.
Proof, (i) Necessity. For each и e C, let
Xu - U n[-u, ul
n
Cu — Xu П G> ||. || u be the gauge of [— w, u] on Xu, and let Tu be the
injection of Xu into X. Then each (Xu, <7W, ||. ||J is a unital normed
Riesz space with the order-unit и, X — (J Xu, and each Xu is an A
ueC
ideal in X. Suppose that X' is the inductive topology of
№-u,Ou, \\.\\u):ugC}
with respect to {Tu, и g C}. Then, by proposition (11.9), N' is locally
solid, and a fortiori ЗГ is coarser than ^b. On the other hand, since
the relative topology on each Xu induced by is coarser than the
norm, topology ||. ||M, it follows that the injection map Tu from
(XM, Cu, ||. ||M) into (X, С, is continuous, and hence from the defini-
tion of inductive topology that 0й is coarser than Therefore, ^b
is the inductive topology of {(Xu, Cu, ||. ||„):u e 0} with respect to
{TpueC}.
(ii) Sufficiency. Let ||. ||M be the order-unit norm on Xu corresponding
to the order-unit и e Xu. In order to show that is the order-bound
topology ^b, it is sufficient to verify that each convex and circled
subset V of X, which absorbs all order-bounded subsets of X, is a iP-
neighbourhood of 0 or, equivalently, each Ty^F) is a ||. |(M-neighbour-
hood of 0 in Xu. Let V be such a set. Since и is an order-unit in Xu and
.144
TOPOLOGICAL RIESZ SPACES
since ||. ||M is the order-unit norm corresponding to u, it follows that the
order-interval [—ад, ад] is a ||. || „-neighbourhood of 0 in Xu. Notice that
Тм([ —ад, ад]) = [—Tu(u), ^(ад)] is an order-interval in X; then, there
exists а Я > 0 such that [ —Tu(u), ^и(ад)] <= XV. We conclude from
[—ад, ад]	that PffV) is a ||. ||„-neighbourhood of 0 in Xu;
hence is the order-bound topology ^b.
We shall now turn our attention to the topological completeness. It
is natural to ask under what conditions an order-complete topological
Riesz space is necessarily topologically complete.
(11.12)	Definition. A topological Riesz space (X, C, ^) is said to
be locally order-complete if there exists a neighbourhood-base at 0 for
consisting of solid and order-complete sets in X.
Obviously each locally order-complete topological Riesz space must
be order-complete.
(11.13)	Proposition (Nakano). Let (X, O, be a topological Riesz
space. If (X, C, is locally order-complete, then each order-interval in
X is complete for
Proof (Schaefer). Since X is a Riesz space, it is enough to verify
that each order-interval of the form [0, ад] (ад e C) is ^-complete. We
first prove that the result holds for the special case when 0 is metrizable.
Choose a countable neighbourhood-base f№Q:n e N] at 0 for & con-
sisting of solid order-complete sets and Wn+1 + K
4-1 C Wn for all n.
Any ^-Cauchy sequence in [0, ад] has a subsequence such that
xn+i ~~xn E Wn+t for all n e N.
Suppose that
yr — supDn:n > r}, zr — inl'D’,,:n > r}.
For fixed r and for any natural number q with q > r,
sup{«n:r < n < q}—xr
is increasing with respect to q and
0 < sup{a?m — xr'.r < n < q} < ад;
it follows from the order-completeness of X that
w = sup{sup{«n—xr\r < n < 7}}
TOPOLOGICAL RIESZ SPACES
146
exists in X. Furthermore we also have
w sup{sup{a;w:r < n < <?}} — xr
sup{«m:n > r}—xr = yr — xr.
On the other hand, notice that
0 < sup{a;n:r < n < q}--x.r < sup{|a?„ — xr\:r < n<,q}
< l^r+i 'xr \ I'l^r+2 ^r+il d~• • • '4~l^
It then follows from the order-completeness of Wr that w e Wr, and
hence that yr g xrXWr. A similar argument shows that zrExrqWr.
Therefore we have, for any r g N,
yrExr-\-Wr and zrExr+Wr.	(11.1)
It is clear that yr j, (гД) and that 0 < yr (zr < u) for all r; then
у = inf{y/.r > 1}, z = sup{gr:r > 1}
exist in X, and y, z are in [0, и]. Since
e Wn+1XWn+1 <= Wn
for all n, we conclude from p Wn = {0} that y—z = 0; hence
neN
inf{yr:r > 1} = sup{^:r > 1} — y.	(11.2)
We now claim that xn converges to у with respect to Indeed we
note that
Zn~~Xn < Z~~Xn = У ХХп < Уп Хи-
Since zn—xn and yn—xn converge to 0 and 0 is locally solid (so locally
order-convex) it follows from theorem (5.1) that xn converges to у
with respect to This proves the result for the metrizable case.
Now let us consider the general case, that is the case when is not
necessarily metrizable. Let % — {Wa: a e D} be a neighbourhood-base
at 0 for Sfi consisting of solid and order-complete sets in X. Define Q
to be the class of all countable collections
“ {Ж<„: g D, n = 1, 2...}
of each of which forms a neighbourhood-base at 0 for a vector
topology, say. Then Q is non-empty, directed by inclusion,
= vj g £1},
146
TOPOLOGICAL RIESZ SPACES
and (X, C, is a pseudo-metrizable (not necessarily Hausdorff)
locally order-complete topological Riesz space. For each a e D, let
W(a0) = n
Л>0
and suppose that
Then Xo and are normal subspaces of X. Since X is order-complete,
then, by corollary (10.11), we have, for each
X = Х^фХ| = X0@X£.
Let P be the /-projection of X onto N%, and let PQ be the /-projection
of X onto Nq. The following conclusions concerning /-projections are
easily verified:
(a)	if c then P^ < P^ and P^ ° P^ — P^;
(b)	0 < Pe < Po — I, where I is the identity map;
(c)	Po = sup{P^: ^efl};
(d)	if uT } и then P^(uT) } P#(u) for each $ eQ;
(e)	if denotes the vector topology on N^, induced by then
Pg is a continuous /-projection of (X, C, ^) into (X^, ^).
Suppose that {жя} is a ^-Cauchy net in [0, u], For each efi,
{^(жй)} is a ^-Cauchy net in X^; since (X|, is a metrizable
locally order-complete topological Riesz space, there exists
№ G [0, P^)]
such that Pg(%a) converges to yv with respect to ; for convenience,
it is denoted by y<$ = .^—lim P^(«a). By (b), у % g [0, X|. Therefore
a
we find a net {y^: g Q} in [0, u], where
У % = ^J-lim Р^(жа).
a
Moreover, the net e fl} has the following properties:
(i) if c then y^ = ^Jg-lim P^K),
a
(ii) v — supfy^:^7 g Q} exists in X and
P^(r) - y^, for any g Q.
In fact, if c: ^g, then X^ cz X^g, and so property (i) follows
from the fact that (X^, is Hausdorff. To prove property (ii), we
observe that 0 < y^ < u; it follows from the order-completeness of
X that	r <z>
v == sup{^: g 12}
TOPOLOGICAL RIESZ SPACES
147
exists in X (0 < v < u). On the other hand, if <= then, by (a),
(e), and property (i), we have
-= B^(^|a-lim P^M) = P^-lim (P^ ° PrJK)
a	a
- Р|а-НтР^(жй) = y^.
Я
We conclude from (d) that
= sup{PA(y^):^ eQ} = sup{P^(y^): cifeD} = y^,
verifying property (ii).
Therefore we complete the proof by showing that xa converges to v
with respect to In fact, since
PM = P^-lim PM) e
Ct
P#(v) is a .^-cluster point of { tP^(xx)}. Note also that P# is a continuous
^-projection of (X, onto (N^, ^L) which vanishes on whence
v is a ^-cluster point of {жй}. Clearly {aq} is a ^-Cauchy net for
each e Q; consequently жй converges to v with respect to for
each e Q. We conclude from
Щ - и 6Й)
that x^ converges to v with respect to tP, and hence that [0, it] is iP-
complete.
Recall that an ordered topological vector space (X, C, tP) is said to
be boundedly order-complete if each increasing net in E which is iP-
bounded has a supremum in E. We now present one of the deepest
results in the theory of topological Riesz spaces concerning the topo-
logical completeness.
(11.14)	Theorem (Nakano). A topological Riesz space (X, С, P)
which is both locally order-complete and boundedly order-complete is
complete for P.
Proof. Let {хт:т c JD} be a ^-Cauchy net. In view of proposition
(11,1), {xf:r c D} and {«7:r e I)} are also ^-Cauchy nets, and hence
it is sufficient to verify that : т e D} is convergent with respect to
P since the convergence of {жг} follows immediately from that of
{W} and {.г."}.
Lor any x g C, the continuity of lattice operations ensures that
{&+ A x-.r e D} is a ^-Cauchy net in the order-interval [0, «]; it then
148
TOPOLOGICAL RIESZ SPACES
follows from proposition (11.13) that there exists Ax g [0, ж] such that
л x converges to A№ with respect to Therefore we have found a
net {Ax:x g (7} in C, where Ax is the limit of A x with respect to
We now claim that {Ax:x g 0} is directed upwards and ^-bounded.
For any x, у e С, x V у g C, by making use of the continuity of lattice
operations, we have
~ ]im{.y!' л (x v y)} — {lim x* А ж} V {lim ж/ л у} Ax V Av
so that {Ax:x g C} is directed upwards. On the other hand, let W be
any solid ^-neighbourhood of 0, and let V be a solid ^-neighbourhood
of 0 such that F+ F+ V W. Since {жг+ : r g D} is a ^-Cauchy net,
there exists r0 g D such that
x^—x^ g F whenever r, t > r0;
for this t0 there exists Ло with > 1 such that x^ g AoF and so
xt =	V +	<= ^0(F+F).
Since xx A x converges to Ax with respect to there exists r0 e D
such that л	I7 .
Ax—x‘ A x G V whenever r > r0.
Take ту g D such that тг > r0 and rr r0. We conclude, from
0 < Ax == (Ax—л x)-\-xt A x < (Ax —xf А ж)+ж+ e F--H0(F+F)
c ^o(^+^+F) cz X0W whenever т > тъ
that {Ax:x g C} is ^-bounded.
According to the hypothesis,
a = sup^/.a: e C}
exists in X. Furthermore, the element a has the following property:
a A у = Av for any у eC.
(11.2)
In fact, by proposition (10.2), we obtain
since
а л у = у Л 8ир{Лд,:ж g С} = supfz/ A Ax:x е G};
у л Ах == у л {lim л «} = 1нп(ж+ л х л у} = Axf,y (х е С)
and since АхЛу < Av for all x g C, it follows that
а Л у = Ay.
TOPOLOGICAL RIESZ SPACES	149
We complete the proof by showing that x^ converges to a with
respect to B8. Let W be any ^-neighbourhood of 0, and let V be a solid
^-neighbourhood of 0 such that F+F <= W. Since {x^’.teB} is a
^-Cauchy net, there exists r0 e D such that
x^—x^, e F whenever т, ?' > r0.	(11.3)
On the other hand, for any t\e D with > r0, let
xt = x'}' V a
then, by equality (11.2),
A^ = a A xt = а л (ж* Va) = a.
Notice also that xAT A xt converges to A^ with respect to then
there exists r2 g D such that
A xr~~a = x+ A xl—AXi e F whenever т > т2. (И.4)
^ncc x^—a = (#+ —«if A aq) +(жгн A aq — a) (t g D)	(11.5)
and since
[ж;1' — A aq|~ \x^ Л Xt — %* A X^ <	(r G P), (11.6)
it then follows from formulae (.11.3), (11.4), (11.5), and (11.6) that
x^-a e F+F JF,
where > r0. This implies that
x+—a g IF whenever т > r0,
and hence {x^'.r e D} converges to a with respect to
It should be noted that the above result is still true whenever & is
non-Hausdorff. For a proof, see, for instance, Peressini (1967).
The following example shows that a topological Riesz space which is
topologically complete need not be boundedly order-complete.
(11.15)	Example. Consider the space c0 consisting of all null
sequences of real numbers with the usual ordering and norm. Then c0
is a В-lattice. Let en = (1, 1,... 1, 0, 0 ...) g c0 and let В ~ {en:n e N}.
Then В is increasing and norm-bounded. It is obvious that В does not
have a supremum (not even any upper bound) in c0. This shows that
c0 is not boundedly order-complete.
It is known from corollary (6.5) that the topological dual of any
locally convex Riesz space (X, C, is an zf-ideal in Xb. Therefore it
is natural to ask whether the strong dual of a locally convex Riesz
n
150
TOPOLOGICAL RIESZ SPACES
space is also a locally convex Riesz space. We give an affirmative
answer of this question as follows.
(11.16)	Theorem. Let (X, 0, be a locally convex Riesz space with
the topological dual X', and let {3(X', X) be the strong topology on X'.
Then (Xz, C, fl(X', X)) is a locally convex Riesz space and locally order-
complete. If, in addition, (X, IP) is infrabarrelled, then (Xf, O', fi(X', X))
is boundedly order-complete and hence X' is complete for @(X', X).
Proof. Since the polar, taken in X', of any solid set in X is solid
in X', and since the solid hull of each ^-bounded set in X is ^-bounded,
then (X', C, @(X', X)) is an order-complete locally convex Riesz
space. Suppose that	is the family consisting of all convex, solid,
^-bounded subsets of X. Then % = {B°:B g is a neighbourhood -
base at 0 for fi(X', X), where B° is the polar of В taken in X'. We now
show that each B° is order-complete. Let {fr} be an increasing not in
which is majorized in X'. Then/ = sup/r exists in X' because (X', O')
is clearly ord er-complete. For any x g B, we obtain
/И <f(\x\) = sup/r(|^|) < 1.
Therefore / g B°, and hence each B° is order-complete; consequently
(X', С, P(X', X)) is locally order-complete.
Suppose now that {X, tP) is infrabarrelled, and that {fr} is а /3(Х', X)-
bounded subset of X' which is directed upwards. Then, by the Alaoglu- 
Rourbaki theorem, {fT} is relatively a(X', X)-compact, and hence
{fT} has a <7(X', X)-cluster point / in X'. Since /4 and since C'
is a(X', X)-closed, it follows that / = sup/r in X'. Therefore
(X', O', fi(X', X)) is boundedly order-complete. Finally, in view of the
Nakano theorem (11,14), X' is complete for /3(Х', X). Therefore the
proof of the theorem is complete.
We shall see in the next chapter that (X', O', а&(Х', X)) is also locally
order-complete for any locally convex Riesz space (X, C, &), where
crs(X', X) is the locally solid topology associated with а(Х', X).
(11.17)	Definition. Let (X, C, &) be a locally convex Riesz space
with the topological dual X'. Then (X', O', fi(X', X)) is called the
Riesz dual of X and (X", C", fi(X", X')) is referred to as the Riesz
bidual of X, where X" = (X', /?(Х', X))' and C" is the dual cone of O'.
TOPOLOGICAL RIESZ SPACES
151
(11.18)	Proposition. Let (X, C, be a locally convex Riesz space
with the Riesz dual (X', С', /3(Х', X}). Then the image X of X into X"
under the evaluation map x —> x, defined by
for all feX',
is a Riesz subspace of X", and the evaluation map is an ^-isomorphism of
X onto X. Furthermore, X <= (X')b, and hence X cz X" П (X')b.
Proof. It is clear that the evaluation map is a bijection of X onto
X. We now show that x+ = (ж)+. Let/be in O'. It follows from proposi-
tion (10.15){a) that
x+(f) = Лж+) = вир{0(ж): 0 < g < /}.
Obviously x is an order-bounded linear functional on X'; in view of
theorem (10.12)(a), we obtain
(£’)+(/) = sup{f(#):0 < g </}.
Therefore x+(f) = ($)+(/), hence x+ = (ж)+ because of X' = C—C,
and thus the evaluation map x -+ x is an /-isomorphism of X onto X.
Consequently X is a Riesz subspace of X". By making use of theorem
(10.12)(b) and of the definition of normal integrals, we have X <= (X')b,
and hence X X" П (X')b. This completes the proof.
In the future, if no confusion can arise, we shall agree not to dis-
tinguish between x and ж; thus X will be identified with its canonical
image X.
The question naturally arises as to whether X is an /-ideal in X". In
Chapter 13, we shall give necessary and sufficient condition to ensure
that this occurs.
We conclude this section with a few examples of topological Riesz
spaces.
(11,19)	Examples, (a) Let T be a non-empty set. The vector space
RT of all real-valued functions on T is an order-complete Riesz space
with the ordering induced by the cone
К = {x g RT:x(fi) > 0 for all t e T}.
Let X be a Riesz subspace of RT, (5 a family of subsets of T such that
every ж, in X, is bounded on each В e (5, and suppose that
T - u {B:B s ©}.
152	TOPOLOGICAL RIESZ SPACES
The topology on X of uniform convergence on members of ® is then a
locally solid topology defined by the Riesz semi-norms {pB:B g ®},
where pB(x) = вир{|ж(/)| :t e B}. If T is a Hausfdorff topological space,
X is the Riesz space G(T) consisting of ah continuous real-valued
functions on T, and if each t g T is interior to some В g (5, then X is
complete by, for example, Bourbaki (1961). If T is a completely
regular Hausdorff space and if (5 is the family of all compact subsets
of T, then the topology on G(T) of uniform convergence on members
of (5 is called the compact-open topology, and denoted by & . In
particular, if T is a compact Hausdorff topological space then the
compact-open topology on C(T) is the uniform topology; in this case,
G(T) is a В-lattice.
(b)	Let (X, O') be a Riesz space with the order-bound dual Ab, and.
let У be a Riesz subspace of Хъ such that У is totaJ over X. For any
f g У П C*, suppose that
p/ж) = /(|ж|) (жсХ).
Then the locally solid topology | u|(A, У) determined by the family of
Riesz semi-norms {pf:f g У n C*} is generally called the Dieudonne
topology (more precisely, the Dieudonne topology induced by У). In
next chapter we shall show that |o‘|(X, У) is the locally solid topology
associated with cr(X, У).
(c)	Let (T, 3d, y) be a totally u-finite measure space, and let
co
{7’n:neN} be a fixed sequence in J? such that T = (J Tn and
n-1
y{Tn) < oo for all n e N. A real-valued function x on T is said to be
locally summable if the restriction of x to each Tn is summable. Two
functions on T are said to be equivalent if their difference is zero except
on a set of //-measure zero. Let Q denote the set of all equivalent
classes of locally summable functions on T. Q has a natural cone G,
defined by (7 = {ж g Н:ж(£) > 0 for all t g T}.
It is easily seen that (Q, C) is a Riesz space; in fact, (Q, G) is order-
complete. For a given set A in Q, we define
Д Ax = {ж g Q:xy g ВЦу) for all у g A},
Ax = {z g Q:zx g D^y) for all ж g Л},
then A and Ax are Aideals in Q; A and Ax are placed in duality by the
bilinear form	f
(x, Z) = И dy Ж G Л Z G Ax.
21
TOPOLOGICAL RIESZ SPACES
353
A is called a Kothe function space, and Ax is referred to as the Kothe
d/ual of A; of course, Ax is also a Kothe function space. It is not hard to
show that A is the Kothe dual of Ax, i.e. A = Axx. Q is the largest
Kothe function space and Dx is the smallest Kothe function space
under the inclusion.
From now on, we always assume that A is a Kothe function space
and that Ax is the Kothe dual of A. If C\ C C\ A, then (A, CA) is a
Riesz space; moreover it is order-complete since A is an /-ideal in Q.
If A* is the algebraic dual of A and Ab is the order-bounded dual of
(Л, <7Л), we then define a mapping ф of Ax into A* by putting
</>(z)(&) = (x, z) for all x g A,
where z g Ax; it is easily seen that ф is an /-isomorphism of Ax into Ab,
and hence </>(Ax) is a Riesz subspace of Ab, consequently Ax can be
regarded as a Riesz subspace of Ab.
According to the above remarks, u(A, Ax) and cr(Ax, A) are locally
o-convex topologies on A and Ax respectively, and Ax is the topological
dual of (A, u(A, Ax)); A is the topological dual of (Ax, u(Ax, A)). If
crs(A, Ax) denotes the locally solid topology on A associated with
cr(A, Ax) (for definition see Chapter 6), then (A, CA, ng(A, Ax)) is a
locally convex Riesz space; in view of corollary (6.5), the topological
dual (A, CA, crs(A, Ax))' of (A, CA, os(A, Ax)) is the /-ideal in Ab
generated by the Kothe dual Ax of A. Let ® be a family consisting of
solid and o*(Ax, A)-bounded sets in Ax which covers Ax and is directed
by inclusion. Then there exists a unique locally solid topology, de-
noted by ^K, on (A, CA) such that {.В0: В G 65} is a neighbourhood-base
at 0 for so (A, CA, is a locally convex Riesz space. This topology
is called a Kothe topology, and (A, CA, ^K) is referred to as a
topological Kothe function space. It should be noted that there is a
finest Kothe topology, denoted by on A obtained from Ax in this
way; this is the one obtained by letting ® be the family of all solid and
cr(Ax, A)-bounded sets in Ax. Also there is a coarsest Kothe topology
on A obtained in the described way from Ax; this is the one obtained by
letting ® be the family of all order-intervals of the form [—-z, z]
(0 < z e Ax). We note that the coarsest Kothe topology on A is pre-
cisely the locally solid topology crs(A, Ax) associated with u(A, Ax).
Coffman (1959) has shown that, by virtue of the Nakano theorem,
every topological Kothe function space is topologically complete.
(d)	Let T be a locally compact Hausdorff space, and let K(T) be the
Riesz space of all real-valued and continuous functions on T with
154
TOPOLOGICAL RIESZ SPACES
compact support, ordered by the positive cone C of non-negative
functions in K(T). For any compact subset L of T, let K(T, L) be the
set of all elements in K(T) with support contained in L. It is easily seen
that K(T, L) is an Д ideal in K^T) and that K(T) is the linear hull of
{K(T, L)'.L is compact subset of T}. For any compact subset L of T,
the norm ||. ||z, defined by
IMIz = sup{|«(£)|:t el} (x gK(T, L)),
is a Riesz norm on K(T, L). The inductive topology on K(T) with
respect to the family of {(K(T, L), ||. ||х):^ is compact in T} and the
injection maps {jL:L is compact in T} is referred to as the measure
topology on K(T). In view of proposition (11.9), is a locally solid
topology on K(T), and hence (K(T), C, 3Q is a (Hausdorff) locally
convex Riesz space. Further we note that (K(T), is bornological
(cf. Schaefer (1966)). The topological dual of (K(T), C, can be
identified with the space MIT} of all Radon measures on T (cf. R. E.
Edwards (1965)), and hence from corollary (7.6) M(T) = K(T)b
because K(T) is ^-complete. Therefore each normal integral on ,K(T)
is a Radon measure on T; but the converse is not true as shown by
Roberts (1964).
12
LOCALLY O-CONVEX RIESZ SPACES
Feom the preceding chapter we have seen that solid sets in a Riesz
space play an important role in the study of topological Riesz spaces.
Although locally o-convex topologies on a Riesz space are, in general,
not locally solid (cf. example (3.15)), some results on locally convex
Riesz spaces can be generalized to the case of locally o-convex Riesz
spaces. On the other hand, we have seen from Chapter 6 that any locally
o-convex topology & can be associated with a locally solid topology
which is the greatest lower bound of all locally solid topologies
that are finer than HF. Now if HF is a locally o-convex topology on a
Riesz space (X, C) and if is a neighbourhood-base at 0 for HF con-
sisting of o-convex circled sets in X, then
- {sk( F): V e - {$( F): F e Я
is a neighbourhood-base at 0 for <^g.
Let (X, C) be a Riesz space with the order-bound dual Xb, and let
У be a Riesz subspace of Xb which is total over X. Recall that the
Dieudonne topology |сг| (X, F) on X, induced by Y, is defined to be the
vector topology determined by the family {pf:Q </g F} of Riesz
semi-norms, where , 4 n
(ж e X).
We now show that |cr| (X, F) is the locally solid topology associated
with the weak topology u(X, F).
(12.1) Proposition. Let (X, C) be a Riesz space with the order-bound
dual Xb, and let F be a Riesz subspace of Xb which is total over X. Then
[cr| (X, F) — <7В(Х, F) and the family
</g F)
forms a neighbourhood-base at 0 for <rs(X, F). Furthermore, the topo-
logical dual of (X, G, o“s(X, F)) is the /-ideal in Xb generated by Y
Therefore, if F is an /-ideal in Xb, then <rs(X, F) is the, coarsest locally
solid topology which is consistent with the duality (X, F).
.156
.LOCALLY O-CONVEX RIESZ SPACES
Proof. Since У is a Riesz subspace of Xb, then
У У n С* — У n C*;
in view of corollary (6.1.2), the weak topology сг(Х, У) on X is locally
o-convex. On the other hand, for any 0 < f e У, if
V = {x еХ:|/(ж)| < 1},
then sk(F) ~ {ж аХ:|_ - [ж|, |ж|] с F}  {ж еХ:/(|ж|) < 1), and it
follows from proposition (10.15) that sk(F) =	Therefore
|cr| (X, У) = crg(X, У) and {[•—/, f]°:0 < f e У} is a neighbourhood -
base at 0 for crg(X, У). The result now follows from corollary (6.5) and
the proof is complete.
(12.2)	Corollary. Lei (X, (7) be, a .Riesz with the order-bound dual
Xb, and let У be an Lideal in Xb which is total over X. Then the following
statements hold:
(a)	a subset M of X is a(X, Y)-bounded if and only if its solid hull
is a(X, Y)-bounded;
(b)	each a(X, Y}-bounded subset of X is uniformly bounded on each
order-interval in Y;
(c)	a subset В of Y is as(X, Yf equicontinuous if and only if it is an
order-bounded subset of У,
(d)	a subset В of Y is с8(У, X)-bounded if and only if SB-o(Y, X)-
bounded.
Proof. Since У is an У-ideal in Хъ, then. us(X, У) is consistent with
the duality {X, Y). The conclusion (a) follows from proposition (11.2)
and the Mackey-Arens theorem, while the conclusion (b) is an imme-
diate consequence of proposition (12.1). The conclusion (d) follows
from the remark (c) of Theorem (10.12). It remains for us to verify (c).
The sufficiency follows immediately from proposition (12.1); to prove
the necessity, we note that the polar B° of B, taken in X, is a <ts(X, У)-
neighbourhood of 0 in X, hence there exists 0 < f g Y such that
<= B°, and hence В cz [—f,f~]. This completes the proof.
(12.3)	Corollary. Let (X, (7) be a Riesz space and let Y be a Riesz
subspace of Xb which is total over X. Then each positive u( У, Xfbounded
subset В of Y is uniformly bounded on each order-interval in X.
LOCALLY O-CONVEX RIESZ SPACES
157
Proof. Elementary.
(12.4)	Proposition. Let P be a locally o-convex topology on a Riesz
space {X, G) and let be the locally solid topology on X associated with
TP. Then the following statements are equivalent:
(a)	C is a locally strict P-cone in (X, P);
(b)	each P-bounded subset of X is ^bounded:
(c)	G is a strict P-cone in (X, P).
Proof. The implication (c) --> (a) is obvious, and the implication
(a) ~> (b) is a restatement of corollary (6.9). The observation that G is
a strict ^-cone in (X, Ps) shows that (b) implies (c).
(12.5)	Corollary. Let P be a locally o-convex topology on a Riesz
space (X, G). If the topological dual X' of (X, G, P) is an /-ideal in Xb
then G is a strict Rd-cone in (X, P).
Proof. Since X' is an /-ideal in Xb, it follows from corollary (6.5)
that the locally solid topology Ps associated with P is consistent with
the duality (X, X'). The result now follows from, proposition (12.4).
A partial converse of the preceding result is the following corollary.
(12.6)	Corollary. Let P be a locally o-convex topology on a Riesz
space (X, G). Suppose that each circled o-convex set in X which absorbs
all IP-bounded subsets of X is a P-neighbourhood of 0 (i.e. (X, С, P) is
o-bomological in the sense of Kist (1968)). Then the following statements
are equivalent:
(a)	P is locally solid;
(b)	the topological dual X' of (X, G, P) is an /-ideal in Xb;
(с)	C is a strict P-cone in (X, P);
(d)	G is a locally strict P-cone in (X, P).
Proof. The implication (a) => (b) follows from corollary (6.5), the
implication (b) ->- (c) and the equivalence of (c) and (d) follow .from
proposition (12.4). It remains to verify that (c) implies (a). Since
(X, G, P) is o-bornological, it is easily seen that no strictly finer locally
o-convex topology on X has the same ^-bounded sets in X, it then
follows from proposition (12.4) and from the definition of PQ that
158
LOCALLY O-CONVEX RIESZ SPACES
and hence that is locally solid. This completes the proof.
It should be noted from example (3,15) that the o-bornological
condition in corollary (12.6) cannot be dropped. We shall see in
Chapter 15 that under one of the assumptions (a), (b), (c), and (d) in
the preceding result, (X, (7, ^*) is a bornological Riesz space.
By a locally o-convex Riesz space we mean a Riesz space (X, C)
equipped with a locally o-convex topology & on X such that the cone
C is 6^-closed.
It is known from Peressini ((1961), theorem 2.4) that if (X, 67, ||. ||) is
a normed Riesz space with the topological dual X', then the weak
topology <r(X, X') is locally solid if and only if X is finite-dimensional.
It is natural to ask under what conditions the Mackey topology
т(Х, X') is locally solid. It will be shown that т(Х, Xх) is locally solid
if and only if it is locally o-convex.
(12.7)	Proposition. Let (X, 67, ^) be a locally o-convex Riesz space
with the topological dual X'. Then the Mackey topology т(Х, X') on X is
locally solid if and only if т(Х, Xх) is a locally o-convex topology on X.
and X' is an /-ideal in Xb.
Proof. See theorem (6.1) and corollary (3.14).
(12.8)	Proposition. Let {(Xa, C\):a g Г} be a family of Riesz
spaces, and let &a be a locally o-convex topology on (Xa, C\) for each
a e Г. Suppose that I =	= П that 0 - JJ
аеГ	аеГ	аеГ
product topology on X). Then	^aS, and therefore the /-ideal in
аеГ
Xb generated by © (Ха, (7Й, ^Jx is precisely the algebraic direct sum of
аеГ
/-ideals in Xb generated by (Ха, (7a, ^a)x (a G Г), where denotes the
locally solid topology on Xa associated with for each a e Г and
denotes the locally solid topology on X associated with (P.
Proof. See theorems (3.19) and (6.1).
(12.9)	Proposition. Let {(Xa, 67a):a g Г) be a family of Riesz
spaces, and let be a locally o-convex topology on (Xa, 67a) for each
a е Г. Suppose that Y = © Xa (the algebraic direct sum of {Xa: a g Г}),
acP
К — Y Y\ JJ Ca, and that = © 3/л (the locally convex sum
a	a
topology of on F). Then	and therefore the /-ideal
a
LOCALLY O-CONVEX RIESZ SPACES
169
in Yb generated by JJ (Хй, tPaf is precisely the product of /-ideals in
aef
Xb generated by (Xe, Ca, &a)' (a g Г), where ^a>& denotes the locally
solid topology on Xa associated with for each a. g Г, and <£% denotes
the locally solid topology on Y associated with SP.
Proof. See theorems (3.16) and (6.1).
We conclude this chapter with a result concerning the locally order-
completeness of (X', C, a3{X’, X)).
(12.10)	Proposition. Let (X, C, be a locally o-convex Riesz
space with the topological dual X'. If X' is an /-ideal in Xb, then
(X', O', crg(X', X)) is always locally order-complete.
Proof. Since X' is an ^-ideal in Xb, it follows that (X', С, аа(Х\ X))
is a locally convex Biesz space. In view of proposition (12.1),
{[—u, u]0:u g C} is a neighbourhood-base at 0 for crg(X', X). In order to
verify this result, it is sufficient to show that each [ — и, ад]0 is order-
complete. Let {fT} be an increasing net in [—ад, ад]0 which is majorized
in Xf. Then/ = sup/T exists in X' because (X', C1) is order-complete.
It is clear that /(w) — sup/r(w) for any w e C; then fT converges to
/ with respect to a(X', X), and hence, from the tf(X', X)-closedness of
[—ад, ад]0, we have/ g [—ад, ад]0. Thus each [—ад, ад]° is order-complete,
consequently (X', С', аа(Х', X)) is locally order-complete.
13
COMPLETENESS EOR THE
DIEUDONNE TOPOLOGY
It is known from proposition (11.18) that a locally convex Riesz space
X can be embedded as a Riesz subspace of X" C\ (X7)k. The following
question naturally arises:
(1)	What condition is necessary and sufficient for the embedding to
preserve the supremum and infimum for infinite subsets of X ?
This also suggests the following two intimately related questions:
(2)	What condition on X (or X7) is necessary and sufficient for X
to be an /-ideal in X'1 П (X7)^ ?
(3)	What condition on X (or X7) is necessary and sufficient for X
to be a normal subspace of (X7)^ ?
This chapter is devoted to answering these and other related ques-
tions. We shall see in particular that the answer to (3) relies on the
converse of the Nakano theorem or, equivalently, on the completeness
property of the Dieudonne topology (see theorem (13.9)).
(13.1)	Theorem (Ando-Luxemburg-Zaanen). Let (X, C, TP) be a
locally convex Riesz space with the topological dual X'. Then the following
statements are equivalent:
(a)	X' cz Xj;
(b)	if wT J. 0 (r g D) then wT converges to 0 with respect to
(c)	the d-ideal in X" generated by X is an order completion of X;
(d)	the supremum and infimum of any subset of X are preserved under
the evaluation map x x of X into X" П (X7)^;
(e)	for any f e X', the Lideal Nf- {хеХ: I/КИ) = 0} is & normal
subspace of X;
(f)	each order-dense Lideal in X is iP-dense in X;
(g)	each ^-closed Lideal in X is a normal subspace of X.
Proof. In view of proposition (5.8), it is clear that (a) o(b).
(a) => (c): Let L be the /-ideal in X" П (X7)^ generated by X. In
order to verify that L is an order-completion of X, it is sufficient to
verify that, for each 0 < u" e L, there exist u0, и e X such that
COMPLETENESS FOR THE DIEUDONNE TOPOLOGY 161
0 < й0 < и" < u. Since X is a Riesz subspace of X" Л (А^')д, the
generated /-ideal L must be the order-convex hull of X.; hence и" < й
for some и e X. Since u" > 0, for this u, we can take a positive real
number A small enough such that (u" ~~ M)+ > 0. For such a A let
yA = (u" — Au). Then, vx, v~^ are all in X" П (JC/J. Let
A {y; GX'w^ip} 0}.
Since is a normal integral, A is a normal subspace of X'; also
X' = ЛфЛа by corollary (10.11). Since > 0, it follows that X' A A
and hence there exists 0 < h e Ad such that w‘|(A) >0. Since
v\ < и" < й, h(u) = u(h) > v^(h) > 0 for this h. Since (a) holds, apply
proposition (10.20) to obtain an element w eX with 0 < w < и such
that h[w) > о and y>(w) = 0 for all ip e with ip | A;
thus, in particular, ip(w) — 0 for all ip e A.
Then Aw g X and 0 < Aw < ад". To verify this, let 0 < ip e X/, and
suppose that ip — where 0 < ipt g A and 0 < ip2 g A". Then
(Ac6)(y>T) = Aip^w) = 0 < u"(yg).
Also, since | v^, we recall from corollary (10.19) that must vanish,
on Ad, and, in particular, that W(Vk) 0. Consequently
(u"~Aa>)(ip2} > (ад"-Аад)(^2) -
=	> 0.
This, together with an earlier established inequality, implies that
(ад" —Aco)(y>) = (ад" — Ac6)(yg) +(ад"— Aa>)(ip2) > 0,
therefore ад" > Aw, as claimed.
(c)	=> (d): Let {ag: r g D} be a subset of X and let x be the supremum
in X of the set {жг}. We assume without loss of generality that 0 e {ag}.
Let L be the /-ideal in X" generated by X. Then x is also the supremum
in L of {жг} since L is an order-completion of X. Finally, since L is an
/-ideal in X", x must in fact be the supremum of {ag} in X". To verify
this, let у be an element of X" majorizing {ag}. Then 0 у hx < x eL,
so у A x g L and is an upper bound of {ag} in L, hence x < x A у since
x is the supremum of {ag} in L, Therefore x < yy and this implies that
x is the supremum of {ag} in X”.
(d)	(e): Let / G X' and let 0 < uT f ад in X, where ur g Nf. Then, by
(d), we have 0 < ur } й in X", i.e.
sup uT{ip} = ад(у’)	(у’ £ X.', ip > 0).
162 COMPLETENESS FOR THE DIEUDONNL TOPOLOGY
In particular, since uT g Nf,
0 sup |/|(^r) - sup uT(|/|) = й(|/|) - |/| (<
showing that и eNf. Therefore Nf is a normal subspace of X.
(e)	=> (f): Let A be an order- dense /-ideal in X. If A is not ^-dense
in X, then there exists и e G such that и ф A. By the separation
theorem, there exists a g e X' such that
g(u) 0 and g(a) = 0 for all a in A.
Let/ - |^|. Then/eX',
f(u) = sup{|^(a;)|: |ж| < и] > |^(w)| > 0,
and
/(a) = 0 for all a in A.
Since A is an /-ideal, it follows that A c Nf. By (e), we then have
{A} cz Nf. However, since A is order-dense in X, {А} = X, that is,
X Nf> contrary to the fact that f(u) Ф 0. Therefore A must be
^-dense in X.
(f)	(g): Let В be a ^-closed /-ideal in X, and let {B} be the normal
subspace in X generated by B. Since X is Archimedean, by proposition
(10.9), we have that {B} = _Bdd and that BG>Bd is order-dense in X. By
(f) it follows that B®Bd is ^-dense in X. In particular, if 0 < и g {B},
then there exists a net {uT: r e D} in such that uT converges to и
with respect to For each т g j9, let wT = иф Ku. Then 0 < wT < u,
wT g B®Bd, and wT converges to и with respect to S? because the lattice
operations are ^-continuous. Since 0 < wT < и g {B} = Bdd, each
wT g J5dd. We shall show that wT g B. In fact, write
гот — w'-f-w" e B@Bd,
where w'r g В and w" g Bd. Then, since В c Bdd, w’T g Bdd,
w” = wr-w; g Bdd-Bdd c Bdd.
But we also have w" eBd; it follows that w" = 0. Hence wT = w'T g B.
Since и is the ,^-limit of {wT}, it follows that и G B. Since В is ^-closed,
и g B. This shows that {В} с: B. Consequently {В} = В and В is a
normal subspace of X.
Finally we show that (g) => (b). Clearly (b) is equivalent to the
following statement:
(b') If 0 < ur f и then uT converges to и with respect to
COMPLETENESS FOR THE DIEUDONNE TOPOLOGY .163
Suppose that 0 < ur f u, and let Ж be a ^-neighbourhood of 0
in X. Take a convex and solid ^-neighbourhood V of 0 such, that
V + V <= W. Choose a real number a with 0 < a < 1 such that
(1 —a)u e V. For each r in the index set D, let
vT — ит-—ам
and let A be the /-ideal in X generated by {vT:r e D}. Since vr | и — aw,
it is clear that u — au e {A}, hence и e {Л}. On the other hand, let A
denote the ^-closed of A. Then A is also an /-ideal and hence must be
a normal subspace in X by (g). Therefore A = {A} and wed, Take an
element w in A such that u—w e V. Then w+ e A and hence
0 < w+ < nvT
‘o
for some positive integer n and some т0 e I). Thus 0 < w+ < nvfo; and
since v~ is disjoint from vf, it must be disjoint from w+ and w1' A u.
ТТрпpp
A U — vfQ v (w+ A u) < и
and
0 <	< u — w+ А и a (u—w+) V 0 < [?Zh —W+| < \u~-w] g V.
Since V is solid, we see that гН e Г. Now, for all т > r0 in D, wc have
u—u. < u—uT ~ и—uT — ашА-аи
7	ro	To	1
= (1 — a)up(uTQ—~au)"— (uT — au)+ < (I — а)и + (иГо — au)"
- (l~a)u+v- e V+V c W.
This implies that uT converges to и with respect to thus (b') is true
and hence (b) is proved.
(1.3.2)	Corollary, For any Riesz space (X, C), the following
statements are equivalent:
(a)
(b)	for each f e Xb, the ^-ideal Nf = {x e X: \f\ (|^|) = 0} is a normal
subspace of X;
(c)	each a(X, X^-closed d-ideal in X is a normal subspace of X.
Proof. It is clear that cr(X, Xb) is a locally o-convex topology on X.
Let aQ(X, Xp) be the locally solid topology on X associated with
a(X, Xg). Since X^ is an /-ideal in Xb, it follows from corollary (6.5)
that Xb — (X, C, as(X, Xb))' and hence that an /-ideal in X is
од(Х, Xb)-closed if and only if it is и(Х, Xb)-closed. The result now
follows immediately from the preceding theorem.
164 COMPLETENESS FOR THE DJEUDONNl^ TOPOLOGY
(13.3)	Corollary. Let (X, C, be a locally o-convex Riesz space
with the topological dual X'. Then X' c: X£ if and only if ur converges to 0
with respect to 33 whenever u.. | 0.
Proof. Let be the locally solid topology on X associated with
3L Then (X, C, tPsy <= Хд if and only if uT converges to 0 with respect
to whenever ur | 0, in view of theorem (13.1). Therefore the result
follows from (c) of corollary (6.9) and corollary (6.5). We note that this
corollary can also be proved directly from proposition (5.8).
The proof of the following proposition is similar to that given in
theorem (13.1), and will be omitted.
(13.4)	Proposition. Let (X, C, 33} be a locally convex Riesz space.
Then the following statements are equivalent:
(a)	X' c Xj; '
(b)	if un j. 0 (n g N) then un converges to 0 with respect to £3;
(c)	the supremum and infimum of any countable subset of X are
preserved under the evaluation map x —> x of X into X" П (X')^.
We are now going to seek some necessary and sufficient condition
ensuring that X can be regarded as an /-ideal in X". Parts of the
following theorem, namely the equivalence of (a), (b), and (d), were
proved by Kawai (1957).
(13.5)	Theorem. Let (X, C, 33} be a locally convex Riesz space with
the topological dual X'. Then the following statements are equivalent:
(a)	X' c= Хд and (X, (7) is order-complete;
(b)	X is 3-isomorphic with an 3-ideal in X" under the evaluation map
xx of X into X" П (X'£;
(c)	the Dieudonne topology од(Х', X) on X' is consistent with the
duality {X, X');
(d)	each order-interval in X is cr(X, X'}-compact;
(e)	each order-bounded subset of X which is directed upwards has a
33-limit.
Furthermore, if (X, C, 33} satisfies one (and hence all} of (a)-(e) then
(X, C, £3} is locally order-complete.
Proof. Recall from corollary (6.5) that the topological dual of
(X', crs(X', X)) is the /-ideal in X" generated by X — X; thus the
COMPLETENESS FOR THE DIEUDONNlS TOPOLOGY 165
equivalence of (b) and (c) is clear. Proposition (12.1) tells us that
o,g(X,J X) is the topology of uniform convergence on order-intervals in.
X, hence the equivalence of (c) and (d) is just a restatement of the
Makey-Arens theorem. Further, by proposition (5.8), a directed
upwards net in X converges with respect to Sd if and only if it does with
respect to <r(X, X'); hence (d) implies (e). Therefore, to complete the
proof, it remains to verify that (a) => (b) and (e) => (a).
(a) => (b): Suppose (a) holds. In view of theorem (13.1), the «f-idcal L
in. X" generated by X is an order-completion of X. However, since X
is already order-complete, it follows that X = L.
(e) => (a): The order completeness of X is obvious since C is ^-closed.
On the other hand, if 0 < u? j 0 in X, for any fixed r0, let wq = inf{w„ uTo}.
Then 0 < wT < urQ 0 < wr < uT, so that wT j, 0, we then have that
0 < uTg —wT f uT(j in X. In view of the assumption (e), there exists
и eX such that urg—-wT converges to и with respect to IP. It follows
from proposition (2.1) that uTg—wT f u, and hence from адТо—| uTg
that и = uTg. Therefore wT converges to 0 with respect to F. Since uT
is decreasing, then uT converges to 0 with respect to and so, by
making use of theorem (13.1), X' cz X^.
Finally, let & be a neighbourhood-base at 0 in (X, ^) consisting of
^-closed solid sets. Then each number V in is order-complete. In
fact, let xT in V be such that xT f < у for some у e X. Since X is order-
complete, there exists x e X such that x = sup xx. It follows from
X' с Хц that xT converges to x with respect to and hence that
x g V. Therefore each V is an order-complete set in X, consequently
(X, C, is locally order-complete. This completes the entire proof
of this theorem.
(13.6)	Corollary. Let (X, C, IP) be a locally o-convex Riesz space
with the topological dual X'. Then the following statements are equivalent:
(a)	X' c: X^ and (X, C) is or der-complete;
(b)	each order-bounded subset of X which is directed upwards has a
iP-limit.
Furthermore, if denotes the locally solid topology associated with IP,
and if (X, C, IP) satisfies one of (a) and (b), then (X, C, Pf) is locally
order-complete.
Proof. It is known from corollary (6.5) that X' c X^ if and only if
(X, C, Ff)' <= X£. From corollary (6.9), ur converges to и with respect
to IP if and only if ur converges to и with respect to <^s whenever
12
166 COMPLETENESS FOR THE DIEUDONNE TOPOLOGY
Therefore the result follows immediately from the preceding
theorem.
(13.7)	Corollary. Let (X, (7) be a Riesz space, Xb total over X,
and suppose that Xnn is the set of all normal integrals on Xb, that is
Xnn = (Xb)b. Then the following statements are equivalent:
(a)	X is order-complete;
(b)	X is I-isomorphic with an I-ideal in Xm under the evaluation
map x -> x defined by x(f) = f(x) for all f e Xb.
Furthermore, if X satisfies one of (a) and (b) then the smallest normal
subspace of Xnn generated by X is precisely Xnn.
Proof. Since Xb is a normal subspace of Xb and certainly an /-ideal,
(X, C, oJs(X, Xb)) is a locally convex Riesz space with the topological
dual Xb, and so (b) implies (a) in view of theorem (13.5). Conversely, if
X is order-complete, then theorem (13.5) shows that X is an /-ideal in
(Xb, X))' and hence in (Xb)b. Note also that X = Xnn and that
Xnn is an /-ideal in (Xb)b; then X is an /-ideal in Xnn, and consequently
(a) implies (b). Finally, since Xb is order-complete and since X is an
/-ideal in Xnn, it follows from corollary (10.22) that {X} — (X0)", where
X° = {f e Xb:«(/) =f(x) — 0 V x e X}. Since Xb is total over X then
X° = {0} and so (X°)n = Xnn; consequently the smallest normal
subspace of Xnn generated by X is exactly Xnn.
(13.8)	Corollary. Let (X, C) be an order-com,plete Riesz space, and
let Xb be total over X. If r(X, Xb) denotes the Mackey topology on X. with
respect to the dual pair {X, Xb), then the following statements are equivalent
and each is equivalent to each of (a)-(g) of theorem (13.1):
(a)	(X, G, r(X, Xb)) is barrelled;
(b)	each ffiX, Xfif-closed I-ideal in X is a normal subspace of Xb;
(c)	if ur | и in X then u7 converges uniformly to и on each tr(Xb, X)-
bounded subset of Xb.
Proof. In view of corollary (13.7), X is an /-ideal in Xnn, then
(Xb, trs(Xb, X))' = X, and hence the solid hull of each <r(Xb, X)-
bounded subset of Xb is u(Xb, X)-bounded because the topologies
и(Хь, X) and Os(Xb, X) are consistent with the dual pair (X, Xb).
Consequently fi(X, Xb) is a locally solid topology on X. Now suppose
that (X, G, r(X, Xb)) is barrelled. Then each u(Xb, X)-bounded subset
of Xb is r(X, Xb)-equicontinuous, and hence r(X, Xb) is exactly
^(X, Xb); consequently (X, C, (fiX, Xb))' = Xb. By theorem (13.1),
COMPLETENESS FOR THE DIEUDONN-Й TOPOLOGY 167
each [fiX, X^)-closed /-ideal in X is a normal subspace of X, thus (a)
implies (b). If uT f и in X and if the statement (b) holds then, in view of
theorem (13.1), uT converges to и with respect to fi(X, X^); in other
words, ur converges uniformly to и on each cr(X^, X)-bounded subset of
X^. Finally suppose that the statement (c) holds. Then wr converges to
0 with respect to /3(X, X^) provided that wT | 0; hence, according to
theorem (13.1), (X, G, (fiX, X^))' c X^, and consequently /?(X, X^)
is consistent with the dual pair (X, X^). Therefore /?(X, X^) is exactly
т(Х, Хц), and so (X, C, r(X, X^)) is barrelled. This completes the proof.
We are now in a position to deal with the final question posed at the
beginning of this chapter, namely: What condition on X (or X') is
necessary and sufficient to ensure that X is a normal subspace of
(X')n ? We shall see that the answer to this question is equivalent to the
completeness of X for the Dieudonne topology. On the other hand, it is
known from example (11.15) that, in general, the converse of Nakano’s
theorem (11.14) fails; therefore it is interesting to find some classes of
locally convex Riesz spaces for which the converse of Nakano’s theorem
holds. The following result of Wong (1969c) shows that the completeness
of X for the Dieudonne topology ensures that the converse of Nakano’s
theorem holds.
(13.9)	Theorem. Let (X, G, HF} be a locally o-convex Riesz space
with the topological dual X'. Then the following statements are equivalent:
(a)	X' <= x£ and (X, G, HF} is boundedly order-complete;
(b)	(X, G, os(X, X7)) is both locally order-complete and boundedly
order-complete;
(с)	X is complete for (rs(X, X');
(d)	each positive a(X, X'}-bounded subset of X which is directed upwards
has a <fiX, X'}-limit;
(e)	each positive P-bounded subset of X which is directed upwards has
a HF-limit.
Furthermore, if (X, C, HF} satisfies one of (a)-(e) then X is complete
for HF Q.
Proof, (a) => (b): Since cr(X, X') and HP are consistent with the
duality (X, X'}, then the locally o-convex Riesz space (X, G, tf(X, X'))
is boundedly order-complete; in particular, (X, C, crs(X, X')) is
boundedly order-complete and X is order-complete. According to
corollary (13.6), (X, G, as(X, X')) is locally order-complete.
(b)	(c): Follows from Nakano’s theorem (11.14),
168 COMPLETENESS FOR THE DIEUDONN'fi TOPOLOGY
(c)	=> (d): Let {адг} be a positive a(X, X'/bounded subset of X which
is directed upwards. Then, for any/ g C, {/(wT)} is a bounded increasing
net of real numbers. It follows that {wj is а <r(X, X7)-Cauchy ne^ and
consequently a ^(X, X')-Cauchy net since u/\. There exists и e X such
that uT converges to и with respect to <rg(X, X') and a fortiori with
respect to u(X, X').
(d)	=> (e): Let {wr} be a positive ^-bounded subset of X which is
directed upwards. The {^r} is cr(X, X')-bounded, and so there exists
и e X such that uT converges to и with respect to u(X, X'); we conclude
from proposition (5.8) that uT converges to и with respect to
(e)	=> (a): It is clear that (X, C, is boundedly order-complete.
On the other hand, suppose ur j, 0. Without loss of generality we can
assume that uT < и for some и g 0. Then {u—uT} is a positive order-
bounded subset of X which is directed upwards, and. so {u—uT} is
^-bounded. There exists е X such that u~-uT converges to w0 with
respect to so, by the closedness of C, u—ur t u0, and hence и = u0
because u — uT f u; consequently uT converges to 0 with respect to SP.
In view of corollary (13.3), X' <= Xb.
To see the final assertion of the theorem, it suffices to remark that
> o's(X, X') and both topologies admit the same topological dual.
The following corollary is a dual result of theorem (13.9).
(13.10)	Corollary. Let (X, C, be a locally convex Riesz space
with the topological dual X'. Then the following statements are equivalent:
(a)	(X', С', a(X', Xf) is boundedly order-complete;
(b)	(X', &, ug(X', X)) is both locally order-complete and boundedly
or der-complete;
(с)	X' is complete for oQ(X', X);
(d)	each positive o(X.', X)-bounded subset of X' which is directed
upwards has а a(X\ Xflimit;
(e)	X' is a normal subspace of X\
Proof. It should be noted that (Xf, Cr, g(X', Xf) is a locally
o-convex Riesz space for which the topological dual of (X', C', a(XX))
is contained in (X')b, and then the equivalence of (a), (b), (c), and (d)
follows immediately from theorem (13.9). It remains to verify the
implications (d) ~-=> (e) => (a).
(d)	(e): Let {fT} be a positive and directed upwards subset of X',
and let / in Xb be such that fr /, The f(u) ~ sup/T(^) for any и e C,
and so {fr} is o(X', X)-bounded. By the statement (d), there exists
g e X' such that fr converges to g with respect to g(X', X), so fr f g
COMPLETENESS FOR THE DIEUDONNE TOPOLOGY 169
since C is a(X', X)-closcd, hence/ = g and/ g X'; consequently X' is a
normal subspace of Уь.
(e)	=> (a): Let {fT:r e D} be a о^У', y/bounded and monotonically
increasing net in X'. Then, for each и e C, {ffu): r e D} is a bounded
and increasing net of real numbers; hence the net converges to a real
number, denoted by/(ад). Then ад ->/(ад) is a well-defined, additive, and
positively homogeneous functional on C. This function / can be extended
to be defined on the whole X. Then/ e Уь and fr | /, and it follows from
(e) that / e X'. Thus (a) holds.
The completion of (X', ав(Х', X)) coincides exactly with the normal
subspace of Уь generated by X', as the following result (due to
Peressini (1967)) shows.
(13.11)	Corollary, Let (X, O, HP) be a locally convex Riesz space
with the topological dual X'. Then the completion of X' for aQ(X', X) is
the normal subspace of Хъ generated by X'.
Proof. Let Y be the normal subspace of Уь generated by X'. Then
о"8(У, У) is a locally solid topology on X such that У is the topological
dual of (У, C, os(X, У)), so, by corollary (13.10), У is complete for
cts(Y, X). It is clear that for each x e X, the linear functional x,
defined by	for аЦ
is a normal integral on У; hence (У, ^(У, .У))' Уь and consequently
(У, os(y, X))' c y*. Furthermore, X' is an У-ideal in У, it follows from
theorem (13.1) that X' is dense in У with respect to <т8(У, X). Since the
topology &s(X', X) on X' is the relative topology induced by и8(У, X),
we conclude that У is the completion of X' for ogpC, X).
(13.12)	Corollary. Let (X, C} HP) be a locally convex Riesz space
with the topological dual X'. Then the following statements are equivalent
and each is equivalent to each of (a), (b), (d), and (e) of theorem (13.9):
(a)	X is a normal subspace of (У')ь;
(b)	X = (X')J;
(с)	X is complete for cts(X, X').
Proof, (a) => (b): Since
I» = {/g Г:ж(/) - /(ж) - 0 for all же!}- {0},
it follows from corollary (10.23) that {Z} — (У')ь, and hence from
statement (a) that X — (У')^-
170 COMPLETENESS FOR THE DIEUDONNE TOPOLOGY
(b)	(c): Note that (X')q is a normal subspace of (X')b. It
follows from the statement (b) that X is the topological dual of
(X', O', as(X', X)), and hence from corollary (13.10) that X is complete
for <rs(X, X').
(c)^>(a): In view of theorems (13.9) and (13.5), X is an /-ideal
in (X')b, it follows from corollary (6.5) that X ~ (X', С', o^X', X))',
and hence from corollary (13.10) that X is a normal subspace of (X')b.
A Riesz space (X, C) is said to be perfect if Xb is total over X and
X = Xnn. Using theorem (13.9) and corollary (13.12), we give some
characterizations of perfectness in terms of topological completeness.
The equivalence of (a) and (e) in the following corollary was given by
Nakano (1950a).
(13.13)	Corollary. For any Riesz space (X, C), if Xb is total over
X, then the following statements are equivalent:
(a)	(X, C) is perfect;
(b)	X is a normal subspace of Xnn;
(с)	X is complete for us(X, Xb);
(d)	each positive <r(X, X^-bounded subset of X which is directed
upwards has a u(X, X^f-limit;
(e)	(X, C, <r(X, Xb)) is boundedly order-complete;
(f)	(X, (7, <ts(X, Xb)) is both locally order-complete and boundedly
order-complete.
Proof. Give X the topology us(X, Xb). Then the equivalence of
(a)-(c) is a restatement of corollary (13.12) and the equivalence of
(c)-(f) follows from theorem (13.9).
(13.14)	Corollary. Let (X, C) be an order-complete Riesz space for
which Xb is total over X. Then Xnn is the completion of X for ffg(X, Xb).
Proof. By making use of corollary (13.7), X can be regarded as an
/-ideal in XnI1, hence, by corollary (6.5), X is the topological dual of the
locally convex Riesz space (Xb, ug(Xb, X)). In view of corollary (13.11),
the completion of X for us(X, Xb) is the normal subspace of Xnn
generated by X, consequently, by corollary (13.7), Xnn is exactly the
completion of X for crs(X, Xb).
The completeness of X' for trg(X', X) ensures that ug(X', X) and
^b(X') have the same topologically bounded sets as shown by the
following result.
COMPLETENESS FOR THE DIEUDONNP TOPOLOGY 171
(13.15)	Proposition. Let (JC, O', ^) be a locally convex Riesz space
with the topological dual X', and let lPb(X') be the order-bound topology on
X'. Suppose that X' is a normal subspace of Xb. Then, for any subset В
of X', the following statements are equivalent:
(a)	В is &b(X'}-bounded;
(b)	В is fl(X', X)-bounded;
(с)	В is aQ(X', Xybounded.
Proof. It is clear that trg(X', J?) is coarser than fi(X', X) and that
fi(X', X) is coarser than ^b(X'). Then the implications (a) => (b) => (c)
are obvious. It remains to verify that (c) implies (a). Without loss of
generality we can assume that В is a convex, solid, and о8(Х', X)-
bounded subset of X'. If j? is not ^b(X')-bounded, there exists a convex
and solid &b(X')-neighbourhood V of 0 in X' such that the assertion
В c 2ZnV is false for each natural number n. For any n, there exists
fn in X' such that \fn\ e В and \fn\ ф 22W F. Let gn = 2~* \fk\. Then
{</„} is а a&{X', J?)-Cauchy sequence in X' because В is convex and
crs(X', X)-bounded. Since Xr is a normal subspace of JVb, by corollary
(13.10) X' is complete for ^(JC7, X), and so there exists g g X' such
that gn converges to g with respect to crs(X', X). Since 0 < gn] and
since C is &8(Х', X)-closed, it follows from proposition (2.1) that
g = sup gn. Since V is a tPb(X^-neighbourhood of 0 in X', there exists
a natural number к such that [0, <?] <= 2fcP. We conclude from
0 < 2~~7с|/й| < gk < g that \fk\ e 22fcP, contradicting the fact that
\fn\ ф 22nV. Therefore В is ^b(X')-bounded, and the proof is complete.
We now present a dual result to proposition (13.15) as follows.
(13.16)	Corollary. Let (X, 0, SR) be a locally convex Riesz space
with the topological dual X', and let /Pb be the order-bound topology on X.
Suppose that X is a normal subspace of (X7)b. Then, for any subset A of
X, the following statements are equivalent:
(a)	A is ^-bounded;
(b)	A is fi(X, X')-bounded;
(c)	A is o(X, X’ybounded.
Proof. Since X' is an Aideal in Wb, then а(Х, X') and cfs(X X')
are consistent with the duality (X, X1), therefore a subset A of X is
o{X, X')-bounded if and only if it is trg(X7, X)-bounded. On the other
hand, since X is a normal subspace of (X7)b, X must be the topological
172 COMPLETENESS FOR THE DIEUDONNP TOPOLOGY
dual of the locally convex Riesz space (X', C, <rs(X', Xf), and so the
result follows from, proposition (13.15).
(13.17)	Proposition. Let (X, C, ^) be a locally convex Riesz space
with the topological dual X'. If X' is complete for в&(Х', X) then it is
complete for @(X', X).
Proof. Let {fr’.T e I)} be a fl(X', X)-Cauchy net in X'. Then it must
be а сгд(Х', X)-Cauchy net because а&(Х', X) is clearly coarser than
{3{X\X). Since X' is complete for сг8(Хл, X), the net {fp.r g D}
(Г$(Х', .X)~converges in X', say to f. (Tn particular, {ft} converges to f
with respect to a(X', X).) We now show that fT converges to f with
respect to /3(Х\ X). Let A be any solid, convex, and a(X, X')-bounded
subset of X. There exists r0 e D such that fT—fTo e whenever
т r0, where is the polar of A taken in X'. Since A0 is <r(X', X)-
closed and since fT converges to/with respect to o'(X’, X), it follows that
/~~/Го e |.4°, and hence that
f-fr =	+ e M° + M« = Л»
whenever r r0. Therefore fT converges to/with, respect to /3(X', X);
consequently X' is complete for P(X', X). This completes the proof.
It will be seen from theorems (11.16), (18.17) and example (15.9)
that the converse of the above result is not true in general.
The following corollary, which should be compared with theorem
(13.9), is a dual result to proposition (13.17).
(13.18)	Corollary. Let (X, C, IP) be a locally convex Riesz space. If
X is complete for crs(X, X1) then it is complete for {3(X, X') and also for SP.
Proof. It is known from corollaries (13.12) and (6.5) that X is the
topological dual of the locally convex Riesz, space (X', O', а&(Х', X));
it follows from proposition (13.17) that X is complete for /?(Х, X').
Finally the completeness of X for P follows from a well-known result.
Of course, crs(X, is the coarsest locally solid topology on X
consistent with the dual pair (X, X') while /?(Х, X') is not consistent
with (X, X'), therefore the preceding corollary is of particular interest.
14
REFLEXIVITY FOR LOCALLY
CONVEX RIESZ SPACES
This chapter is concerned with a study of the interrelation between
reflexivity and order. It is known from Komura (1964) that there are
reflexive locally convex spaces which are not topologically complete.
But for locally convex Riesz spaces completeness is a consequence of
semi-reflexivity as shown by the following result due to Wong (1969c).
(14.1)	Theorem. For any locally convex Riesz space (X, C, the
following statements are equivalent:
(a)	X is semi-reflexive;
(b)	(X, C, tfs(X, X')) is both locally and boundedly order-complete, and
X" c: (X')£;
(с)	X is complete for аа(Х, X'), and X" <= (X')^;
(d)	each positive a(X, X'fbounded subset of X which is directed
upwards has a <r(X, X'flimit, and X" <= (X')^;
(e)	each positive ^-bounded subset of X which is directed upwards
has a Sd-limit, and X" <= (X')^;
(f)	(X, C, ^) is boundedly order-complete, X' c: X^ and X" <= (X')^;
(g)	X is a normal subspace of (X')^ and X" c: (X')„.
Furthermore, if (X, C, tF) satisfies one of (a)-(g) then X is complete for
SP and also for [3(X, X').
Proof. We have only to verify the implications (a)	(b)	(g)
(a). Other equivalences follow from theorem (13.9). Observe that X is
always a Riesz subspace of (X')^. It follows from the semi-reflexivity of
X that X" = X с (Х')д and hence from proposition (12.10) that
(X, С, a&(X, X')) is locally order-complete. Let {wT} be a од(Х, X')-
bounded subset of X which is directed upwards. Since <Tg(X, X') is
consistent with the duality (X, X'), then {/q} is ^-bounded and hence
the polar ({4q})° of the set {адг}, taken in X', is a /5(X', X)-neighbourhood
of 0 in X/. According to the Alaoglu-Bourbaki theorem, {uT} is relatively
cr(X, X')-compact, and hence {адг} has a n(X, X')-cluster point и in X,
174 REFLEXIVITY FOR LOCALLY CONVEX RIESZ SPACES
consequently ur u. Therefore, (X, C, trs(X, X')) is boundedly order-
complete; this proves the implication (a) => (b). The implication
(b) => (g) is a consequence of corollary (13.12). To see the implication
(g)^> (a), let
X° = {h g X': h(x) = 0 for all x in X)
and	(I? = {/ g X': <£(/) = 0 for all ф in X"}.
Then X° = {0}	(Xz/)°; since X and X" are /’-ideals in (Xz)b it follows
from corollary (10.23) that {X} 2 {X"}. By (g), X = {X}. Conse-
quently X 3 {X"} 3 X" and so X = X". This shows that X is semi-
reflexive and proves the implication (g) => (a).
The final assertion that X is complete for & and also for /?(X, Xz)
is a consequence of corollary (13.18), therefore the proof is complete.
Remark. Since X" is the topological dual of the locally convex
Riesz space (X', G', /3(XZ, X)), then the condition that X" cz (Xz)b in
theorem. (14.1) can be replaced by any one of the equivalent properties
of theorem (13.1).
Since semi-reflexivity and reflexivity are equivalent for normed
vector spaces, we obtain an immediate consequence of the preceding
result as follows.
(.14.2) Corollary (Ogasawara). Let (X, G, ||. ||) be a normed Riesz
space. Then the following statements are equivalent and each is equivalent
to each of (b)-(g) of theorem (14.1);
(i) (X, G, ||, ||) is reflexive;
(ii) (X, G, || -1|) is boundedly order-complete, X' <= Xb and X" (Xz)b.
(14.3)	Corollary. Let X be a locally convex topology on a Riesz
space (X, O') such that the topological dual X' of (X, X) is an dideal in
Хъ. Then the following statements are equivalent and each is equivalent to
each of (b)-(g) of theorem (14.1);
(a)	X is semi-reflexive;
(b)	X is complete for <rs(X, X'), and X” <= (X')b,
Proof. Since X' is an /-ideal in Xb, it follows that u(X, X') is a
locally o-convex topology on X, and hence that us(X, X') and X are
consistent with the duality (X, X'). On the other hand, since semi-
reflexivity and boundedly order-completeness depend only on the
duality, the result now follows from theorem (14.1).
REFLEXIVITY FOR LOCALLY CONVEX RIESZ SPACES 175
(14.4)	Corollary. Let (X, G) be a Riesz space, Xb total over (X, G),
and let LT be a locally convex topology (not necessarily locally solid) on X
which is consistent with the duality (X, Xb). Then the following statements
are equivalent and each is equivalent to (b)-(g) of theorem (14.1);
(a)	X is semi-reflexive;
(b)	Xb = Xb, X is perfect, and (Хъ0, /3(Х*, X))' <= Xnil.
Proof. (a)=>(b): Let us denote the topology o's(X, Xb). Then
(X, 0's)' = Xb. By (a), (X} (/g) is semi-reflexive. By the implication
(a) => (f) of theorem (14,1), (X, G, 0's) is boundedly order-complete,
Xb <= Xb and (Xb, ^(Xb, Xb))' c (Xb)b. Consequently, Xb - Xb and
(Xb, Д(ХЬ, X))' <== (Xb)b = Xnn. Further, by (c) of theorem (14.1), X is
o's(X, Xb)-complete and it follows from corollary (13.13) that X is
perfect.
(b)	(a): By the first two properties stated in (b), it follows from
corollary (13.13) that (X, og) is complete. The third property stated in
(b) implies that (Xb, /?(ХЬ, X))' cz (Xb)n. By the implication (c) => (a)
of theorem (14.1), we conclude that X is semi-reflexive.
(14.5)	Corollary, Let (X, G) be a Riesz space, and let X be a locally
convex topology on X such that the topological dual X' of (X, X) is an
/-ideal in X)\ If (X, X) is reflexive then X is a locally solid topology,
consequently X is complete for o's(X, X') and also for X.
Proof. It is known that cr(X, X') is a locally o-convex topology on
X. Since X' is an /-ideal in Xb, it follows from corollary (6.5) that
(X, G, us(X, X'))' = X/, andhencefrom theorem (11.16) that the strong
dual (X', C', fi(X', X)) of (X, X) is a locally convex Riesz space.
Since (X, G, X) is reflexive, (X, С, X) is the strong dual of (Xz, G',
/3(Х', X)) so, by theorem (11.16) again, (X, G, X) is a locally
convex Riesz space, consequently X is a locally solid topology. Finally,
in view of theorem (14.1), X is complete for Og(X, %') an(^ als0
for X.
Theorem (14.1) leads to the following characterizations of reflexivity
for locally convex Riesz spaces.
(14.6)	Theorem. For any locally convex Riesz space (X, G, X) with
the topological dual X’, the reflexivity of (X, С, X) is equivalent to the
following three conditions:
176 REFLEXIVITY FOR LOCALLY CONVEX RIESZ SPACES
(a)	is the Mackey topology т(Х, X'};
(b)	if ur J, 0 in X then uT converges to 0 with respect to fi(X, X'), and if
fr J, 0 in X' thenfT converges to 0 with respect to [}(X', X);
(с)	(X, С, а(Х, X')) and (X', C, &(X', X)) are boundedly order-
complete.
(Remark. The statement that if uT | 0 in X then ur converges to 0
with respect to fi(X, X') in theorem (14.6)(b) can be replaced by one of
(a)-(g) of theorem (13.1); similarly for the statement that if/r | 0 in
X' thcn/r converges to 0 with respect to f3(X', X) in theorem (14.6)(b).
Also the statement that (X, С, a(X, X')) is boundedly order-complete
in theorem (14.6)(c) is equivalent to each of (a)-(e) of theorem (13.9),
and there is a similar equivalence for the statement that (X', C'cfX', X))
is boundedly order-complete in theorem (14.6)(c).)
Proof. It is well known that (X, C, tP) is reflexive if and only if
is т(Х, X') and both (X, С, a(X, X')) and (X', O', /3(Х', X)) are semi-
reflexive. Then the necessity follows from theorems (14.1), (13.5), and
(13.1) and & — (3(X, X'). Conversely, if uT 0 in X, then uT must
converge to 0 with respect to SP since & is, in general, coarser than
fi(X, X') and hence (X, О, а(Х, X')) is semi-reflexive in view of
theorems (13.9) and (13.1); on the other hand, because of corollary
(13.10) and the fact that X" - X £ (X')b, (X', O', ft(X', X)) must be
semi-reflexive. Therefore (X, C, ^) is reflexive, and the proof is
complete.
Let (E1; Ef) be a dual pair. Recall that Et is said to be semi-reflexive
with respect to E2 if Ek — (E2> fl(E2> Ef))' and that the dual pair (E1} Ef)
is said to be reflexive if Er is semi-reflexive with respect to E2 and E2 is
semi-reflexive with respect to E3. For further information about the
reflexivity of a dual pair, we refer the reader to Kothe (1969).
(14.7)	Proposition. For any Riesz space (X, C), if Xb is total over
X then X is semi-reflexive with respect to Xb if and only if the following
two conditions hold:
(a)	X is perfect;
(b)	iffr J, 0 in Xb thenfT converges to 0 with respect to /3(Xb, X).
Proof. Since Xb is a normal subspace of Xb, it follows from
corollary (6.5) that Xb = (X, 0, crs(X, Xb))'. Therefore the semi-
reflexivity of X with respect to XJ( is equivalent to the semi-reflexivity
REFLEXIVITY FOR LOCALLY CONVEX RIESZ SPACES 177
of (X, C, o*s(X, X^)), and this is the case if and only if X is complete for
tfs(X, 1ц) and (X^, fi(X^ X))' c consequently the result now
follows from theorem (13.1) and corollary (13.13).
Remark. The condition (a) in proposition (14.7) can be replaced by
one of (a)-(f) of corollary (13.13), and the condition (b) in proposition
(14.7) is equivalent to each of (a)-(g) of theorem (13.1).
(14.8)	Corollary. For any Riesz space (X, C), if X^ is total over X,
then the dual pair {X, X^) is reflexive if and only if the following two
conditions hold:
(a)	X is perfect;
(b)	if ur [ 0 in X then ur converges to 0 with respect to fi(X, X^) and if
fT I 0 in Хд thenfT converges to 0 with respect to fi(X\, X).
Proof. The necessity is obvious. To prove the sufficiency, we first
note that the perfectness of X^ is a direct consequence of the perfectness
of X and so, by proposition (14.7), X^ is also semi-reflexive with respect
to X. Consequently the dual pair (X, X£) is reflexive.
15
BORNOLOGICAL AND
INFRABARRELLED RIESZ SPACES
The remaining four chapters of this book are devoted to studying some
important classes of locally convex Riesz spaces and their relationship.
Let (E, ^) be a locally convex space. Recall that (E, tP) is bornological
if each convex circled subset of E which absorbs all ^-bounded subsets
of E is a ^-neighbourhood of 0, (X, is barrelled if each barrel in E
is a ^-neighbourhood of 0, and that (E, tP) is infrabarrelled if each
barrel in E which absorbs all ^-bounded subsets of E is a ^-neighbour-
hood of 0. Clearly bornological spaces and barrelled spaces are infra-
barrelled. A locally convex Riesz space (X, C, ^) is called a bornological
Riesz space if (X, ^) is bornological; it is called an infrabarrelled Riesz
space if (X, ^) is infrabarrelled. It is known from proposition (11.2)(c)
that if SP is a locally solid topology on (X, C) then the solid hull of each
^-bounded subset of X is ^-bounded. Therefore the question arises
naturally whether the converse of proposition (11.2)(c) is true, namely:
If & is a locally convex topology on (X, (7) such that the solid hull of
each ^-bounded set in E is ^-bounded, is & a locally solid topology?
Example (3.15) shows that the answer to the above question is, in
general, negative. However, we have the following theorem.
(15.1)	Theorem. Let 3? be a locally convex topology on (X, C}such that
(X, ^) is bornological. If the solid hull of each Abounded subset of X
is ^-bounded, then SP is a locally solid topology.
Proof. We first note that each order-bounded subset of X is
^-bounded because C is generating and the solid hull of each и e C is
[— u, u]. Let be a neighbourhood-base at 0 for tP consisting of convex
and circled sets in X, and suppose that
= {sk(F):Fe^Z},
where sk(F) is the solid kernel of V. In view of proposition (10.5)(d),
each sk(F) absorbs all order-bounded subsets of X, and is certainly
absorbing. On the other hand, it is easily seen that sk(ZF) = A sk(F)
INFRABARRELLED RIESZ SPACES
179
л v’ sk(F) n sk(lF) - sk(F n W),
and that sk(JF) £ sk(F) whenever W £ V. There exists a unique
locally solid topology, ЗГ say, on X such that is a neighbourhood-base
at 0 for ЗГ. Clearly ST is the greatest lower bound of all locally solid
topologies on X which are liner than 33.
We now claim that any ^-bounded subset of X is ^-bounded. In
fact, let A be a ^-bounded subset of X. Then A is contained in a solid
^-bounded subset В of X. For this B, there exists F in such that
В £ nV for some natural number n. Since В is solid and sk(F) is the
largest solid subset of F under the inclusion, it follows that
A £ В c n sk(F) for this n, This shows that A is ^"-bounded.
Finally, since (X, 33) is bornological or, equivalently, no strictly finer
locally convex topology on X has the same topologically bounded sets,
we conclude that 3? and 3F coincide, and hence that is locally solid.
This completes the proof.
(15.2)	Corollary. Let 33 be a locally convex topology on {X, C), and
let Xtb be the topologically bounded dual of (X, ^). If the solid hull of each
^-bounded subset of X is 33-bounded then the Mackey topology t(X, Xtb)
is a locally solid topology on X. In particular, the bornological space
associated with a locally convex Biesz space is always a locally convex
Biesz space.
Proof. Since the locally convex topologies 33 and t(X, Xtb) have the
same topologically bounded sets in X, it follows that the solid hull of
each r(X, Xtb)-bounded subset of X is r(X, Xtb)-bounded, and hence
the result follows from theorem (15.1),
A semi-norm p on an ordered topological vector space is said to be
topologically bounded if it sends every topologically bounded set to a
bounded set; it is said to be order-bounded if it sends every order-bounded
set to a bounded set. In terms of the order structure, we are able to give
some characterizations of bornological Riesz spaces as follows.
(15.3)	Proposition. For any locally convex B,iesz space (X, G, 33)
the following statements are equivalent:
(a)	(X, G, 33) is bornological;
(b)	each decomposable circled convex set in X which absorbs all 33-
bounded subsets of X is a 33-neighbourhood of 0;
180
BORNOLOGICAL AND
(c)	each circled o-convex set in X which absorbs all tP-bounded subsets
of X is a tP-neighbourhood of 0;
(d)	each convex solid set in X which absorbs all ^-bounded subsets
of X is a ^-neighbourhood of 0;
(e)	each topologically bounded, Riesz semi-norm p on X is -con-
tinuous.
(Remark. Before giving the proof of this proposition, we note that,
since О is a strict ^-cone in (X, tP), a subset В of X absorbs all
^-bounded subsets of X if and only if it absorbs all positive ^-bounded
sets in X, therefore, in the statements (b), (c), and (d) of the proposition,
‘^-bounded subsets of X’ may be replaced by ‘positive ^-bounded
subsets of X’.)
Proof. The implication (a)	(b) is trivial. It is noted that for any
convex circled set V in X, co( — (V П С) U (V П (7)) is a decompos-
able subset of X for which
co(-(F Л С) и (V n C)) s V,
aild C n {co( —(F П С) и (V Л О))} - V П C.
It then follows that (b) implies (c). Suppose that V is a convex and solid
subset of X and that W is the order-convex hull of B, i.e.
W = (BXC) n (B-C),
where В - co(-(F n (7) и (F n (7)). Then IFnC'-FnC'-
В П (7, W £ 2F and IF is o-convex and circled; consequently Fabsorbs
all ^-bounded subsets of X if and only if IF absorbs all ^-bounded
subsets of X since C is a strict ^-cone in (X, &); consequently (c)
implies (d). Since a subset F of X absorbs each ^-bounded set in X
if and only if the solid kernel sk( F) of F absorbs all ^-bounded sets in
X, it follows that (d) implies (a). Observe that the implication (d) => (e)
follows from the fact that W = {x eX:p(x) < 1} is a convex solid
subset of X which absorbs all ^-bounded subsets of X. Finally, since
the gauge of a convex, solid, and absorbing subset of X must be a
Riesz semi-norm, then (e) implies (d). Therefore the proof is complete.
It is well known that subspaces of a bornological, locally convex
space are, in general, not bornological with respect to the relative
topology. But the following result shows that any Z-ideal in a borno-
logical Riesz space must be bornological with respect to the relative
topology.
INFRABARRELLED RIESZ SPACES
181
(15.4)	Corollaby. Any d-ideal in a bornological Riesz space is
bornological with respect to the relative topology.
Proof. Let J be an /-ideal in a bornological Riesz space (X, C, ,P),
and let V be a convex solid subset of J which absorbs all ^-bounded
subsets of J. In view of the preceding proposition, we have only to show
that V is a neighbourhood of 0 in the subspace J.
Let
U - {xeX:y e V whenever 0 < у < |ж| and у e J}.
Then U is a convex solid set in X such that U П J = V. Further, U
absorbs all ^-bounded subsets В of X. To verify this, we may assume
without loss of generality that В is solid. Now, if В Ф nU for each
positive integer n, then there exists bnG В such that bn fnU. Hence
there is yn e J with 0 < yn < \bn^n\ such that yn ф V. Since В is solid,
the set {nyn}f^ is contained in В П J, but fails to be absorbed by V,
contrary to our assumption on V. Therefore U must absorb all tP-
bounded sets in X. Since X is bornological, it follows that U is a
^-neighbourhood of 0 in X. Since V — TJ C\ J, V is a neighbourhood
of 0 in J with respect to the relative topology.
Similar to the case of bornological spaces, we shall show that each
bornological Riesz space is the inductive limit of a family of normed
Riesz spaces with respect to /-homomorphisms.
(15.5)	Proposition. Every bornological Riesz space (X, C, ^) is the
inductive limit of a family of normed Riesz spaces (and of В-lattices if X is
guasi-complete for IP} with respect to ^-homomorphisms; the cardinality
of this family can be so chosen as the cardinality of any fundamental
system of tP-bounded sets in X.
Proof. Let Ad denote the family of all ^-bounded, convex, solid,
and ^-closed sets in X. According to propositions (11.2) and (11.3), Ad
is a fundamental system of the family consisting of all ^-bounded
subsets of X. For each В g Ad, suppose that XB = (J nB and that
n
pB is the gauge of В on XB. Since В is solid and convex, XB is an /-ideal
in X, and pB is a Riesz semi-norm on XB. Observe that the relative
topology on XB induced by A? is coarser than the semi-norm topology
pB\ it follows that (XB, CB, pB) is a normed Riesz space, where
CB = C Oi XB. Obviously each injection jB of XB into X is an /-
homomorphism, and X — U {XB-.B g Ad}. By a well-known result (cf.
13
182
BORNOLOGICAL AND
Schaefer (1966)), & is the inductive topology with respect to the
families {(XB, pB\.B e SS} and {jB’. В e ^}, therefore (X, G, ^) is the
inductive limit of a family {(XB, GB, pB): В e of normed Riesz
spaces with respect to /-homomorphisms {jB:B e F§}. This completes
the proof.
A subset V of a locally convex Riesz space (X, C, is called a
solid barrel if it is ^-closed, convex, solid, and absorbing. It is known
from proposition (11.3) that a barreJ V absorbs all ^-bounded subsets
of X if and only if the solid barrel sk( F) absorbs all ^-bounded sets
in X. This deduces the following characterization of infrabarrelled Riesz
spaces.
(15.6)	Proposition. For any locally convex Biesz space (X, C, 01),
the following statements are equivalent:
(a)	(X, C, is infrabarrelled;
(b)	each solid barrel in X which absorbs all positive ^-bounded subsets
of X is a ^-neighbourhood of 0;
(c)	each solid barrel in X which absorbs all tP-bounded subsets of X is a
SP-neighbourhood of 0;
(d)	each topologically bounded lower semi-continuous Biesz semi-norm
on X is IP-continuous;
(e)	each positive /?(X', Xfbounded subset of X' is S?-equicontinuous.
The proof of this result is similar to that given in proposition (15.3)
and will be left to the reader.
(15.7)	Corollary. Any P-ideal in an infrabarrelled Biesz space is
infrabarrelled with respect to the relative topology.
Proof. Let J be an /-ideal in an infrabarrelled Riesz space (X, G,
and let Г be a solid barrel in the subspace J which absorbs all
bounded subsets of J. Let
U — {x e X:y e V whenever 0 < у < ja?| and у eJ}.
As in the proof of corollary (15.4), U is a convex solid set in X such that
F = V. П J and U absorbs all ^-bounded subsets of X. Further, U is
^-closed. In fact, let x e U and suppose that is a net in U which is
^-convergent to x. Let у eJ be such that 0 < у < ]ж|. Then
0 < у Л |ят| < |жт| g U and each у л |жт| е J since J is a /-ideal. By the
definition of U, it follows that у Л |жт| e V. By the continuity of the
INFRABARRELLED RIESZ SPACES
183
lattice operations, we conclude that у К [жг| converges to у Л |«| = у,
hence у е V = V. This shows that х g U and hence that U is ^-closed.
Therefore U is a solid barrel in X which absorbs all ^-bounded sets
in X. Hence U is a ^-neighbourhood of 0 in X. Consequently,
V = U И J is a neighbourhood of 0 in the subspace J. This implies that
J is infrabarrelled.
It should be noted that the quotient Riesz space and the locally
convex direct sum of infrabarrelled Riesz spaces are infrabarrclled;
the quotient Riesz space and the locally convex direct sum of borno-
logical Riesz spaces are bornological.
Combining propositions (15.3) and (15.6) we have the following result.
(15.8)	Proposition. Let (X, (7, ^) be an infrabarrelled Riesz space.
Then the following statements are equivalent:
(a)	(X, G, is bornological;
(b)	each topologically bounded monotonic semi-norm on X is lower
semi-continuous;
(c)	each topologically bounded Riesz semi-norm on X is lower semi-
continuous.
It is known from proposition (7.1) that if is the order-bound
topology on (X, G) then (X, C, ^b) is bornological. The question
naturally arises whether the topology on a bornological Riesz space
is necessarily the order-bound topology. The following example gives a
negative answer.
(15.9)	Example. Let X be the Banach space of all continuous real-
valued functions on [0, 1] with, the supremum norm defined by
||«j| = max{|ir(i)j :le[0, 1]} and let
G = {x g X ;x(t) > 0 for all t e [0, 1]}.
Then (X, G, ||. ||) is a Banach lattice. Suppose that J is the vector
subspace consisting of all elements ж in X which vanish in a neighbour-
hood (depending on x) of t = 0, and that Cr = G П J. Then J is an
/-ideal in X, and (J, Cr, ||. [|) is a normed Riesz space and a fortiori a
bornological Riesz space. Let
V — {x eJ-.n |.r(rr<)| < 1 for all natural numbers n > 1}.
Then V has the following properties:
(a)	V is not a ||. ||-neighbourhood of 0 in J;
(b)	V is a solid barrel in (J, Cr, ||. ||).
184
BORNOLOGICAL AND
In fact, in order to verify the assertion (a), it is sufficient to show that
for any natural number R > 1, there exists xR in J with ||жл|| < 1/7?
such that xR ф V; it then follows that 0 is not an interior point of 7;
consequently V is not a ||. ||-neighbourhood of 0. Consider two closed
disjoint subsets [0, 1/47?] and [l/(7? + l), 1] of [0, 1]; by Urysohn’s
lemma, there exists a continuous real-valued function, xR say, on [0, 1]
with range in [0, 1/7?] such, that
xR(t) = 0 (t g [0, 1/47?]) and xR(t) --= 1/7? (t e {1/(7? +1), 1]).
Clearly xR gJ and ||+j| < 1/7?. On the other hand, since
/ I 1	1
ж+7?+1/ = R > 7?+“l ’
it follows that xR ф V. This proves our assertion (a). It is easily seen that
V is convex and ]. |[-closed. Let x be in J, and lot with 0 < oq. < 1
be such that x(t) — 0 for all t g [0, аж]. If we choose 2 = ||ж||-1, then
2.x eV, and so V is absorbing. It remains to show that V is solid. Let x
belong to V, and let у in J be such that |?/| < ]ж|. It follows from
n l+v1)! <	< 1
that у e V, and hence that V is solid. This proves the second assertion.
Finally since a solid set is absorbing if and only if it absorbs all order-
bounded sets, this fact implies that V is a ^-neighbourhood of 0 in Js
and hence from (a) that the norm, topology on J is not the order-bound
topology ^b.
We are now in a position to establish some necessary and sufficient
conditions for the topology on a bornological Riesz space to be the
order-bound topology.
(15.10)	Proposition. For any bornological Riesz space (X, G, &),
the following statements are equivalent:
(a)	tF coincides with the order-bound topology
(b)	each order -bounded semi-norm on X is topologically bounded;
(c)	each monotone semi-norm on X is topologically bounded;
(d)	each Riesz semi-norm on X is topologically bounded;
(e)	each positive o(Xb, Xfbounded subset of Xb is tF-equicontinuous.
Proof. The implications (b) => (c) => (d) are clear. If p is an order-
bounded semi-norm on X, then the semi-norm p, defined by
p(x) = sup{p(?/):0 < у < И}, (ж gX),
TNFRABARRELLED RIESZ SPACES
185
is a Riesz semi-norm on X for which p(x) < 2p(x) for all x g X. There-
fore, if the statement (d) holds then each order-bounded semi-norm on
X must be topologically bounded; consequently the statements (b), (c),
and (d) are equivalent. On the other hand, if the statement (e) holds,
then Xb = X' and so = r(X, Xb) because (X, 0, SP) is bornological;
thus (e) implies (a). It remains for us to verify the implications (a) =>
(b)	(e). For any order-bounded semi-norm q on X, the set
V =- {ж eX:q(x) < 1} is convex, circled, and also absorbs all order-
bounded subsets of X. Now if SP = ^b, then V is a ^-neighbourhood of
0, hence q is ^-continuous, and a fortiori topologically bounded,
therefore (a) implies (b).
(b) => (e): Let В be a positive cr(Xb, X)-bounded subset of Xb and
V = {жеХ:/(|ж|) < 1 for all f e B}.
Then V is a solid convex and absorbing set in X, and hence absorbs all
order-bounded sets in X. By (b), it follows easily that V absorbs all
topologically bounded subsets of X and hence must be a ^-neighbour-
hood of 0 since is bornological.
Corollary (15.2) leads naturally to the following question: Under
what conditions is the order-bound topology on a locally convex Riesz
space (X, C, S?} the topology on the bornological space associated with
(X, C, SP) ? By using the preceding result, we are able to give an answer
as follows.
(15.11)	Corollary. Let (X, C, SP} be a locally convex Riesz space
with the topologically bounded dual Xtb. Then the following statements are
equivalent:
(a)	Xtb : Xb;
(b)	r(X, Xtb) is the order-bound topology gP^ on X;
(c)	each order-bounded semi-norm on X is topologically bounded;
(d)	each monotone semi-norm on X is topologically bounded;
(e)	each Riesz semi-norm on X is topologically bounded;
(f)	each positive <г(Хь, Xfbounded subset of Xb is r(X, X^fequi-
continuous.
Proof. It should be noted that (X, C, t(X, Xtb)) is a bornological
Riesz space by corollary (15.2), and that the locally solid topologies £P
and r(X, Xtb) have the same topologically bounded sets. Therefore
the result follows immediately from the preceding proposition.
186
INFRABARRELLED RIESZSPACES
(15.12)	Corollary. Let (X, (7, be a locally convex Riesz space
with the topologically bounded dual Xtb. If each increasing tP-Cauchy
sequence in X has an upper bound {in particular, if X is monotonically
sequentially tP-complete), then Xtb = Xb. If, in addition, {X, SP) is
bornological then S? is the order-bound topology.
Proof. If p is a Riesz semi-norm on X which, is not topologically
bounded, then there exists a positive, ^-bounded, convex subset В of
X such that 0 g В and p{wn) > 22n for all n > 1, where wn g B. It is
n
clear that 2 2~fcw7c is an increasing ^-Cauchy sequence in X; and so
n
there exists w g X such that 2~kwk < w. We now conclude from
p(w) > 2~np(wn) > 2n for all n > 1
that p is not bounded on the order-interval [0, w], which gives a
contradiction. Therefore, in view of corollary (15.11), Xtb = Xb, and
the proof is complete.
16
THE STRUCTURE OF ORDER»
INFRABARRELLED RIESZ SPACES
AND ITS SIMPLEST PROPERTIES
It is known, that a locally convex space equipped with the finest locally
convex topology is barrelled, and that the order-bound topology on a
Riesz space is the finest locally solid topology. This suggests the
following question: If <^b is the order-bound topology on (X, G), is
(X, С, ^b) barrelled ? Unfortunately the following example shows that
a locally convex Riesz space, equipped with the order-bound topology,
may not be barrelled.
(16,1) Example. Let be the Banach lattice of all bounded real
sequences, with the pointwise ordering and the supremum-norm ||.||.
For n = 1, 2,... let en be the sequence having 1 in the mth coordinate
and 0 elsewhere. The subspace Eo of generated by the en consists of
all finite sequences. Let e be the sequence having 1 in every coordinate,
and let	E - {x+Xe\x gX0, A g R},
with the supremum-norm ||. || and the ordering inherited from . Then
E is an order-unit normed space with order unit e, and so the norm
topology ||. [| is the order-bound topology. Also it is easily seen that E
is a Riesz space. We show that E is not barrelled. Let
70 = {(cq, a2,...) eX0:|uJ < 1/n for each n},
and let	V - {x + Ae-.x g Fo, |A| < 1}.
Then Fo is a relatively closed, convex, circled, and absorbing subset of
Eo, and contains (l/%)eK for all n. Since [ — 1,1] is a compact subset of R,
it follows that V is a barrel in E. We show that V is not a ||. ||-neigh-
bourhood of 0 in E. Let e > 0. Choose a positive integer m such
that llm < e and put
Then || у || = 8. We claim that у ф V; suppose not, there exists x g Vo
and A e [ — 1, 1] such that у = x-J-Ле. Since x = eew —eem+1—Ae g Fo
188 THE STRUCTURE OF ORDER-INFRABARRELLED
by considering the ?nth and (wH-l)th coordinates of x, we have
1 1
18 — ЛI	'—	I — s — I	—— <
m	m 1
It then, follows that ।	।
£ < ~ —+—~v) < — ,
2 \m m +1/ m
contrary to the choice of m. This shows that у ф V. Since ||?/|| = e and
since s is arbitrary, it follows that V is not a (|. || -neighbourhood of 0 in
E. Therefore, E is not barrelled.
(16.2)	Definition. A locally convex Riesz space (X, C, (?) is called
an order-infrabarrelled Riesz space if each barrel in (X, ^) which
absorbs all order-bounded subsets of X is a ^-neighbourhood of 0.
It should be noted that if (X, C, I?) is a locally convex Riesz space
with the topological dual X' and if us(X', X) is the locally solid topology
on X' associated with u(X', X), then <rs(X', X) is coarser than the strong
topology fi(X', X) because each order-bounded subset of X is ^f-
bounded; consequently each /3(X', X)-bounded subset of X' is
as(Xf, X)-bounded.
(16.3)	Theorem. Let (X, C, (?) be a locally convex Riesz space with
the topological dual X'. Then the following statements are equivalent:
(a)	(X, C, (?) is order-infrabarrelled;
(b)	each solid barrel in (X, C, (?) is a (?-neighbourhood of 0;
(c)	every order-bounded, lower semi-continuous semi-norm on X is
(?-continuous;
(d)	every lower semi-continuous Riesz semi-norm on X is (?-continuous;
(e)	each barrel in (X, (?) which absorbs all relative uniform null-
sequences in X is a (?-neighbourhood of 0;
(f)	each a3(X', X)-bounded subset of X' is (?-equicontinuous;
(g)	each positive c^X', X)-bounded subset of X' is (?-equicontinuous.
Proof, A semi-norm p on X is order-bounded and lower semi-
continuous if and only if the set V ~{xe X:p(x) < 1} is a barrel in
(X, C, (?) which absorbs all order-bounded subsets of X; this means
that (a) and (c) are equivalent. A semi-norm q on X is a lower semi-
continuous Riesz semi-norm if and only if the set U = [xeX: q(x) < 1}
is a solid barrel in (X, C, (?); it then follows that (b) and (d) are
RIESZ SPACES AND ITS SIMPLEST PROPERTIES 189
equivalent. On the other hand, a subset W of X is a barrel in (X, G, hP)
which absorbs all order-bounded sets in X if and only if the polar W°
of W, taken in X', is од(Х', X)-bounded, therefore (a) is equivalent to
(f). Observe that O' is a strict ^-cone in (X', crs(X', X)) and that each
positive a(X', X)-bounded subset of X' is од(Х', X)-bounded; it then
follows that (f) and (g) are equivalent. It is known from lemma (7.2)
that a barrel in (X, (7, ^) absorbs all order-bounded subsets of X if
and only if it absorbs all relative uniform null-sequences in X, then (a)
is equivalent to (e). It is clear that (a) implies (b); we therefore complete
the proof by showing that (b) implies (a). Let W be any barrel in
(X, C, which absorbs all order-bounded subsets of X. By making use
of proposition (11.3), the solid kernel sk(PF) of W is a solid barrel in
(X, C, £P), hence sk(lF) is a ^-neighbourhood of 0 by the statement
(b), and therefore IF is a ^-neighbourhood of 0 ; consequently (X, C,
is ordcr-infrabarrelled.
For any locally convex Riesz space (X, C, A#), /?|ff|(X, X') denotes the
topology on X which is determined by the family
{B°:B is a8(X', X)-bounded},
where jB° is the polar of B, taken in X. It is clear that & is coarser than
(16.4)	Coeollary. A locally convex Riesz space (X, G, ^) is order-
infrabarrelled if and only if & coincides with p\afX, X'), and this is the
case if and only if & is the topology of uniform convergence on the convex,
solid, crs(X', Xfbounded sets in X'.
Proof. This follows from theorem (16.3).
(16.5)	Corollary, For any order-infrabarrelled Riesz space
(X, C, the dual cone O' of C is a strict &-cone in (X', a(X', X)) if
and only if it is a Sd-cone in (X', a(X', X)).
Proof. By making use of theorem (16.3), each positive a(X', X)-
bounded subset of X' is ^-equicontinuous, the result now follows from
proposition (4.8).
A locally convex Riesz space (X, C, tP) is called a barrelled Riesz
space if the locally convex space (X, hP) is barrelled.
The following result gives many examples of order-infrabarrelled
Riesz spaces.
14
190 THE STRUCTURE OF ORDER-INFRABARRELLED
(16.6)	Corollary. Barrelled Riesz spaces are order-infrabarrelled,
and order-infrabarrelled Riesz spaces are infrabarrelled. Further, a locally
convex Riesz space equipped with the order-bound topology is always
order-infrabarrelled.
Example (15.9) shows that a bornological Riesz space, and hence an
infrabarrelled Riesz space is, in general, not order-infrabarrelled; while
example (16.1) indicates that the class of all barrelled Riesz space is
properly contained in the class of all order-infrabarrelled Riesz spaces.
Now the topology of an order-infrabarrelled Riesz is the topology of a
bornological Riesz space, as Nachbin (1954) and Shirota (1954) have
shown.
Before giving other characterizations for which a locally convex
Riesz space is order-infrabarrelled, we require the following notation:
let (X, C, be a locally convex Riesz space with the topological dual
X' and let X^ = (X', С', o8(X', X))'. It is known from, corollary (6.5)
that X'^ is the Aideal in (X')b generated by X. If we define the mapping
of X into X^ by putting
(«)(/) = /И for ah f in X'
and if is the positive cone in Xj" j consisting of all positive crs(X', X)-
continuous linear functionals on X', then is an injective ^-homo-
morphism of (X, C) into (X|"|, C^j). Moreover, is a relative open
mapping of (X, C, into (X|" (,	/3(X"ffj, X')); namely, for any
circled convex ^-neighbourhood V of 0, Т^^Е) is a relative neighbour-
hood of 0 in J|ffl(X) induced by (3(X”a\, X').
(16.7)	Theorem. For any locally convex Riesz space (X, C, ^), the
/-homomorphism of (X, C, into (X|"|, Cp.|, /?(Х|"|, X')) is
continuous if and only if (X, C, is an order-infrabarrelled Riesz space.
Proof. Let В be any од(Х', X)-bounded set in X'. Then the polar
,B0(X|^|) of B, taken in X"ff|, is а /S(X|"(, X')-neighbourhood of 0 in
X|" |. If e|ff| is continuous, then e^(B°(X|" |)) is a ^-neighbourhood of 0
in X. If B° denotes the polar of B, taken in X, it follows from
e^(.B0(Xf;,)) = B«
that В is ^-equicontinuous; and hence from theorem. (16.3) (X, С,
is order-infrabarrelled.
RIESZ SPACES AND ITS SIMPLEST PROPERTIES 191
Conversely, let F" be any f(X'^p X')-neighbourhood of 0 in X"CTp
and let В be a (7S(X', X)-bounded subset of X' such that B°(X^|) Q V",
where B0(Xj" |) is the polar of B, taken in X^. Since (X, C, X} is order-
infrabarrelled then, in view of theorem (16.3), В is X-equicontinuous,
and so the polar B° of B, taken in X, is a ^-neighbourhood of 0. The
continuity of e^i follows from
e|ff|(B<>) В°ВД П си(Х)
and w = n
Therefore the proof is complete.
Since X"ff| is the /-ideal in (X')to generated by X, it follows from
theorem (13.1) that X^ is the order-completion of X if and only if
X' £ Xl
(16.8)	Corollary. For any locally convex Biesz space (X, С, X), the
following statements are equivalent:
(a)	(X, C, X) and (X|"p C”a^ f{X'^, X')) are topologically isomorphic
and ^-isomorphic under the mapping e^p-
(b)	(X, (7, X) is an order-infrabarrelled order-complete Biesz space
and X' £ Xj(.
Proof. (b)=>(a): ejfr| is certainly an /-isomorphism and an open
mapping from (X, C, X) onto the subspace е^(Х) of
(Xfffp Cfffp piX^, X')).
Further, by (b), ep|(X) is order-complete, and hence coincides with its
order-completion X^ (the condition X' £ Xj( is used here (cf. theorem
(13.1)). In other words, eiffj is onto X^. Finally, since X is order-
infrabarrelled, it follows from theorem (16.7) that is continuous and
consequently topologically isomorphic to ©^[(X) = X"^.
In view of theorems (13.1) and (16.7), the implication (a)	(b) is
trivial.
In order to give another characterization of order-infrabarrelled
Riesz spaces in terms of the closed-graph theorem, we recall the follow-
ing well-known terminology: If (E, and (F, X) are topological
vector spaces and if T is a linear mapping of E into F, then T is said to
be nearly continuous if, for each X-neighbourhood U of 0 in F, the
^-closure T~1(U’) of T“1(U') is aX-neighbourhood of 0 in В; T is said
to be nearly open if T(F) is an X-neighbourhood of 0 in F whenever F
192 THE STRUCTURE OF ORDER-INERABARRELLED
is a ^“-neighbourhood of 0 in E, for each ^/-neighbourhood U of 0 in F.
A locally convex space (E, hF} is said to be fully complete (or В-complete,
a Ptak space} if a subspace Q of E' is o(E', Enclosed whenever Q П A
is o‘(E', Enclosed in A for each ^-equicontinuous subset A of E'. It is
well known that every Frechet space is a Ptak space.
(16.9)	Theorem. A locally convex Biesz, space (X, C, &} is order-
infrabarrelled if and only if for any Ptak space (F, hF} the following
statement holds: if T is a linear mapping of X into F such that
(a)	T is order-bounded and
(b)	the graph of T is closed
then T is continuous.
Proof. Suppose that T is order-bounded and that V is a circled
convex «У-neighbourhood of 0 in F. Then T~ \ V) is a barrel in (X, C, F}
which absorbs all order-bounded subsets of X, hence T is nearly
continuous provided, that (X, C, F} is order-infrabarrelled, and so the
necessity follows from a well-known result due to Ptak (cf. Robertson
and Robertson (1964, p. 115)).
Conversely, let IF be a solid barrel in (X, C, &}, p the gauge of IF,
and let J = jp ^(O). Then p is a lower semi-continuous (Riesz) semi-
norm on (X, F}, thus J is a ^-closed subspace of X (in fact, J is a
^-closed /-ideal in X), consequently (X/J, p} is a normed vector space.
Let (У, p} denote the completion of (X/J, p}, and let ф be the quotient
mapping of (X, C, SF) into (X/J,p}. If В is any order-bounded subset
of X, there exists Я > 0 such that p(b} < Я for all b g B, it then follows
that ф is an order-bounded linear mapping of X into (У, p}.
By the following lemma (16.10), ф has a closed graph. Hence, in view
of the hypothesis of the sufficiency, ф is continuous. In particular,
</>"x(S) is a neighbourhood of 0 in (X, F), where 2 - {?/ e У  $(&} < !}•
Notice that ^>~1(S) s W; thus IF must be a neighbourhood of 0 in
(X, F}. This shows that (X, C, F} is order-infrabarrelled.
(16.10)	Lemma. Let p be a lower semi-continuous semi-norm on a
locally convex space (E, FT}, let (F, p) denote the completion of the normed
space (Е/^"х(0), p}, and let ф be the quotient mapping of (E, FT} onto
(Elp-^Q), p}. Then the graph of ф is closed (with respect to the product
topology of ST and the norm topology p).
RIESZ SPACES AND ITS SIMPLEST PROPERTIES 193
Proof. Suppose that xT converges to x in (E, and that ф(хт)
converges to у in (F, p). Then, for any s > 0, there exists x' eE such
that р(у — ф(х')) < e/2. The lower semi-continuity of p with respect to
ST and the contin uity of p with respect to the norm topology on F show
фЬ,ф\:с)---ф{х'У) --p(x-x') < lim inf_p(a?r—«')
- lim inf р(ф(хт)-ф(х')) =- р(у-ф(х')У
We conclude from
р(?/-/(ж)) < р(у-ф(х'))-\-р(ф(х')-ф{х)) < £
that у = ф(х), and hence that the graph of ф is closed. This completes
the proof.
It should be noted that if in theorem (16.9) we omit (b), we obtain a
characterization for to be the order-bound topology (cf. corollary
(7.4)), while if we omit (a) and keep (b), we obtain a characterization of
barrelled Riesz spaces, in view of lemma (16.10).
We conclude this chapter with a result about the topological dual of
order-infrabarrelled Riesz spaces.
(16.11)	Theorem, For any order-infrabarrelled Biesz space (X, C, ^),
the topological dual X' of X is a normal subspace of Хъ; consequently X'
is complete for оё(Х', X) and also for fl(X', X). Further, (X', C, fl(X', X))
is boundedly order-complete.
Proof. Let fT in C be such that fr ] f in Xb for some f 6 Xb. It is
required to show that/belongs to X'. Observe that
f(u) = sup/(ад) for any и in C.
It follows that {/} is tf^X', X)-bounded, and hence from theorem (16.3)
that {fT} is an ^-equicontinuous subset of X'. In view of the Alaoglu-
Bourbaki theorem, {/} has a o(X', X)-cluster point, say g, in X'. Since
/f and since C is o(X.', X)-closed, it follows that/ f g, and hence from
fT]f that g = / Therefore f g X/ and X' must be a normal subspace
of Xb.
The completeness of X' for crs(X', X) and also for fi(X', X) is then
a direct consequence of corollary (13.10) and proposition (13.17), and
the bounded order-completeness of (X', С", /(X', X)) follows from
theorem (11.16) because (X, C, 0s) must be infrabarrelled.
PERMANENCE PROPERTIES OF
ORDER-INFRABARRELLED RIESZ SPACES
We have seen from corollaries (15.4) and (15.7) that /-ideals in a
bornological Riesz space are also bornological with respect to the
relative topology, and those in an infrabarrelled Riesz space are
infrabarrelled with respect to the relative topology; but example (15.9)
shows that this is not true for barrelled and order-infrabarrelled Riesz
spaces. We shall see below that if there are some additional conditions
about some sort of completeness, then the hereditary property is still
satisfied for barrelled and order-infrabarrelled Riesz spaces.
We recall that an /-ideal J in a Riesz space (X, C) is a u-normal
subspace of X if it follows from 0 < un j и in X with u^eJ for all
natural numbers n that и e J.
(17.1)	Theorem. Let (X, C, be an order-infrabarrelled, a-order-
complete Riesz space, and let J be a o-normal subspace of X. Then J is
order-infrabarrelled with respect to the relative topology.
Proof. Let V be a solid barrel in the subspace J and suppose that
U — {ж e X: у eV whenever 0 < у < |ж( and у eJ}.
Then U is a ^-closed convex solid set in X such that U ГУ J = V.
Further, U must be absorbing in X. Otherwise, there exists an element
x of C which fails to be absorbed by U. Hence, for each positive integer
1
n, there exists ynEJ such that 0 < yn < - % but yn ф V. By the
n
m-order-completeness of X, у = sup{n7/w:n = 1, 2,...} exists in X. Since
J is a cr-normal subspace of X, it is clear that у e J. Thus {nyff^ is
contained in the order-interval [0, y\ in J and is not absorbed by V;
this is absurd since F is a solid barrel in J. The contradition established
shows that U absorbs every element in X; hence U is a solid barrel in
X. Since (X, C, 0s) is order-infrabarrelled, U must then be a neighbour-
hood of 0 in (X, ^). Since V = U ГУ J, it follows that F is a neighbour-
hood of 0 in the subspace J. This shows that J is order-infrabarrelled in
its own right.
ORDER-INFRABARRELLED RIESZ SPACES
195
(17.2)	Proposition. Let (X, C, 3?) and (Y, K, be locally convex
Riesz spaces, and let T be a positive continuous linear mapping of X into
Y. If (X, C, is order-infrabarrelled and if T is nearly open then
(У, K, is order-infrabarrelled.
Proof. Let V be any barrel in (Y, K, ЗР} which absorbs all order-
bounded subsets of Y. Then T“1( J7) is a barrel in (X, C, ^) and absorbs
all order-bounded subsets of X, so	is a ^-neighbourhood of 0
in X; the near-openess of T implies that the ^-closure T(T-1(I7)) of
У(/Г^1('У)) is a ^-neighbourhood of 0 in У. We conclude from
V ~-= V
that V is a ^-neighbourhood of 0 in У. Therefore (Y, K, 3") is an
order-infrabarrelled Riesz space.
As a special case of the preceding result we have the following
corollary.
(17.3)	Corollary. Let (X, C, ^) be an order-infrabarrelled Riesz
space, and let J be a ^-closed I-ideal in X. Then the quotient Riesz space
(X[J,	^j) is order-infrabarrelled.
The property of being order-infrabarrelled is preserved under the
formation of inductive topologies with respect to lattice homo-
morphisms as the following result shows.
(17.4)	Proposition. Let (X, (?) be a Riesz space and let
{{Xa,Ca, ^):аеГ}
be a family of locally convex Riesz spaces. Suppose that Ta is an I-
homomorphism of Xx into X (a <= Г), and that X is the linear hull of
и{Тя(Хй): a e Г}. If 3? denotes the inductive topology on X with respect
to {Xa} and {Ta}, and if each (Хй, Ca, 3?f} is order-infrabarrelled, then
each solid barrel in (X, C, is a &-neighbourhood of 0. If, in addition,
& is Hausdorff then (X, C, 3°) is order-infrabarrelled.
Proof. Let V be any solid barrel in (X, C, 3й). Since each If is a
continuous ^-homomorphism of (Xa, Ca, 3f) into (X, C, ^), it follows
from proposition (10.24)(f) that IfffVf) is a solid barrel in (Хя, Ga, 3^)
and hence that T“1(l7) is a ^-neighbourhood of 0 in Хй. Consequently
У is a ^-neighbourhood of 0 in X.
196
PERMANENCE PROPERTIES OF
(17.6)	Corollary. The locally convex direct sum of a family of
order-infrabarrelled Riesz spaces is an order-infrabarrelled Riesz space.
Proof. Follows from corollary (11.10) and the preceding result.
It is worthwhile to remark that the quotient Riesz space and the
locally convex direct sum. of barrelled Riesz spaces are barrelled.
(17.6)	Proposition. The completion of an order-infrabarrelled
Riesz space is a barrelled Riesz space.
Proof. Since order-infrabarrelled Riesz spaces must be infra-
barrelled and since, in view of proposition (11.6), the completion of a
locally convex Riesz space is also a locally convex Riesz space, it follows
from a well-known result that the completion of an order-infrabarrelled
Riesz space is barrelled.
(17.7)	Proposition. Let (X, C, ^) and (F, K, be locally convex
Riesz spaces and let T be a lattice homomorphism of X into Y. Then the
following statements hold:
(a)	if (X, C, ^) is order-infrabarrelled then T is nearly continuous;
(b)	if (F, K, .Xj is order-infrabarrelled and if T is surjective then T
is nearly open.
Proof, (a) If V is any tX-closed, convex, solid -neighbourhood of 0
in F, then T”1(F) is a convex solid subset of X which absorbs all
order-bounded subsets of X and hence, by proposition (11.3)(a),
T~1(F) is a solid barrel in (X, C, X); consequently T-1(F) is a
neighbourhood of 0 in X. Therefore T is nearly continuous.
(b)	Let U be a ^-closed convex solid ^-neighbourhood of 0 in X.
Since T is surjective, it follows from proposition (10.24)(e) that T(U)
is a convex, solid, and. absorbing subset of F, and hence that T(U) is a
solid barrel in (F, К, «X). Therefore, in view of the hypothesis, T(17)
is a ^-neighbourhood of 0 in F and so T is nearly open.
18
RELATIONSHIP BETWEEN BARRELLED,
ORDER-INFRABARRELLED, AND
INFRABARRELLED RIESZ SPACES
In Chapter 15 we have given some conditions for infrabarrelled Riesz
spaces to be bornological and for the topology on bornological Riesz
spaces to be the order-bound topology. It is known from corollary
(16.6) that locally convex Riesz spaces equipped with the order-bound
topology are order-infrabarrelled and, from the example constructed
by Nachbin and Shirota, that the converse is, in general, not true;
therefore the following question naturally arises:
(1)	Let (A, C, ^) be an order-infrabarrelled Riesz space. What
condition on X (or X') is necessary and sufficient for the topology &
to be the order-bound topology ?
Since barrelled Riesz spaces are order-infrabarrelled, and since the
example (16.1) shows that order-infrabarrelled Riesz spaces are, in
general, not barrelled, this leads to the following question:
(2)	What condition on X (or X') is necessary and sufficient for an
order-infrabarrelled Riesz space (X, G, to be barrelled ?
Also the class of order-infrabarrelled Riesz spaces is properly con-
tained in the class of infrabarrelled Riesz spaces, in view of corollary
(16.6); therefore it is interesting for us to answer the following natural
problem:
(3)	What condition on X (or X') is necessary and sufficient for an
infrabarrelled Riesz space (X, C, to be order-infrabarrelled ?
The last chapter of this book is devoted to answering these questions;
we shall begin with a discussion of problems raised by question (1).
(18.1)	Theorem. For an order-infrabarrelled Riesz space (X, C, ^),
the following statements are equivalent:
(a)	is the order-bound topology;
(b)	each order-bounded semi-norm on X is lower semi-continuous;
(c)	each monotone semi-norm on X is lower semi-continuous;
(d)	each Riesz semi-norm on X is lower semi-continuous;
(e)	each positive tr(Xb, Xf bounded subset of Xb is &-equicontinuous.
198
RELATIONSHIP BETWEEN BARRELLED,
Proof. The implications (a) => (b)	(c) => (d) are easy; wo prove
the implications (d)	(e)	(a) as follows. Suppose that the statement
(d) holds. We then show that Xb — X'. For any 0 < f e Xb, let
pf(x) = /(И) for any x in X.
Then pf is a Riesz semi-norm on. X, and. so pf is lower semi-continuous.
Since (X, C, P) is order-infrabarrelled, it follows from theorem (16.3)
that pf is ^-continuous. We conclude from
{x eX'.pf(x} < 1} с {ж eX:\f(x)\ < 1}
that/is ^-continuous, and hence that Xb = X'. On the other hand, if
В is any positive <r(Xb, X)-bounded subset of Xb, then it is Ой(Хь, X)-
bounded, and so В is ^-equicontinuous in view of theorem (16.3).
Therefore (d) implies (e). If the statement (e) holds, then Xb = X' and
P is the Mackey topology t(X, Xb). Hence ГР is the order-bound
topology, consequently (e) implies (a). This completes the proof.
It is known from example (16.9) that the relative topology on an.
/-ideal induced by the order-bound topology need not be the order-
bound topology. In the next result we give some sufficient conditions
for this sort of hereditary property.
(18.2)	Proposition. Let (X,C, be an а-order-complete, locally
convex Riesz space, and let J Ъе a cf-normal subspace of X. If tP is the
order-bound topology then the relative topology on J induced by Pb is
also the order-bound topology.
Proof. Let V be a circled convex set in J which absorbs all order-
bounded subsets of J. We have to show that V is a neighbourhood of 0
in the subspace J. We can assume without loss of generality that V is
solid (if necessary, consider the solid kernel of F). Now, as in the proof
of theorem (17.1), let
U = {x g X: у gV whenever 0 < у < |ж| and у e J}.
Then U is a convex, solid set in X such that U ГУ J — V. Further, since
X is (У-order-complete and J is a tf-normal subspace, it follows from an
argument given in the proof of theorem (17.1) that 17 must be absorbing.
Consequently, U is a circled convex set in X which absorbs all order-
bounded subsets of X; hence Г7 must be a ^-neighbourhood of 0 in
(X, 7^) since ГР = Pb. Since V — U ГУ J, it then follows that V is a
neighbourhood of 0 in the subspace J, as required to be shown.
ORDER-TNFR AB ARRELLED RIESZ SPACES 199
We are now in a position to deal with, the second question posed at
the beginning of this chapter, namely: What condition on X (or X') is
necessary and sufficient for order-infrabarrelled Riesz spaces to be
barrelled? We shall see that the concept of ^-cones as well as the
geometric properties of solid sets play an important role in these
considerations.
(18.3)	Theorem. Let (X, C, be an order-infrabarrelled Riesz
space with the topological dual X'. Then the following statements are
equivalent:
(a)	(X, G, df is barrelled;
(b)	each lower semi-continuous semi-norm on X is order-bounded;
(c)	each lower semi-continuous semi-norm on X is dominated by a
lower semi-continuous Riesz semi-norm defined on X;
(d)	O' is a dd-cone in (X', а(Х', X));
(e)	the solid hull of each u(X', Xfbounded subset of X' is still cr(X'; X)-
bounded.
Proof. The implication (a)	(b) follows from the fact that each
order-bounded subset of X is ^-bounded and that each lower semi-
continuous semi-norm on a barrelled space must be continuous. By
making use of theorem (16.3), (c) implies (a). Suppose now that p is a
lower semi-continuous semi-norm on X, and that the statement (b)
holds. Bor each x in X, we define
p(xj = sup{p(w)ffi < u < Ml-
Since p is bounded on order-bounded sets in X, it follows that p is
finite on X. It is clear that p is a Riesz semi-norm on X and that
p{x} < 2р(ж) for all x g X. If we can show that p is lower semi-con-
tinuous, then q = 2p is the required Riesz semi-norm. Suppose that xr
converges to x in (X, and that p(xT) < p for all r and for some
p > 0. For any s > 0, there exists и in X with 0 < и < |ж| such that
p(u) > p(^) — 8. Suppose that uT = inf{iq then uT converges to
и = inf {ад, |ж|} with respect to and 0 < ur < |«J. It follows from
p(uT) < Р(хт) < У that p(u) < p because p is lower semi-continuous,
and hence that p{x) < р(и)У-& < p-\-e. Therefore р(ж) < p, and so p
is lower semi-continuous. This shows that (b) => (c); therefore state-
ments (a), (b), and (c) are mutually equivalent. Note that a subset
В of X' is ns(X\ X)-bounded if and only if SB is a(X', X)-bounded;
hence by theorem (16.3), (a)o(e). Also it is trivial that (e) => (d).
Conversely, suppose (d) holds. Then, by proposition (4.8), C must be a
200 RELATIONSHIP BETWEEN BARRELLED,
strict <^-cone in (X', a(X', X)). Hence if В is a o'(X/, X)-boundcd
subset of X/, then there exists an o-convex circled a(X', X)-bounded
subset A of X' such that В с. А Л C' —A. C\ O'. Consequently the solid
hull SB of В must be contained in 2 A. This shows in particular that SB is
(/(X', X) bounded, and hence that (d)	(e).
Since a locally convex Riesz space equipped with the order-bound
topology must be order-infrabarrelled, we record a simple consequence
of the preceding theorem.
(18.4)	Corollaey. Let the order-bound dual Xb of a Riesz space
(X, C) be total over X, and let be. the order-bound topology on X. Then
(X, C, £Pb) is barrelled if and only if C* is a StLcone in (Xb, ^(Х1*, X)),
i.e. if and only if the conditions in theorem (18.3) hold.
Before giving another characterization for order-infrabarrelled Riesz
spaces to be barrelled, we need the following result.
(18.5)	Proposition. Let (X, C, be a locally convex Riesz space
with the topological dual X', and let X"^ = (Xх, O', crs(X', X))'. Then the
following statements are equivalent:
(a)	each n(X', Xybounded subset of X' is a8(X', X)-bounded;
(b)	the topology fi(X, X') on X is the relative topology induced by
^(X|"|, X'), and the a(X', Xfclosure of each ffs(X', Xfbounded subset of
X' is crs(X', Xybounded.
Proof. (a)=>(b): Recall that the /3(X, X')-topology on X is the
topology of uniform convergence on the family of all a(X', X)-
bounded subsets of X', and that the relative fi(X"a\, X')-topology on X
is the topology on X of uniform convergence on the family of all
as(X', X)-bounded subsets of X'. By (a),	= ^2; it follows that the
two topologies (3(X, X') and relative /RX'^, X') must coincide on X.
This proves the first assertion in (b). Further, the second assertion in
(b) is a trivial consequence of (a).
(b)	(a): Let В be any a(X', X)-bounded set in X'. Then the polar
B° of B, taken in X, is a j3(X, X'^neighbourhood of 0 in X, and it
follows from (b) that there exists a circled convex crs(X', X)-bounded
set A in X' such that A° = Л°(Х|" |) л X c B°, where JL° and H°(X|(r|)
are the polars of A taken respectively in X and X^. Notice that
В £ В00 £ Л00. In view of the bipolar theorem, J.00 is the <т(Х, X')-
closure of A; it follows from (b) that Л00 is also и3(Х', X)-bounded, and
a fortiori, В is us(X', X)-bounded.
ORDER-INFRABARRELLED RIESZ SPACES
201
As a direct consequence of proposition (18.5), theorem (16.3), and of
the well-known fact that a locally convex space (E, X') is barrelled if
and only if each a{E', X)-bounded set in E' is ^-equicontinuous, we
have
(18.6)	Corollary. Let (X, C, Tf be an order-infrabarrelled Riesz
space. Then it is barrelled if and only if the topology fi(X, X') on X is the
relative topology on X induced by fl(X”a|, X'} and the a{X', Xfclosure
of each од(Х', Xfbounded subset of X' is <7S(X', Xfbounded, i.e. if and
only if the conditions in theorem (18.3) hold.
Let (X, С, TP) be a locally convex Biesz space, and let и be in C. It
is easily seen that Xu ---= (J n[—u, w] is the Aideal in X generated by
n
u. If pu denotes the gauge of [ —u, u] on Xu and suppose that
Cu — С П Xu, then (XM, Cu, pf is a normed Riesz space, и is an order-
unit in Xu, pu is the order-unit norm, and hence the relative topology
on Xu induced by T? is coarser than the norm topology pu. We shall
see that the completeness of (Xu, pf is one of the sufficient conditions
for order-infrabarrelled Riesz spaces to be barrelled (proposition (18.8)),
but the completeness of (Xu, pf can be characterized by the funda-
mental u-order-completeness of (Xu, pf as shown in the following.
(18.7)	Lemma. For any locally convex Riesz space (X, C, TP) and for
any и in C, the normed Riesz space (XM, Cu, pf is complete if and only
if it is fundamentally a-order-complete.
Proof. The necessity is clear. For the sufficiency, we note from
theorem (8.9) that (XM, Cu,pf is monotonically sequentially complete;
and hence, in view of theorem (8.8) (Xu, Gu, pf is complete.
(18.8)	Proposition. Eel (X, C, Tf be an order-infrabarrelled Riesz
space. For any и g C, if Xu is complete for the normpu then (X, G, Tf is
barrelled.
Proof. Let F be any barrel in (X, C, Tf, and let В be any order-
bounded. subset of X. There exists и in G such that В £ [— и, и]. Since
Xu is complete for pu, then (XM, Cu, pf is barrelled. It is clear that
F П Xu is a barrel in (Xu, Gu, pf since the relative topology on XM
induced by T? is coarser than the norm topology pu. Consequently
202
RELATIONSHIP BETWEEN BARRELLED,
V Ci Xu absorbs [~u, u]; in particular, V absorbs B. Therefore V is a
^-neighbourhood of 0, and thus (X, C, B) is barrelled.
(18.9)	Corollary. Let (X, C, ^) be an order-infrabarrelled Riesz
space. If (X, C) is cr-order-complete then (Xu, Cu, pu) is complete for each
и g C, consequently (X, G, IB) is barrelled.
Proof. Let {wn} be an increasing ^U-Cauchy sequence in Xu. Then
{wn} is a pw-bounded subset of Xu, there exists Л > 0 such that
wn g Я[--u, u] for all natural numbers n. By the cr-order-completeness
of (X, G), w - sup wn exists in X. Since Xu is an ^-ideal in X, we
conclude from —Xu < w < Xu that w g X,, and hence that (X„, G,., pA
is fundamentally cr-order-complete. The conclusions now follow from
Iemma (18.7) and proposition (18.8).
(18.10)	Corollary. Bor any locally convex Riesz space (X, C, IB)
with the topological dual X', if PT is any locally solid topology on X', then
(Xf, G', B") is barrelled if and only if it is order-infrabarrelled.
This is a direct consequence of corollary (18.9).
(18.11)	Corollary. A locally convex Riesz space (X, G, IB) is
distinguished if and only if {X', C', (3(X', X)) is order-infrabarrelled.
Proof. It is well known (cf. Kothe 1969) that (X, G, tB) is dis-
tinguished if and only if (X', G', fi(X', X)) is barrelled. The result now
follows immediately from corollary (18.10).
One of the sufficient conditions for the hereditary property of
barrelled Riesz spaces is easily deduced.
(18.12)	Corollary. Let (X, G, IB) be a barrelled, а-order-complete
Riesz space, and let J be a a-normal subspace of X. Then J is a barrelled
Riesz space with respect to the relative topology induced by tB.
Proof. It should be noted that J is cr-order-complete. In view of
theorem (17.1) and corollary (18.9), J is a barrelled Riesz space with
respect to the relative topology induced by &.
As an immediate consequence of corollaries (18.9) and (16.6), we have
(18.13)	Corollary. Let Xb be total over (X, C), and let tBb be the
order-bound topology on X. If X is cr-order-complete, then (X, C, lBb) is a
barrelled Riesz space.
ORDER-INFRABARRELLED RIESZ SPACES
203
We shall seek some classes of locally convex Riesz spaces (W, C, 0?)
for which (Xu, pu) is complete for any ueC.
(18.14)	Proposition. For any locally convex Riesz space {X, G, 0s),
if (X, C, 0?) is fundamentally a-order-complete then (Xu, pu) is complete.
If, in addition, (X, G, 0?) is order-infrabarrelled then (Xu, pf) is barrelled.
Proof. It is enough to verify that (Alu, Gu, pu) is fundamentally
o'-order-complete. Let {wn} be an increasing pw-Cauchy sequence in Xu.
Then {wn} is an increasing ^-Cauchy sequence in X and wn g I[—u, u]
for some 2 > 0. It follows from the fundamental c-order-complete-
ness of (X, C, 0?) that w ~~ sup wn exists in X. It is clear that
w g 2[ — и, и]. On the other hand, since Xu is an if-ideal in X, we conclude
that w g Xu, and hence that (Xu, Cu, pu) is fundamentally o'-order-
complete. The result now follows from lemma (18.7) and. proposition
(18.8).
(18.15)	Corollary. For any locally convex Riesz space (X, C, J3),
if {X, G, 0P) is monotonically sequentially complete then (Xu, pu) is
complete for each и e G. If, in addition, (X, G, 0?) is order-infrabarrelled
then (Xu, pu) is barrelled.
Proof. Since G is ^-closed, the result now is a direct consequence of
lemma (8.6) and proposition (18.14).
(18.16)	Proposition. Let (X, G, 0?) be a locally convex Riesz space,
and let и be in G. Then (Xu, Gu, pu) is complete if and only if it is
monotonically sequentially complete.
Proof. Since Cu gives an open decomposition in (XM, pu), the result
follows from theorem (8.8).
We now turn our attention to the third question posed at the
beginning of this chapter, that is: what condition on X (or X') is
necessary and sufficient for infrabarrelled Riesz spaces to be order-
infrabarrelled? We shall see that some sort of completeness plays an
important role in these considerations.
(18.17)	Theorem. For any infrabarrelled Riesz space (X, G, 00), the
following statements are equivalent:
(a)	(X, C, 0?) is order-infrabarrelled;
204
RELATIONSHIP BETWEEN BARRELLED,
(b)	each lower semi-continuous Riesz semi-norm on X is topologically
bounded;
(c)	each positive a(X', Xybounded subset of X' is @{X’, Xybounded;
(d)	(X', С', а(Х, X}} is boundedly order-complete;
(e)	(X', O', crg(X', X)) is both boundedly order-complete and locally
order-complete;
(f)	X' is complete for cfs(X', X};
(g)	each positive ct(X', Xybounded subset of X' which is directed
upwards has a a{X’, Xyiimit;
(h)	X' is a normal subspace of Xb.
Proof. The equivalence of (a), (b), and (c) follows from proposition
(15.6) and theorem (16.3), and the equivalence of (d)-(h) follows from
corollary (13.10). In view of theorem (16.11), (a) => (h). If (X, C, &} is
infrabarrelled then, by proposition (13.15), (h) => (a). Therefore the
proof is complete.
Since each bornological space is infrabarrelled, we obtain the
following corollary.
(18.18)	CoEOLLAEy. For any bornological Riesz space {X, C, &}, if
it satisfies one {and hence all} of (b)-(h) in theorem (18.17), then (X, (7, &}
is order-infrabarrelled.
The following result can be proved by a similar argument to that
given in the proof of proposition (18.5).
(18.19)	Proposition. Let (X, C, &} be a locally convex Riesz space
with the topological dual X’, and let X^ = (X', C, us(X', X))'. Then the
following statements are equivalent:
(a)	each ofiX', Xybounded subset of X' is fl(X', Xybounded;
(b)	the topology X'} on X^ is the relative topology induced
by fi{X", X'), and the a8{X', Xyclosure of each fi(X', Xybounded subset
of X' is fi(X', Xybounded.
As an immediate consequence of theorem (18.17) and the preceding
proposition, we have the following corollary.
(18.20)	Corollary. Let (X, (7, &} be an infrabarrelled Riesz space.
Then it is order-infrabarrelled if and only if the topology fi{X"a(, X') on
X"^ is the relative topology induced by @(Х”, X'), and the us(X', X)-
closure of each fi(X', Xybounded subset of X' is {3(Xf, Xybounded, that
is, if and only if the conditions in theorem (18.17) hold.
ORDER-INFRABARRELLED RIESZ SPACES
205
Since each normed Riesz space is infrabarrelled, we obtain the
following corollary.
(18.21)	Corollary. Let L be a normed Riesz space with the topo-
logical dual L'. If L is order-infrabarrelled then the topology ft(L"a\, IT) on
is normable.
(18.22)	Corollary. Let {X, C) be an order-complete Riesz space,
and let Xb be total over X. Then crs(X, Xb) coincides with fi(X, Xb) if and
only if (X, C, ofiX, Xb)) is infrabarrelled.
Proof. The condition is clearly necessary. To prove its sufficiency,
observe that (X, C, us(X, Xb))' = Xb is a normal subspace of Xb. It
follows from theorem (18.17) that (X, (7, Оц(Х, Xb)) is order-infra-
barrelled and hence, from corollary (18.9), that (X, C, <?s(X, Xb)) is
barrelled; consequently os{X, Xb) and /?(X, Xb) coincide.
Combining theorems (18.3) and (18.17) we have the following very
interesting result.
(18.23)	Theorem. For any infrabarrelled Riesz space (X, C, ^), the
following statements are equivalent:
(a)	(X, CJ, ^) is barrelled;
(b)	(X', O', a{X', X)) is boundedly order-complete and C is a d$-cone
in (X', <j{X', X));
(с)	X' is complete for trH(X', X) and the solid hull of each n(X', X)-
bounded subset of X' is a{X', X)-bounded;
(d)	X' is a normal subspace of Xb and C' is a HR-cone in (X', <r(X', X)).
Remark. The condition that X’ be a normal subspace of Xb in the
preceding result can be replaced by any one of the equivalent properties
listed in theorem (18.17), and the condition that C be a ^-cone in
(X', cr(X', X)) can be replaced by any one of the equivalent properties
listed in theorem (18.3).
(18.24)	Corollary. For any bornological Riesz space (X, C, ^), if
it satisfies one {and hence all) of (b), (c), and (d) in theorem (18.23), then
(X, C, HP) is barrelled.
15
NOTES ON THE BIBLIOGRAPHY
Chapters 1	& 2
The results of these two chapters, in particular, theorems (1.10), (1.12), (1.15),
(1.17), and (2.11) should bo considered very fundamental and important for the
study of the theory of ordered topological vector spaces. The positive extension
problem for linear functionals was first studied by Krein and Rutman (1948);
the general characterisation for linear functionals admitting positive extension,
as that given in theorem (1.12), is due to Namioka (1957) and Bauer (1957, 1958).
Theorem (2.8) is the work of many hands, e.g. Weston (1957b), Namioka (1957),
Schaefer (1966), and Bauer (1957, 1958). Tho generalization of the Hahn-Banach
theorem, given in theorem (1.15), is due to Bonsall and is very useful for our
investigation of the duality problems for ordered vector spaces. Theorem (1.17)
is essentially due to Jameson (1970). Theorem (2.11), in the present form, is
taken from an article of Ng and Duhoux (1973), while parts are implicitly given
in earlier papers of Ng (1970), Wong (1970a) (1973a), and Duhoux (1972a).
Corollary (2.12) is given by Jameson (1970) with a different proof; but see also
Grosberg and Krein (1939).
Chapters 3, 4, and 5
In the study of an ordered locally convex space (£?, O, 0), two conditions have
played an important role in our discussion: one condition is to say that the cone
is ‘large’ enough to give an open decomposition, property and the other is to say
that the cone is ‘small’ enough such that & admits a neighbourhood-base at 0
consisting of order-convex sets. These two conditions are respectively equivalent
to saying that & is locally decomposable and locally o-convex. Krein appears to
bo the first one to consider locally o-convex Banach spaces, and the duality
theorem (5.15) was proved in a 1939 joint paper with Grosberg (see Krein and
Grosberg (1939)). The result was generalized to general locally convex spaces
by Bonsall (1957), and Schaefer (1966) studied the duality of locally o-convox
spaces and ^-cones. Dually, Bonsall (1955) introduced locally decomposable
normed spaces and the concept was extended by Jameson (1970) (where ho
used the term ‘open decomposition’), Wong (1973c), and Duhoux (1972a). The
dual characterization of such spaces was independently obtained by Andd (1962)
and Ellis (1964). The construction of the associated locally o-convex topology
is essentially due to Namioka (1957) and that of to Wong and Cheung (1971).
The dual characterizations of and (in particular, the dual charac-
terization of locally decomposable spaces) are given by Ng and Duhoux (1973b).
The equivalence of (a) and (b) in theorem (3.8) is due to Klee and the (c)
equivalence is duo to Jameson (1970). Tho short proof presented here for this
theorem as well as that of theorem (3.9) is taken from Ng (1973b). The concept
of nearly open decomposition is due independently to Wong and Duhoux (1972b);
and, in particular, theorem (3.11) and corollary (3.13) are taken from the latter.
Propositions (4.1) and (4.3) are duo to Wong and Cheung (1971). Theorems
(5.1) and (5.4) are due to Namioka (1957) and Schaefer (1966). For other
NOTES ON THE BIBLIOGRAPHY
207
equivalent properties for normality see Riedl (1964). Theorem (5.9) is a funda-
mental duality result between normal cones and ^-cones; it is due to Schaefer
(1966) but part (ii) is also implicitly contained in Bonsall (1957). The proof
presented here is taken from Ng and Duhoux (1973), while other short proofs
were also given by Wong (1970a) and Duhoux (1972a). Theorem (5.16) is dual
to theorem (5.15) of Grosberg and Krein and is due to Ellis (1964) (whore he
assumes that the space E is complete, and Ng (1973b) observes later that the
completeness is automatic from the other assumptions by applying a generalized
open mapping theorem). A. somewhat loss strong form of theorem (5.16) was
earlier obtained by Ando (1962) (where ho did not calculate the constants). The
proofs of theorems (5.15) and (5.16) are taken from Ng (1970) and (1973b). An-
other related paper: Kist (1958).
Chapter 6
The concept of solid sets in a general ordered vector space was introduced by
Ng (1971b) and Duhoux (1972a). Theorem (6.1) seems to be new. Theorem (6.3)
is a generalization of Nachbin’s result (1965) on. vector lattices, in part due to
Wong (1973c) and Duhoux (1972a). For the Banach space ease, theorem (6.12)
was proved by Davies (1968); for the present form, see Ng and Duhoux (1973).
Other related papers: Wong (1969a) and Wong and Cheung (1971).
Chapter 7
The construction of the order-bound topology .^b is due to Namioka (1957)
and Schaefer (1966), while the dual characterization of ^b is given in (1972a) by
Wong. Theorem (7.3), duo to Wong (1972a), can be regarded as a general form for
studying the continuity of positive linear mappings. Corollaries (7.6) and (7.8)
are due to Schaefer; (7.7) is established by Namioka (1957) and Klee; and (7.9)
was deduced by Ng (1973a). Theorems (7.10), (7.12), and (7.14) were proved by
Wong (1972a), but (7.14) was earlier obtained by Namioka in (1957) in the vector
lattice case.
Chapter 8
The study of the relationship between order completeness and topological com-
pleteness can be broken down into two stages. The first stage is to establish some
sufficient conditions ensuring that the monotonically sequential completeness
implies the completeness; this had been done by Jameson (1970) for the metriz-
able case (cf. theorem (8.8)). The second stage is to establish some sufficient con-
ditions ensuring that the order completeness implies the monotonically sequential
completeness. This has been done by Wong for the metrizable case (cf. theorem
(8.9)). With the exception of several results pointed out in the text, all results
in this section are taken from an article of Wong (1972b). Other related papers:
Duhoux (1972a), and Ng (1972a).
Chapter 9
The notion of order-unit norm is essentially duo bo Kadison (1950) and that of
base-norm to Edwards (1964) and Ellis (1964). Much of the theory developed in
208
NOTES ON THE BIBLIOGRAPHY
this section was initiated by them. In. particular, Edwards is the first to note that
each compact convex set can be affine-homoomorphically embedded in a Banach
dual space with the w*-topology, and establishes the duality theorem (cf.
theorem (9.10)). A dual result (cf. theorem (9.8)) was given by Ellis (1964).
The notion of approximate order-unit was suggested by (7*-algebra theory and
was introduced by Ng (1969a); ho proved, theorems (9.6), (9.9), and (9.15).
The concept of the Lb-condition and results from lemma (9.24) to theorem (9.28)
wore cited in an article of Ng (1972b). In the case of a partially ordered Banach
space with closed cone, theorem (9.7) was proved independently by Asimow
(1968) and Ng (1969a), and the theorem in the present form was noted in. the
joint paper of Ng and Duhoux (1973); in that joint paper further generalizations
of some results of this section were also discussed. The implication (a) => (b) in
theorem (9.20) is a famous theorem of Riesz (1940) and Ando proves the much
more difficult implication (b) => (a). Effros (.1967) calls an ordered Banach
space E with closed cone a simplex space if the dual E' is an Л^-space. A
intrinsic characterization (equivalent to those presented in corollary (9.22)) of
simplex spaces was independently given by Davies (1967) and. Ng (unpublished)
at about the same time in 1966, by virtue of a powerful separation theorem of
Edwards (1965). A large portion of the materials presented in this section can be
found in Ng (1969a).
Chapter 10
Most of the material of this chapter can be regarded as mathematical folklore.
The solid hull and the solid kernel (absolute core in the terminology of Roberts
(1952)) were introduced by Roberts (1952). The important result of proposition
(10.10) concerning tho basic relation between normal subspaces and order
direct sums in Riesz spaces was cited in an article of Riesz (1940). The concepts
of normal integrals and integrals were introduced in an article of Nakano (1950a),
and so was proposition (10.17). Systematic and extensive treatments of the theory
of Riesz spaces can be found in the book of Luxemburg and Zaanen (1971).
Chapters 11 and 12
The early theory of Banach lattices was studied by F, Riesz, Frendenthal,
Birkhoff (1961), Kakutani (1941, 1942b), Krein (1940), and Nakano (1950a),
while Roberts (1952) seems to be the first to investigate the duality theory for
locally convex Riesz spaces. Theorem (11.14) concerning the completeness of
topological Riesz spaces is duo to Nakano (195. 0), and so is proposition (11.13),
but the proof that we have presented here for (11.13) is due to Schaefer (1960).
A part of theorem (11.16), namely that the strong dual of a locally convex
Riesz space X reflects the properties of X, was introduced by Kawai (1957);
while the second assertion in theorem (11.16) on the completeness of the strong
dual of infrabarrelled Riesz spaces was proved by Wong (1969b), it is a generali-
zation of Schaefer’s result (1960). Most of the results in Chapter 12 are taken from
the articles of Wong (1969a) and (1969b).
Other papers or books related to the subject matter of Chapter 11: Jameson
(1970), Peressini (1967), Coffman (1956, 1959), Gordon (1960), andKuller (1958).
NOTES ON THE BIBLIOGRAPHY
209
Chapter. 13
The equivalence of (b) and (e) in theorem (13.1) was found, by Ando, and the
other equivalent properties in theorem (13.1) were proved by Luxemburg and
Zaanen. Kawai (1957) and Wong (1969b) found the criterion, for X to be an
Z-ideal in X"; their results are presented in theorem (13.5). A necessary and
sufficient condition for X to be a normal subspaco of X" was proved by Wong
(1969c). Corollary (13.7) was cited earlier in the article of Nakano (1950a).
Theorem (13.9) on the completeness for the Dieudonne topology was established
by Wong (1969c), and it generalizes results of Goffman (1956), Porossini (1967),
and Schoafer (1960). Corollary (13.11) is duo to Perossini (1967). Results (13.15) •
(13.18) are taken from an article of Wong (1973b).
Chapter 14
This chapter is concerned with a study of the interrelation between reflexivity
and order. Corollary (14.2) is due to Ogasawara. Corollary (14.5) was cited in
an article of Schaefer (1960). Theorems (14.1) and (14.6) were found by Wong
(1969c).
Chapter 15
Kawai (1957) proved that every bornological Riesz space is the inductive
limit of a family of normed Riesz spaces (cf. proposition (15.5)); also he proved
corollary (15.4). (15.1)--(15.3) and propositions (15.6)-(15.10) are taken from
Wong (1970b). Other related paper: Warner (1960).
Chapters 16, 17, and 18
It is known that a locally convex space equipped with the finest locally convex
topology is barrelled, and that the order-bound topology is the finest locally
solid topology. However example (16.1), due to Ng (1971b), shows that locally
convex Riesz spaces equipped with the order-bound topology may not be
barrelled. The class of order-infrabarrelled Riesz spaces, on the one hand,
includes the class of locally convex Riesz spaces equipped with the order-bound
topology, and on the other hand, it behaves as and plays a role similar to barrelled
spaces in the theory of locally convex spaces. The class was introduced and
studied by Wong in (1969d) and (1973c). In particular, he gave (1969d, 1973c)
various characterizations for spaces in the class, for example that presented in
Chapter 16, studied (1969d) the permanence properties of such spaces, presented
here in Chapter 17, and established (1969d) some interrelationship between
various classes of locally convex Riesz spaces, for example that presented here
in Chapter 18. Some of his work was extended in Ng (1971b) to weakly Riesz
spaces.
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INDEX
Absolute
monotone seminorm, 6
order-convex, 6
value, 114
Absolutely
convex hull, 1
dominated, 8
Affine function, 96
Almost-Archimedean, 3
Almost <y(E', jS)-closed, 28
AL-space, 102
A ZA-space, 102
A/A-space, 102
AL00-space, 102
AM-space, 102
Antisymmetric, 2
Approximate order-unit, 12
normed space, 85
seminorm, 12
Archimedean, 3
Banach lattice (B-laltice), 101
Barrelled Riesz space, 189
Base
norm, 88
normed space, 88
seminorm, 88
Б-complete space, 192
Bornological Riesz space, 178
Boundedly
order-complete, 77; 147
cr-order-complete, 77
Cap, 98
Circled, 1
Cofinal subspace, 11
Compact-open topology, 152
Complement, 1
Cone, 1
^-cone, 43
local ^-cone, 43
strict ^-cone, 43
locally strict ^-cone, 43
^-cone, 43
strict ^-cone, 43
dual cone, 9
normal cone, 48
а-normal cone, 51
Convex, 1
circled hull, 1
Decomposable, 7
kernel, 7
Decreasing, 16
Dieudonne topology, 152
Directed
downwards, 16
upwards, 16
Disjoint, 114
Dual
cone, 9
ordering, 9
.Empty set, 1
Equivalent, 152
Erechet lattice (H-Iattice), 140
Full hull, 6
Fully complete space, 192
Fundamental system, 43
Fundamentally cr-order-complete, 69; 77
Gauge, 1
Generating, 2
«-generating, 32
nearly «-generating, 32
strictly generating, 32
Greatest lower bound, 113
Hypercomplete space, 28
Increasing, 16
Infrabarrelled Riesz space, 178
Integral, 127
Kothe
dual, 153
function space, 153
topology, 153
Lattice-ideal (Z-ideal), 118
generated by, 118
Lattice homomorphism (Z-homomorph-
ism), 132
Z-isomorphism, 132
Z-propeotion, 123
ZLbounded, 71
Z1-order-summable, 77
Lp-condition (i), 108
.ZA-condition (ii), 108
L^-condition, 111
Least upper bound, 75
216
INDEX
Local
^-cone, 43
null-sequence, 72
Locally
bounded with respect to, 4 3
convex direct sum topology, 40
convex Riesz space, 136
convex vector lattice, 136
decomposable space, 31
decomposable topology associated with,
33
full topology, 48
full space, 48
full topology associated with, 56
o-convex Riesz space, 158
o-convex topology, 50
o-convex space, 50
order-complote Riesz space, 144
order-convex topology, 48
order-convex topology associated with,
56
order-convex space, 48
solid spaco, 61
solid topology, 61
solid topology associated with, 61
strict ^-cone, 43
summable function, 152
Majorized, 5
Measure topology, 154
Minkowski functional, 1
Minorized, 5
Monotone, 6
Monotonically sequentially complete, 77
Nearly
a-generating, 32
continuous, 191
open, 191
-open decomposition, 30
-open decomposition property, 30
Negative part, 114
Normal
a-normal cone, 5].
cone, 48
integral, 127
subspace, 16; 118
<7-normal subspace, 16; 118
tf-normal subspace generated by, 119
Normed lattice (normed Riesz space), 101
o-convex, 6
Open decomposition, 30
property, 30
Order
bound dual, 10
bound topology (order topology), 67
bounded (linear) functional, 9
bounded (linear) mapping, 68
bounded semi-norm, 179
bounded set, 5
complete, 16
completion, 134
continuous, 127
о-continuous, 127
convex hull, 6
convex set, 5
dual, 10
infrabarrelled Riesz spaco, J 88
interval, 5
unit, 12
unit normed spaco, 86
unit semi-norm, 12
Ordered
Banach space, 29
convex space, 18
direct sum, 17	<
normed space, 31; 85
topological vector space, 18
vector space, 2
tf-order-complote, 16
Perfect space, 170
Polar, 13
Positive
functional, 9
linear mapping, 68
part, 114
sot, 5
Positive-order-convex set, 6
Positively
dominated set, 8
generated set, 7
Ptak spaco, 192
Proper cone, 2
Quotient Riesz space, 140
Reflexive
dual pair, 176
ordering, .1
Relative
ordering, 4
uniform null-sequence, 67
Riesz
bidual, 150
decomposition property, 9; 75; 113
dual, 150
norm, 101
seminorm, 61
space, 9; 75
subspace, 118
subspace generated by, 118
INDEX
217
Saturated family, 52
Semi-decomposable semi-norm, 30
Semi-norm, 1
Semi-reflexive with respect to, 176
Simplex space, 208
Solid
barrel, 182
hull, 116
kernel, 116
sot, 8
Strict
^-cono, 43
^-cone, 43
Strictly
generating cone, 32
positive linear functional, 87
Sublinear functional, 1
Symmetric, 1
ST -determined family, 71
Theorem of
Ando-Ellis, 55
Andd-Luxemburg-Zaanen, 160
Bonsall, 13
Davies, 64
Edwards, 95
Grosberg—Krein, 54.
Jameson, 14
Klee, 34
Luxemburg-Zaanen, 125
Nakano, 147
Riesz, 10
Riesz--Ando, 106
Schaefer, 52
Topological
Kothe function space, 153
Riesz space (topological vector lattice),
136
Topologically bounded
seminorm, 179
dual, 18
Topology of uniform convergence
on local null-sequences, 73
on relative uniform null-sequences, 73
Transitive, 1
Unital normed Riesz space, 143
Universal cap, 98
Vector
lattice, 9
(partial) ordering, 2
topology with the open decomposition
property associated with, 33
Weakly Riesz space, 75