Lecture Notes in
Mathematics
Edited by A. Dold and B. Eckmann
955
Gerhard Gierz
Bundles of
Topological Vector Spaces
and Their Duality
Springer-Verlag
Berlin Heidelberg New York 1982
Author
Gerhard Gierz
Department of Mathematics, University of California
Riverside, CA 92521, USA
AMS Subject Classifications (1980): 46E10, 46E15, 46E40, 46H 25,
46В 20, 55R25, 28C20
ISBN 3-540-11610-9 Springer-Verlag Berlin Heidelberg NewYork
ISBN 0-387-11610-9 Springer-Verlag NewYork Heidelberg Berlin
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Contents
Introduction ................................................ 1
Notational remarks .......................................... 7
1.	Basic definitions..........................-.................. 8
2.	Full bundles and bundles with completely regular
base spaces ...	22
3.	Bundles with locally paracompact base spaces	....	28
4.	Stone - WeierstraB theorems for bundles ..................... 39
5.	An alternative description of spaces of sections:
Function modules ...	44
6.	Some algebraic aspects of ^-spaces............................60
7.	A third description of spaces of sections: С(X)-
-convex modules ...	62
8.	С (X)-submodules of Г (p).....................................80
9.	Quotients of bundles of C(X)-modules ........................ 86
10.	Morphisms between bundles ................................... 95
11.	Bundles of operators ........................................112
12	Excursion: Continuous lattices, and bundles ................ 136
13.	M-structure and bundles......................................144
14.	An adequate M-theory for ^-spaces............................154
15,	Duality......................................................159
16.	The closure of the "unit ball" of a bundle and
separation axioms ....	183
17.	Locally trivial bundles: A definition ...................... 200
18.	Local linear independence .................................. 202
19.	The space Mod (Г (p) ,C (X)).................................209
2c • Internal duality of C(X)-modules.............................232
IV
21. The dual space Г(p)1 of a space of sections ....	252
Appendix:	Integral representation of linear functionals
on a space of sections (by Gerhard Gierz
and Klaus Keimel)....................................260
References.................................................284
Index .....................................................291
Introduction.
In the present notes we are dealing with topological vector spaces
which vary continuously over a topological space. Among the first
authors formulating this idea were Godement [Go 49], Kaplansky
[Ka 51], Gelfand and Naimark. In these early papers, they axiomatized
the idea of subdirect continuous representation of Banach spaces. To
be precise, they considered spaces E of functions a defined on a
topological space X with values in given Banach spaces Ex, x e X,
satisfying axioms like
(1)	The function x -> || a (x) || : X + JR is (upper semi-) continuous
and bounded for every a e E.
(2)	The space E is complete in the norm || a || = sup || a(x) || .
xeX
(3)	Ex={a(x) : a e E} for every x e X.
(4)	E is a (X)-module relative to the multiplication (f,a) -* f*a:
С, (X) *E -> E, where (f«a)(x) := f(x)«a(x) and where С, (X) denotes
r>	b
the algebra of all continuous and bounded scalar valued
functions on X.
In the following years, L. Nachbin, S. Machado and J.B.Prolla
gave a similar definition for locally convex spaces. They, however,
started from an approximation theoretical point of view.
The authors mentioned above were very well aware of the fact that
their notion of continuous decomposition was closely related to
continuous sections in fibre bundles. In fact, Fell [Fe 61], Dixmier and
Douady [DD 63] as well as Dauns and Hofmann [DH 68] succeeded in
giving a fibre bundle theoretical formulation of the axioms (1) - (4).
2
A third interesting aspect of spaces satisfying axioms (1) - (4) has
its origin in sheaf theory and intuitionistic logic (see the Lecture
Notes in Mathematics 753).
Originally, the work on the present notes was begun in order to give
a useful description of the space of all continuous linear functio-
nals on a space of sections in a bundle (or, equivalently, on a topo-
logical vector space satisfying axioms (1) - (4)). However, I have to
admit that I did not succeed to my own satisfaction. Let me explain
what my intention was at the beginning:
The best known example of a space satisfying axioms (1) - (4) is the
space С^(Х) of all continuous and bounded real-valued functions it-
self. If we assume for a moment that the base space X is compact, then
the dual space of C(X) consists of all finite regular Borel measures
on X acting on С(X) by integration. How should this fact generalize to
a vector space E satisfying axioms (1) - (4)? Suppose that we start
with a family Ф = (ф )	, where each Ф is a continuous linear func-
X X t x	X
tional on E . Then every a e E defines a real-valued mapping Ф(а) by
Ф(а)(x) := ф (a(x)). Suppose moreover, that ц is a measure on X and
that Ф(а) is bounded and ц-integrable for every a e E. Then we may
define a linear functional ф on E by
Ф (a) : = f Ф (a) «dji.
X
Now the following questions arise:
1)	Under which conditions on the family (ф ) is the function
X X€ X
Ф(а) integrable with respect to ц for every a e E?
2)	Is every continuous linear functional representable in the form
3
ф(а) = J Ф( а) «dp ?
X
3)	If so, how does one add j Ф(а) «dp and J Т(а)>dv, i.e. how is the
X	X
algebraic structure of the dual space E1 of E reflected in this
integral representation of linear functionals?
A first answer to problem 1 of course would be that the mapping Ф(а)
is Borel measurable for every а e E. In this case, the family (фх)ХеХ
yields a (bounded) linear mapping from E into the vector space of all
bounded Borel measurable functions on X, equipped with the supremum
norm. It is easy to see that <I>(f-a) = f^(a) for every f e С (X) and
every а e E, i.e. Ф is a C(X)-module homomorphism.
It is very tempting to postulate that Ф(а) is even continuous for the
following reason: The space C(X) is the space of sections in the
most simple bundle which can be thought of. Therefore and because
C(X) acts on every space of sections by multiplication, this space
should play the role of the field of the real numbers in the category
of all spaces of sections in bundles over x. In this case, we would
obtain a representation of linear functionals ф : E +B by an "internal
linear functional" Ф : E -* С(X), i.e. by a continuous С(X)-module
homomorphism into the "internal field" C(X) and a measure p on X, i.e.
an "external" linear functional on С(X).
Unfortunately, examples show that the linear functionals represented
in this way do not even form a linear subspace of the dual space E1
and that the linear span of these functionals does not have to be
dense in E1.
I am aware of the fact that mathematicians like Burden and Mulvey
viewing bundles from a point of view of sheaves, toposes and logic
do not agree with my choice of the "internal real numbers? Therefore,
they will not be suprised that I was not able to carry out my program
in full generality (nor am I now) and it is certainly worthwhile
4
to check to what extend a use of their internal real numbers would lead
to better results.
Problem 2 is solvable provided that X is a compact metric space (see
section 21). An example of R.Evans (FU Berlin) conversely shows that
this problem has a positive solution if and only if every finite
regular Borel measure on X admits a strong lifting.
A discussion of the third problem leads to tensor products over С(X).
Indeed, under certain (strong)restrictions, the dual space E' of E
may be identified with a certain tensor product over С(X) between the
space M(X) of all finite regular Borel measures on X and the space
Mod(E,C(X)) of all continuous C(X)-module homomorphisms from E into
C(X). Interpreting this result, we may say that the "external" dual of
a bundle is obtained by tensoring the "internal" dual of the bundle
with the "external" dual of the trivial bundle which has C(X) as its
set of sections.
Having now revealed my original intentions, I should also say what
I was able to achieve:
Firstly, I found it convenient to gather some known information from
the literature for later references, and that is what is done in the
first 10 sections. The informed reader will hardly find anything new
here, an exception are perhaps the results concerning bundles of
^-spaces, which present the common aspects of bundles of topological
vector lattices, Banach algebras, C*-algebras etc. Most of the other
results here originated from papers of J.Dauns, K.H.Hofmann, L.Nach-
bin, S.Machado, J.B.Prolla, H.Moller, E.Behrends, E.M.Alfsen, E.G.
Effros, A.Douady and L.Dal Soglio-Herault and I apologize to all the
others which are not mentioned here explicitely. To make these notes
5
more self-contained, I included the proofs.
In section 11 we start with the development of a duality for bundles.
Here the "dual unit ball" of the "unit ball" of a bundle of Banach
spaces is introduced and it is shown how the upper semicontinuity
(resp. continuity) of the norm of the bundle is reflected in this
dual unit ball. Moreover, we discuss the relation between "stalkwise"
convex subsets of the bundle and "stalkwise" convex subsets of the
dual unit ball.
In sections 12, 13 and 14 we apply the results from section 11 to
subbundles and quotients of bundles and discuss morphisms between
bundles in general.
In paragraph 15 we take a closer look at the topology of the bundle
space. Especially, we study the strength of separation in the bundle
space and its relation to the closure of the "unit ball".
The theorem saying that every bundle with a Hausdorff bundle space
whose stalks are of a fixed finite dimension n and whose base space
is locally compact is in fact locally trivial stands in the center
of sections 16 and 17.
In paragraph 18 we consider spaces of bounded linear operators with
values in a space of sections in a bundle and prove a representation
theorem for these spaces. An application of this representation
theorem to spaces of compact operators yields a result concerning the
approximation property of spaces of sections.
The study of the space of continuous C(X)-module homomorphisms into
6
C(X) is carried out in section 19. The main result presented here
says that the space of sections in a "separable" bundle of Banach
spaces with a compact base space and continuous norm admits "enough"
continuous C(X)-module homomorphisms into С (X) .
In section 20 we investigate to what extend the theorem of Mackey and
Arens holds "internally" in the category of C(X)-modules
The last section is devoted to a treatment of the three problems
mentioned above. The main part of this paragraph is taken from a
joint work of Klaus Keimel and myself done in 1976 which was
never publisched.
I am grateful to Klaus Keimel who always found the time for
helpful conversations.
Notational remarks.
Ж = ]R, (E	is the field of real or complex numbers.
X	always denotes a topological space.
С(X)	stands for the algebra of all continuous
Ж-valued functions on X.
С, (X)	denotes all continuous and bounded Ж-valued
b
functions on X.
conv M	is the convex hull of M.
conv M	abbreviates the closed convex hull of M.
extr M	stands for the extreme points of M.
Compact and locally compact spaces are always understood to be Haus-
dorff, all С, (X)-modules are unital and all topological vector spaces
b	-	-..-	= —
appearing in these notes are supposed to be locally convex.

1. Basic definitions
In many applications of functional analysis, the objects occuring
there are not only topological vector spaces, but carry some
extra structure turning them into algebras, vector lattices,
C*~ algebras etc. As in these notes we would like to deal with all
of them at the same time, we invent the following definition:
1 . 1 Definition. A type т is a mapping т : I И from an index
set I into the positive integers (including 0). A (topological)
il-spaee of type т is a pair (E,F), where E is a (topological)
vector space and where F = (f.). is a family of (continuous)
т ( i )
mappings f^ : E -> E.
An fi-B-space is a topological fi-space such that the underlying
topological vector space is a Banach space.□
We shall often forget the type т and the family F and speak simply
of the fi-space E.
1.2 Examples (i) Let I = {1} and define т(1) = 2. Let (E,F) be
an fi-space of type t. Then F = (f^) and f^ : ExE -* E is a contin-
uous mapping. Instead of f^ (a,b) we shall write a^b. If E happens
to be a Banach space and if • satisfies the equations
Ha-bH < ||a|| ||b||
a- (b + c) = a*b + a-c
(a + b)’C = a>c + b«c
(k-a)"b = k>(a«b)	for all к e Ж
a* (b>c) = (a’b)-c
then E is a Banach algebra.
9
(ii) Let Ж = (E and let I = {1,2}. Define r(1) = 2 and r(2) = 1 and
let (E,F) be an fi-B-space of type t. In this case we have F = (f^,f2>.
The mapping f^ : E*E -* E will again be written as multiplication and
instead of f?(a) we shall write a*.
If • and * satisfy the equations of example (i) and if in addition
(k-a)* = k-a*
(a + b)* = a* + b*
(a"b)* = b*-a*
a** = a
Ha*-a|| = ||a||2
Ila* || = ||a||
then E is called a C* - algebra.
(iii) If we let Ж = ]R, I = {1,2} and т(1) = т(2) = 2, then we may
define Banach lattices in a similar manner.
We now proceed with the central definition of the whole paper:
1.3 Definition. Let p : E -* X be a mapping between two sets Eand X.
If x e X is an element of X, then the preimage p (x) =: Ex of x
is called the stalk over x.
The n-fold stalkwise product of p is defined to be the set
V E = {(а1,...,an> e En : p(a )
= P(an)}.
If there are mappings
add : Ev E -* E
seal : ЖхЕ -> E
D : X -> E
such that
10
poadd(а, В) = p(a)
p°scal(r,a) = p(a)
p»O(x) = x
and such that for every x e X the restrictions of add to ExxEx and
seal to JKxEx turn Ex into a vector space with respect to the
operations a + В := add(a,В) and r-a := seal(r,a) which has O(x)
as a zero, then the triple (E,p,X) is called a fibred vector space.
If т : I + B is a type and if F =	is a family of mappings
r ( i )
f£ : V E -> E
such that p»f.(a ... a , ..) = p(a,), then (E,p,X) is called a
11	т (. i)	1
fibred il-space.
Now let А с X be a subset of X. A selection over A is a mapping
a : A -* E such that p»o = id,. If a, and	are two selections over
A 1	2
A and if к e Ж is a scalar, then we may define the sum + a2 of
and ^2 coordinatewise by
(ct-j + a2) (a) := a1 (a) + a2 (a)	for all a e A.
Similary, the product k>a is defined by
(k • a) (a) := к• (a (a) )	for all a e A.
With these operations, the selections over A form a vector space.
If (E,p,X) is even a fibred fi-space and if a.,...,a ... are selections
r	1	т(i)
over A, then f (a-j, • • •, <\ jjj ) defined by
fi (a1, • • •,	(i) )	:= f ± (o-j (a)	<a>> for all a e A
11
is also a selection over A. In this case, the set of all selections
over A form an fi-space. This fi-space is exactly the product П E of
aeA
the ^-spaces E .
3
Let (E,p,X)
be a fibred vector space. A
mapping v : E +1R is called
a зетгпогт,
provided that for every x e
is a seminorm on the vector space
EX in
X the mapping v : E
' x
the usual sense.
A familiy of seminorms (v.). T is
J J eJ
there is a j
all pairs j Q, j
said to be directed, if for
e J such that for all a e E we
e J
have v .
J o
J
J
If (v.). is a family of seminorms on (E,p,X), then a selection
J J € J
a over A c x is called (v.).	- bounded (or just bounded, of no
J J e J
confusion about the family of seminorms in question is possible), if
sup v.(a(a)) is finite for every j e J. With E- (p) we denote the set
aeA J	A
of all bounded selections over A. If X = A, then we shall use the
symbol E(p) instead of Ev(p)„ □
X
The following remark ist immediate:
1.4	Proposition. For every А с X the set Ед(р) is a subspace of
II E . Moreover, for every j e j ,the mapping 0. : E, (p) -* 1R :
a<A a	J A
a -* sup v.(a(a)) is a seminorm on E,(p). If in addition the restric-
aeA J	A
tion of v. to the stalks E , a e A, is a norm on E . then is
J	a	a	j
a norm on Ед (p), □
If the sets E and X carry topologies and if p : E -* X is continuous,
then every continuous and bounded selection is called a section. With
Гд(р) с Ед(р) we denote the subset of all sections. As above, we
write r(p) instead of ?x(p). If the domain of a section a is open,
12
then a is called a local section.
1.5	Definition. Let т be a type. A bundle of ^-spaces of type т
is a fibred fi-space (E,p,X) together with a directed family
of seminorms on E such that
I)	E and X carry topologies and the mappings p : E -* X,
)
add : EvE -* E, seal : XxE -> E, 0 : X -* E and f. : V E -* E
n	J
are continuous, where V E carries the topology which is in-
induced by the product topology on En.
II)	The c-tubes, i.e. the sets of the form
T(U,a,e,j) := {a e E : p(a) e U and v-(a - a(p(a)) < e}
where U с X is open, a e i^tp), e > 0 and j e J, form a base
for the topology on E.
Ill)	For every choice of j e J, every a e E and every e > 0 there
is an open neighborhood U of p(a) and a section a e Гд(р) such
that Vj(a - a(p(a))) < e.
IV)	We have a = O(p(a)) if and only if (a) = 0 for all j e J.
X is called the base space and E is called the bundle space
1.6	Consequences. (i) The mapping p is open.
(Indeed, we have p(T(U,a,e,j)) = U and the sets of this form are
a base for the topology by axiom II.)
(ii) The seminorms v. : E -* 1R are upper semicontinuous.
(This follows immediately from (]-=»,e[) = T (X,0, e, j ) .)
(ill) For every j e J and every continuous selection a : A E the
mapping x -> Vj(a(x)) : A -> 1R is upper semicontinuous.
(iv)	If А с X is quasicompact, then every continuous selection
a : A + E is bounded.
13
(v)	For every А с X the set Гд(р) is a linear subspace of Ед(р).
Moreover, if A is quasicompact, then Г, (p) is an il-subspace of П E .
A	aeA a
(Indeed, a1 + a2 = add(alZa2) is continuous whenever a and a2 are
continuous and r»a = scal(r,a) is continuous whenever a is. Whence
Гд(р) is a linear subspace of Ед(р), as the triangle inequality
yields the boundedness of 0 + a2-
Now suppose that A c x is quasicompact and let a-| ' • • • ' aT ( e Гд(р).
Then f.(a,,...,a , . ,) is continuous by axiom I and therefore bounded
11 т (1)	-1
by (iv), i.e. f±(a1,...,aT(ip belongs to Гд(р).)
(vi)	Let f e С^(Х) and let a e Гд(р). If we define a selection f«a
by (f«a)(a) = f(a)*a(a) for all a e. &, then we have f«a e Гд(р).
der this multiplication, Гд(р) becomes a (A) - module.
(vii)	For every a e Гу(p), where U с X is open, and for every x e U
the family
{T(V,U/V , e,j,):x e V с и, V open, e > 0 and j e J}
is a neighborhood base at a(x).
(To prove this assertion, let 0 ba any open set around a(x). By axiom
II we may assume that 0 = T(W,o',e1,j). Let r := v.(a(x) - a1(x))	<
< e1. By (iii) we can pick an open set V around x which is contained
in W and satisfies v . (a(y) - a1 (y)) < (e' + r) for all у e V. If
J	*
a is an element of E such that p(a) e V and v. (a - a (p (a)) ) < ,(e'-r)
J	z
then from the triangle inequality we obtain v.(a - a1(p(a))) < e1,
i.e. T(V,a^v ,j(e' - r),j) c 0.)
(viii) The stalks E* ,x e X_, equipped with the induced topologies,
are locally convex topological vector spaces. Moreover, on E^ the
topology induced by E and the topology generated by the sets
{6 : p(6) = x and Vj(a - 6) < e}, where a e E, j e J and e > 0,
agree.
14
a local section a e Fy(p)
triangle inequality yields
Vj(ao - 6) < e}.)
(ix) The seminorms (0.) . T
= sup v.(a(a))
аЛ J
If X is
(Indeed, the topology induced by E is certainly coarser. Pick any
aQ e E. Then, in the second topology, a typical open neighborhood of
aQ looks like {6 : p(B) = x and (aQ - 6) < e} for suitable j e J
and e > 0. By axiom III we can find an open neighborhood U of x and
uch that Vj(aQ - a(x)) < e/2. Now the
aQ e Ex n T(U,a,e/2,j) c {6 : p(B) = x and
on Г-(p) defined by O.(a)
A	J
define a locally convex Hausdorff topology on Гд(р)-
quasicompact, then with respect to this topology, Г(р) is a topologi-
T ( i )	•
oat £l~spaoe3 i.e. the operations : Г (p)	Г(р) are contin-
uous .
т ( i )
(We have to show that for every (a1 ,. . . ,	) e Ftp) , every
Jo e J and every e > 0 there is a 6 > 0 and a j e J such that for
т ( £ )
every r(i) - tuple (p1,...,p^p e Г (p) satisfying
the inequality
\(fi(p1......рт(1)’ " fi(ai......<JT(i))) < c
holds.
Let ap.,,,0^^) e Ftp) , j ° e J and e > 0 be given. As the mapping
f. : V E + E is continuous, the set f.(T(X,f.(a.,...,a ,.,),e/2,
1	t(i)	1	1	1	T(1)
j )) is open in V E and contains the t(i) - tuple (a^x),...,
°T(i)^X^ f°r ever¥ x e X.
Fix x e X for a moment. As V E carries the topology induced by the
product topology on En,
we can find 6.	. . . ., 6 , . . „ > 0
1,x	т(i ) , x
elements
j 1 ,x ' • • •' JT ( £ ) ,x e J and open neighborhoods U	of x
such that
15
т ( i )	т ( i )
JL T(Uk,x'ak/U '6k,x'jk,x) n V E c
К — I	I f X	*
C f7 (T(X,f.(a
i	i
ta /• '),e/2 , j ) )
(we have to use (vii) at this point!) Let U = U n ... n U
X 1/X	T 1 ) f x
6 =? min {6	, . . ., 6 , . .	} and choose j„ e J such that v • v (a) , . . .
X	1/X	H 1 J / X	X	J . Г x
v.	(a) < v. (a) for all a e E. Then we have
JT(i)'X	Jx
r (i)	t(i)
П T(U ,ak/u ,6 ,j ) n V E c f 1(T(X,f (a.,.
. _ 1 X К-/ U X X	1	II

By construction, the open sets U , x e X, cover X. As X is quasi-
compact, there are x ,...,x e X such that X = U u...u U . Define
!	n	X1	xn
6 = min {6x '•••'5x } and choose J
1	n
< v . (a) for all a e E.
J 1
Now assume that the elements p^..
e J such that v. (a) ,..,v .	(a)
JJ v
',pT(i) e Г(р) satisfy the in-
equalities 0. (p - a ),...,0. (p ,., - a ,.J < 6. Let у be any
J 1	'	1	J-j t•, 1 )	T11 J
element of X. Then there is a к e {1,...,n} such that у e U
xk
This yields the following inequality:
v. (p, (y) - a, (y)) £ V. (p, (y) - a (y) )
J x.	J 1
£ 0.^ - 01)
< 6
< 6 for all 1 < к < t(i),
xk
i.e. (p1 (y) , . . . , pt( . j (y) ) e f71 (T (X,f . (a1 , . . . ,ат( . ^) , e/2, j 1) or
Vj (fj (p1 (y) t  • • t pt( i j (y)) - f£ (a1 (y) ,... ,aT(i j (y))) < e/2. To our
satisfaction, this implies the inequality
O.Jf. (Pi,...,pT(. }) - f.(ai,...,aT(i))) < e/2 < e. )
(x) If a, p e Г,(p) and if 0.(a) < 1, 0.(p) < 13 then
л	J	J
Oj(f«a + (1 - f)*p) < 1 for all f e (A) with 0 < f < 1.
(This is an easy calculation using the definitions.)
16
(xi) The mapping add : EvE + E is open.
(Indeed, the map Л : (a, 6) + (a+ 8,8) : EvE -* EvE is a continuous bi-
jection having the continuous inverse Л : (a, B) -> (a - В, B) :
EvE -> EvE and thus is a homeomorphismen. It is straightforward to
see that the first projection л : (a, В) : EvE + E is open.
Whence the mapping add = is open, too.)D
1.7 Remarks. (i) If A с x is not quasicompact, then Гд(р) does not
have to be an fi-space, as the example (1.8(ii)) below shows. Hence
the question, whether or not Гд(р) is even a topological fi-space in
general makes only sense if the subset A c x is quasicompact.
If every section a e Гд(р) can *эе exten^e^ to a global section
a e Г(p) and if A is in fact quasicompact, then a closer look at the
proof of (1.6(ix)) shows that Гд(р) is indeed a topological fi-space.
The same remains true if X is quasicompact and A с x is an arbitrary
subset.
We shall return to the problem of extending sections in paragraph 4.
(ii) The property (1.6(x)) above was called С (X)-convexity by
K.H.Hofmann in [Ho 75] and L.Nachbin in [Na 59].
1.8 Examples. (i) Let E be a topological fi-space of a certain type
т : I -* П with operations (f.) . _. Moreover, let (ц.)  T be an up-
i lel	jeJ
directed family of seminorms on E which generates the topology on E,
i.e. the sets {a e E : Uj (a) < e}, where e > 0 and j e J, form a
neighborhood base at 0 e E. Furthermore, let X be any topological
space. We then can define a bundle of [2-spaces in the following
manner:
Let E := XxE be equipped with the product topology and let p : E * X
be the first projection. Then, up to a natural homeomorphism, we have
n
V E = XxE . If we define the operations add, seal, 0 and (fp^ j
17
"pointwise", i.e.
add	:	(x,a,B) *	(x,a + B)	;	EvE	->	E
seal	:	(r,(x,a))	(x,r-a)	;	EvE	->	E
О	:	x -> (x,O)	;	EvE	->	E
t( i )
fi : (x'a1......“T(i)> (x'fi(al.........aT(i))) ; V E -> E
then (E,p,X) is a fibred fi-space.
Moreover, if we define seminorms v. : E •» В on E by v. (x,a) := ц. (a) ,
J	J	J
then (E,p,X) is a bundle of fi-spaces, called the trivial bundle
with base space X, stalk E and seminorms (v.) T. To verify the
J J € J
axioms II and III, we have only to note that a e Гд(р) is a section
if and only if there is a bounded continuous map £ : A -* E such that
a(x) = (x,E(x)) for all x e A, where A с x is any subset.
Especially, we conclude that Гд(р) is algebraically and topologically
isomorphic to С^(А,Е), where С^(А,Е) denotes the topological vector
space of all bounded E-valued continuous functions, equipped with
the topology of uniform convergence on A.
(ii)	Let E be a Banach space and let f : E + E be any continuous
map which is not bounded. (Clearly, f cannot be linearl) Let A с E
be a bounded set such that f(A) is not bounded. Then E together with
the operation f is a topological fi-space. We now form the trivial
bundle with base space A and stalk E. As we just remarked, we have
the isomorphy Г(p) = Cb(A,E).
Moreover, if г : A -* E denotes the inclusion map, then г belongs to
С^(А,Е), but f«t does not, as this mapping is not bounded. Whence
Ftp) does not have to be an fi-space in general.
(iii)	However, Let us remark that if E is a Banach algebra, a Banach
lattice or a C*-algebra, then С^(Х,Е) is of the same type for every
topological space X.
18
(iv)	Let p : E -*X be a bundle of fi-spaces and let А с X be a subset.
Moreover, let Ед := p (A) and let рд be the restriction of p to Ед.
Restricting the operations add, seal, 0 and (f.). to EvEa, IKxE
T ( 1 )
and V Ед resp. we obtain a new bundle (Ед,рд,А), called the re-
striction of (E,p,X) to A. It is clear that Г(рд) = Гд(р).
(v)	If M is a differentiable manifold and if p : T + N is the tangent
bundle, then we also have a bundle in the sense of (1.5), if we take
as a family of seminorms the usual Euclidean metric on the stalks.
These bundles behave especially nice: They are locally trivial in the
sense that every point x in the base space M has a neighborhood U
such that the restriction (Tof (T,p,U) to U is isomorphic to
a trivial bundle. We shall return to the precise definition and to
a discussion of trivial bundles in a later paragraph.
Obviously, if p : E -> X is a bundle with seminorms (v.).	, then the
J jeJ
seminorms 0^ : E (p) -> 1R induce also a locally convex Hausdorff
topology on E (p). Moreover, we have:
1.9 Proposition. If p : E -> X is a bundle, then p(p) is closed
in E (p) .
Proof. Let a be an element of E(p) which belongs to the closure of
Г(p). We have to show that a is continuous.
Let us start with an element xq e X and let 0 be an open neighbor-
hood of a(xQ). By axiom 1.5.II we may assume that 0 = {a e E :
: p(a) e U and (a - p(p(a))) < e}, where U с X is an open set
around xQ, p e гц (p) is a local section, e > 0 and where	is an
appropriate seminorm. Let 6 = (e - \)j(o(xQ) - p (xQ) ) ) . As a belongs
to the closure of Г (p), we can find a section д' such that
(a - a1) < <$• Hence we have
19
vj(a'(xo>
p(xQ)) < Vj(a'(xo) - p(xq)) + vUa(xQ) - p(xQ))
< 0. (o' - a) + e - 4’6
J
< 6 + e - 4 • 6
- e - 3 • 6
< e - 2-6,
i.e. a1(xq) e {a e E : p(a) e U and v,(a - p(p(a))) < e - 26}. As the
latter set is open and as o' is continuous, we can find an open neigh-
borhood V с и of xq such that v Да'(x) - p(x)) < e - 6 for all x e V.
Whence for every x e V we have
v.(a(x) - p(x)) < v.(a'(x) - p(x)) + v.(a(x) - a'(x))
J	J	J
< v. (a1(x) - p(x)) + 9. (a1 - a)
J	J
<e-6+6=e,
i.e. a(V) c 0. This shows the continuity of a.Q
Before we attack questions of completeness of Г(р), we need a little
observation: Let x e X. Then we have an evaluation map
ex : E(p) * Ex
а -* а (x)
and we shall denote the restriction of ex to Ftp) with the same
symbol . As in both cases we have the inequality Vj(a(x)) < 0 Да) ,
these mappings are continuous and linear. We shall see later, that in
a large number of cases they are also quotient maps, i.e. they
will be open and surjective.
1.10 Proposition. Let p :E -* X be a bundle. If all stalks Ex,
x e X. are complete (quasicomplete, semicomplete), then so are Ftp)
and E(p).
20
Proof. As r(p) is closed in E(p), it is enough to prove these
assertions for E(p). Because in all three cases the proofs are ana-
logous, we only give a proof for the case where all stalks are quasi
complete.
Thus, let A c E(p) be a closed bounded subset and let	j be
a Cauchy net in A. We have to show that the net	T converges.
Firstly, let Ax := ex(A)• As ex is continuous, the set Ax is bounded
in Ex and so is its closure. Now the quasicompletness of Ex yields
that lim a.(x) exists in E . Define a selection
iel 1
a : X -> E
by	x -> lim a, (x) .
iel
Then a is bounded: Let v. be one of the seminorms on E. Then we
J
find an i e I such that v. (a.	- a. ) <1 for all i.,in > i .
о	j 11 i2	1' 2 о
Using the triangle inequality and setting i2 = iQ, we obtain
v.(a. (x)) < 0.(a. ) + 1 for all i. > i . Now let x e X be arbitrary
J i1	J £о	1 о
Then we have v.(a. (x) ) < 0.(a. ) + 1 and therefore v.(a(x)) =
J 11	J 1o	J
= v.(lim a,(x)) = lim v.(a.(x)) < 0.(a. ) + 1. As this holds for all
J iel	iel J 1	J 1o
x e X, we conclude that 0.(a) < O.(a. ) + 1 and whence a is bounded.
J	J i
Finally, the net (Oj^iel converges to a: Indeed, let e > 0 and let
v. be one of the seminorms. Then 0.(a. - a. ) < e/2 for sufficently
J	j i1 i2
large i.,i9 e I, i.e. v.(a. (x) - a. (x)) < e/2 for large i.,i9 and
i	j 11	±2	i z
all x e X. Sending i2 to "infinity", we may conclude that
v.(a. (x) - a (x)) < e/2 for sufficently large i. and all x e X, i.e.
J 11	I
$.(a.	- a) s e/2 < e for sufficently large i1. This completes the
J i -|	।
proof. □
One of the most important classes of bundles are those for which
every stalk is a Banach space. More precisely, we give the following
definition:
21
1.11 Definition. Let т be a type. A Banaoh bundle of il-spaoes
p : E -* X is a bundle of fi-spaces such that
(i) the family (v.) . of seminorms on E consists of one
J J eJ
element || • || only (which then automatically induces a
norm on each stalk).
(ii) every stalk Ex , x e x, is a Banach space in the
norm induced by || • ||. □
1.12 Proposition. If p :E -* X is a Banaoh bundle, then Г(p) is a
Banaoh space in the norm given by ||a|| := SUP || cr (x) II- 0
xeX
2. Full bundles and bundles with completely regular base space
In the following two paragraphs,
This is possible, if we restrict
spaces X.
Let us start with a definition:
we try to simplify definition (1.5)
ourselves to more special base
2.1 Definition.
A bundle p : E -> X
is called full (locally full),
if for every a e
there is a (local)
section a such that a = a(p(a)).
E
□
2.2 Proposition. if X is completely regular, if a : U -> E is a
local section and if x e U, then there is a global section a : X -> E
such that a(x) = a(x).
Proof. Let U с x be open and let a : U -> E be a local section. As
X is completely regular, we can find a neighborhood V c U of x such
that the closure V of V is still contained in U. Moreover, we can
find a continuous mapping f : X -* [0,1] c ]R such that f (x) = 1 and
such that f vanishes on the complement of V. We now define
a : X -* E
_	r f (x) • a (x) if x e U
by	a(x) = <	_
1 0	if x e X \ V
This definition makes sense, as on U n (X \ V) we have f(x) = 0, i.e.
f(x)«a(x) = 0. Clearly, a is continuous, as it is continuous on the
open sets U and X \ V. □
23
2.3	Proposition. If X is completely regular, then every locally
full bundle over X is full. □
If we restrict ourselves to locally full bundles, we need no longer
the whole strenght of axiom (1.5.III). This means: If we make the
third axiom in (1.5) stronger, then we may virtually weaken axiom II.
To be precise:
2.4	Theorem. Let (E,p,X) be a fibred il-space of type r together
with a directed family (v.)of seminorms on E such that the
J jeJ
foI lowing axioms are satisfied:
I)	E and X carry topologies and the mappings p : E -* X,
t(i)
0 : X -> E, add : EvE -* E, seal : IKxE -* E and f^ :	\/ E E
are continuous.
II)	Given x e X, then the sets of the form
{a e E : p(a) e U, (a) < e}
where U is an open neighborhood of x, e > 0 and j e J , are
an open neighborhood base at 0(x) e E.
Ill)	Given a e E, then there	is an open	neighborhood U of p(a)
and a local section a :	U -> E such	that a(p(a)) = a.
IV)	a = O(p(a)) if and only	if v^. (a) =	0 for every j e J.
Then (E,p,X) is a bundle of il-spaces, which is, of course, locally
full.
Proof. We only have to check axiom (1.5.II)
Firstly, note that for every local section a : U -> E the mapping
Ta : p 1 (U) -> p 1 (U)
a -> a + a (p (a))
is continuous (being the composition of
24
a -* (a,p(a)) -> (a,a(p(a))	-* a + a(p(a))
where Ey := p (U) as in (1.8.iv) ). Moreover, the mapping T_a is a
continuous inverse for . Thus, is a homeomorphismen.
Now fix a e E and choose any local section a : U -* E such that
a(p(a)) =	a. Then the homeomorphism Tff	transports	the	open	neighbor-
hood base	{{В	e E : Vj(B) < e, p(£3) e V}	: e	> 0,	j	e	J, p(a) e V с и
V open.} of 0(p(a)) onto the open neighborhood base
{ {£3 eE:	- a(p(8)) < e and p(£3)	e	V}:	p(a)	e	V	с и,	V open,
e > 0 and	j e	J} of T (0(p(a))) = a.D
Now, of course, the question arises, which bundles are locally full.
It turns out that under certain restrictions on the completeness of
the stalks, all bundles with a locally regular base space are locally
full and hence full by (2.3)
2.5 Proposition. Let p : E -* X be a bundle with seminorms (v.) . T
J JeJ
and with a completely regular base space X. If x e X is fixed and
if cx : Г(p) -* Ex is the evaluation map, then for every a e ех(Г(р))
с E and every j e J we have v . (a) = inf {$. (a) : a(x) = a, a e Г(p) }
X	.1	.1
Proof. Clearly, we always have the inequality
v.(a) < inf {0.(a) : a(x) = a, a e Г (p) }.
J	J
To verify the other inequality, pick an a e Ex, let e > 0 and assume
that there is a o' e Ftp) such that a'(x) = a. By (1.6.(iii)), the
mapping у -> v. (o*(y) ) : X + В is upper semicontinuous. As v.(a'(x)) =
J	J
= Vj(a) < Vj(a) + e, we may find an open neighborhood U of x such
that Vj(a1(y)) <	(a) + e for all у e U. Now the fact that X is com-
pletely regular yields a continuous function f : X -> [0,1] c ]R with
25
f(х) = 1 and f(X \ U) = {0}. Define a = f-a1. Then for every у e U
we have v.(a
J
may conclude
(y) ) = f(y)<\).(a'(y)) < v.(a) + e
that v . (a(y) ) = v.(f
J	J
whence 0 . (a) < v.(a)
J	J	J
arbitrary, the proof is complete.
J ""
(y) -a1 (y) )
and for у e X \ U we
vjfO'O1 (y) ) =0 <
+ e. As a (x) = a and as e > 0 was
□
2.6	Corollary. If p : E -* X is a bundle with a completely regular
base space X, then the evaluation maps e : Г (p) -* E , x e X, are
X	x
open onto their images. □
2.7	Corollary. If p : E -* X is a (locally) full bundle with a
completely regular base space X, then for every x e X the evaluation
map ex : Г (p) -* Ex is a quotient map. □
2.8	Definition. Let (E,p,X) be a fibred vector space, let (v.) . _
J jeJ
be a family of seminorms on E and assume that the base space X
carries a topology. If every point x e X has an open neighborhood
U with the property that {j e J :	(a) / 0 for some a e p (U)}
is countable,then the family of seminorms (v.). is called locally
J J e J
countable. □
If p : E -* X is a bundle with a locally countable family	°f
seminorms, then every point x e X has a neighborhood U such that
Гу(р) is metrizable. Moreover, every stalk Ex is metrizable in the
topology induced by E. This allows us to apply Banach's homomorphism
theorem in the proof of the following result:
2.9	Theorem. Let p : E -> X be a bundle with a completely regular
base space X and a locally countable family of seminorms (\>J
If all st-alks Ex_, x e X, are complete, then p : E -* X is a full
bundle and the evaluation maps ex : Г(p) -* Ex are quotient maps.
26
Proof. By (2.3) it is enough to Show that p : E ~ X is locally full.
Thus, let x e X and let U be an open	neighborhood of	x with the
property that {j e j :	(a) / 0	for	some a e p 1 (U)	} is countable.
Then Гу(р) is metrizable	and, by	(1.10), the vector	space Гу(р) is
complete. Let ex : Гд(р)	Ex be	the	evaluation map.	Then (1.5.Ill)
and (2.2) applied to the bundle p : p (U) -> U, show that the image
of ex is dense in Ex> Moreover, by (2.6) , the mapping ex is a topo-
logical homomorphism . Whence, by Banach's homomorphism theorem,
the image of ex is closed. Thus, we may conclude that the mapping
ex is surjective and therefore the bundle p : E -* X is locally
full. □
2.10	Corollary (Dupre'). If X is a completely regular topological space
then every bundle of Banach spaces p : E X with base space X is
full. Moreover, all the evaluation maps ex : Г(p) -* Ex are quotient
maps of Banach spaces, i.e. 11 a 11 = inf { )|a|) : ° e Г (p) and a(p(a)) =
= a}.	□
We conclude this section with two results which we need for later
references:
2.11	Proposition. Let p : E + X be a bundle with a completely
regular base space X. Then {f-a : f e (X), f(x) =0, a e Г(p)}
is dense in {a e Г(p) : a(x) = 0}.
Proof. Let a e Г(p) be such that a(x) = 0, let v be one of the semi-
norms belonging to the bundle and suppose that e > 0 is given. We
shall complete the proof by constructing a continuous function
f e С, (X) with f(x) = 0 and O.(f*a - a) < e.
ь	J
First of all, let U be an open neighborhood of x such that v (a(y)) <
< e/2 for all у e U. Such an open set exists as v (a(x)) = 0 and as
27
the mapping у -* \)Да(у) is upper semicontinuous. As X is completely
regular, there is a continuous mapping f : X -> [0,1] c ]R such that
f(x) = 0 and f(X \ U) = {1}. An easy calculation shows that in fact
we have v^. (f*a - a) < e/2 < e. □
2.12	Proposition. Let p : E -* X be a bundle with an arbitrary base
space X. If all stalks are finite dimensional, then p : E -* X is
locally full.
Proof. Let x e X and let Sx := {a(x) : a e Гу(p) for some neighbor-
hood U of x}. By axiom (1.5.Ill), the set Sx is a dense subspace of
the stalk Ex. As Ex is finite dimensional, we conclude that Sx = Ех-П
3. Bundles with locally paracompact base spaces
The definition of bundles given in (1.5) is rather complicated. The
most annoying axiom is the postulate (1.5.II), because in many
applications we would like to use bundles to describe topological
vector spaces as spaces of sections in a bundle. Hence it is
unsatisfactory to use sections already in the definition of bundles.
But if the base space is locally paracompact, if the family of semi-
norms is locally countable and if the stalks are complete, the
existence of "enough" local sections follows from the other axioms.
3.1 Definition. A topological space X is called locally para-
compact, if every x e X has at least one closed and paracompact
neighborhood. □
It can be shown that every locally paracompact space is completely
regular. On the other hand, every locally compact space, every
paracompact space and every locally metrisable space is locally
paracompact. Moreover, in a locally paracompact space every point
has a neighborhood base of closed and paracompact sets.
The central result of this section is stated as follows:
3.2 Theorem. Let (E,p,X) be a fibered vector space, let (v.). be a
directed family of seminorms on E and assume that E and X carry
topologies such that
(0) p is open and continuous.
(I)	the mappings add : EvE -* E and seal : IKxE -* E are contin-
uous .
29
(II)	If Ox e Ex is the О-element of the stalk Ex, then the sets
of the form {a e E : p(a) e U and Vj (a)< e} form an open
neighborhood base at 0x, where U runs through all open
neighborhoods of x, e > 0 and j e J.
(Ill)	a = O(p(a)) if and only if v (a) = 0 for all j e J.
If all stalks are semicomplete in the topology induced Ъу E and if
X is locally paracompact and |J| - 1 or if X is locally compact and
if the family (Vj)j j is locally countable, then (E,p,X) is a full
bundle.
This theorem has orginally been proved by A.Douady and L.Dal Soglio-
-Herault for bundles of Banach spaces (see the appendix of [Fe 77]).
Our version here is, up to some corrections and modifications, due
to H.Mfiller (see [Мб 78]).
We shall prove(3.2) in several steps. Firstly, we shall assume that
the family (v.)• , is countable and develop some results in this case.
3 J
Hence we may assume that J =JN and that n < m implies v (a) < \>m(a)
for all a e E. Moreover, we shall always assume that X is completely
regular.
3.3	Let us agree that we call a subset U с E an e-n-thin set, if
for all (a,8) e UxU n (EvE) we have v (a - 8) < e.
Fix an arbitrary a e E. If n £ M and if e > 0 are given, then a has
an e-n-thin neighborhood.
(Indeed, let V = {a e. E : v (a) < e}. By axiom (3.2.II) , the set
V с E is open. As the mapping (6,6') -* £3 - £31 : EvE + E is continuous
by (3.2.1) and as (a,a) is mapped onto O(p(a)) e V under this map,
there is a neighborhood U of a such that UxU n (EvE) is mapped into
V. This set U then will be e-n-thin.)
3.4	Let a e E, let Л be a directed set and let (e.). be a net of
30
strictly positive numbers such that lim e =0. Further, assume that
A e Л л
for every n e M there is an
e-n-thin neighborhood n of a such
that
A^ < A2 and n^ < n2
The sets {p(U ) :
Л । n
imply U	c U .
A2' 2 A1 ' 1
A e Л and n e И} form a neighborhood
base of p(a).
Then the family {U^ n
A e Л and n £ B) is a neighborhood base of a.
(Put x = p(a) and V = p(U ). Then it is easy to see that the
А	А
sets 0	= {В e E : v (B) < e, and p(B) eV }, A e Л and n e TJ,
A pl	И	А	Л г n
are a neighborhood base at 0(x).
Now let W be any neighborhood of a. Since the mapping add is contin-
uous and since add(a,0(x)) = a, we can find a neighborhood W1 of a
such that add(W'xO n EvE) c W, if only A and n are large enough.
Л г П
As p is open by (3.2.0) and as W1 n U is a neighborhood of a, we
A r n
may choose A1 > A and n1 > n such that V ,	, c p(W' n U )•
Л F	Л F Г1
We claim that и. ,	, c W.
A ,n
Firstly, we have U ,	, c U ,	, n p 1(p(W' n U )). Let В e U ,	,
Л f-11	Л f Г1	Af11	A f
Then В belongs to U . Moreover, there is an element B1 e W1 n U
Л F	A F 11
such that p(B) = p(B')- Now U. is e-n-thin, i.e. v (В ~ B1) < e-i
Л Fn	A	n	A
and hence В - В1 e 0	. This implies В = В1 + (В- B') =
A f Г1
= add(B,(B - B')) e add(W'xOA „ n EvE) c W, i.e. U, , c W.)
A F	A F 11
3.5	let a : X -> E be a selection. We say that a is e-n-continuous,
provided that
(i)	The mapping x -> \>n(a(x)) : X -> IR is bounded.
(ii)	For every x e X there is a neighborhood V of x and an
e-n-thin neighborhood U of a(x) such that f(V) c U.
3.6	If о : X + E is a selection which is e-n-continuous for every
e > 0 and every n e TJ, then a is bounded and continuous.
31
Clearly, a is bounded by (3.5(i)). Now fix x e X. For every pair
(m,n) of natural numbers let U' be an 1-n-thin neighborhood of
m,n	m
a(x) such that a(p(U' )) c u' (such an U' exists by (3.5(ii)).
m,n m,n	m,n
Define
U = n {U' . : l,k s n}
n	l,k	J
Then U is -1-n-thin, fulfilles still the inclusion a (p (U )) cU , but
n n	n n
we have in addition the relation U , „ с и .
n+1 n
Next, let (V ) be a decreasing neighborhood base at x and let
Л Л z A
I = TJxA. For every i = (n,X) we define e- = — and U. = p (V ) n U .
in	i f n	Л n
Clearly, the net	converges to 0, all the set IT n are
e^-n-thin and the sets of the form p(1Л n) form a neighborhood base
at x. Whence, by (3.4), the sets (U^ n>n form a neighborhood base
at a(x). Moreover, by definition of the set U. we have
i ,n
a(p(U^ n)) c n for every iel and every n e TJ, thus the map a is
continouos at x.)
3.7	Let f„,...,f : X +1K be continuous IK-valued functions on X
1 n
such that
n
У f.(x) / 0 for all x e X
i=1	1
Then the mapping
n
Ф :	V E -> E
(a1(...,an) -* У fi(p(ai))*ai
is open and continuous.
(The continuity of ф follows easily from the continuity of the map-
pings add and seal.
n
To show the openess of ф, let (a^,--.,an) e V E and let x = p(a.j) =
=...= p(an). We may assume without loss of generality that f^ (x) / 0.
Let V be an open neighborhood of x such that f^ (у) / 0 for all у e V.
32
n -1 n	n
As V E n p (V) is open in v E , it is enough to show that the
n	1 n
restriction of ф to V E n p (V) is open.
Define a mapping
n	1	n _.
T : V Enp” (V)n + V E np (V)n
n
(	, . . . , aR) -*	( Д f (p ( a^) ) •	» «2 » • • • » o^) •
Evidently, T is continuous and has the continuous inverse
T : V Enp (V)n + V E n p (V)n
-1	n
(a^,...,an)	->	(f^(p(a^)) "(a. - У f . (p ( a^) ))•	, «2 » • • • » c^)
i=2
and therefore is a homeomorphism.
By definition of the
л : (,. . . , an) ->
topology, the restriction of the first projection
n
: V E + E is open. As we have ф = the
mapping Ф is open as well.)
3.8 Let ff : X -»]R be continuous real-valued functions, let
1 n
A.,...,A c E be e-m-thin subsets of E and let
1	n
n
M := sup { У |f±(x) | : x e p(A1) n .•• n p(Ar) }.
i=1
n	n
If Ф : V E •» E is defined as in (3.7) , then Ф( V E n A^ x...xAr) is
M* e-m-thin.
n
(Indeed, let а, В e Ф ( V E n A^x... xAn) and let x = p (a) = p (£3) •
Then we may find elements a-»B- e A., 1 < i < n, such that
n	n1	1	1
a = У f• (x) -a. and g = у f. (x)»g.. Since the sets A. are e-n-thin,
i=1	11	i=1	1	1	1
we know	that \>п(а^	- fh)	< e	for	all	i e {1,...,n}.	This	yields the
inequality
n
vn(a - 8) = vn( J fi(x).(ai - B±) )
n	n
< У I f, (x) I - V (a. - 8.) < У I f _. (x) I - e < M« e.)
i=1	nil i=1 i
33
From (3.7) and (3.8) it is easy to conclude
3.9	If f.,...,f	: X -> [0,1 1 c IR are continuous functions such
n 1
that У f.(x) = 1 and if aa : X -* E are e-n-continuous
i=1	1	n	1 n
selections, then У f.«a. is a e-n-continuous selection, too.
i i
3.10	Let e > 0 and let (a ) с S(p) be a sequence of e-n-contin-
m mg
uous selections and let a : X -> E be a selections such that
lim 9 (a_ - a) =0. Then a is а 2•e-n'-continuous selection foi
_ n m
1П->оо
every n' < n.
(Firstly, by hypothesis there is a positive integer m such that
9п(% -	< e/2
i.e.
v„(a (x) - a(x)) < e/2 for all x e X.
n m
Fix xq e X and let U be an open neighborhood of xq and let V be an
e-n-thin neighborhood of a (x ) such that о (U) с V. Moreover,
mo	m
define
W := {a + 8 : p (a) = p (8) e U, a e V and vn (B) < e/2}
As the set {a : vn(a) < e/2 and p(a) e U} is open, the set W is open
by (3.7). Moreover, {a : v (a) < e/2 and p(a) e U} is e-n-thin,
thus W is 2«e-n-thin by (3.8) . Further, for every x e V we have
a(x) = am(x) + (a(x) - c?m(x) ) e w, i.e. a(V) c W. Finally, for
all n' <, n and all a e E we have v . (a) < v (a) . This implies that
n	n
the set W is 2«e-n-thin, too.
Now let m e IN be a natural number such that 9 (о - a) < 1 . Then
n m
the triangle inequality yields for every x e X the relation
vn, (a (x) ) < vn(a(x) )
S Jn(°m(x) ’ a(x)) + Vn(am(x))
S 1 + Vam(x))-
34
As х + v (a (x) ) is bounded, so is x -> v  (a(x)). Thus, the selection
n m	n
a is 2•e-n'-continuous.)
From (3.6) and (3.10) we may deduce:
3.11	Let (on)neN be a sequence of selections such that cn is
e -n-continuous. If the sequence (e ) converges to 0 and if
П	П П € U
a : X + E is a selection such that lim <) (a - a) = 0 for every
n-*°° m П
m e JN, then a is a continuous and bounded section.
3.12	If e >0, ifn e IN and if a e E are given, then there is an
e-n-continuous selection a : X -* E such that a(p(a)) = a.
(Firstly, by (3.3) the element a has an open e-n-thin neighborhood
V. Let V := V n {g : v ( B) < vn (a) + "И- Then V is still an open
e-n-thin neighborhood of a. Furthermore, let U = p(V) and let
x = p(a). By the axiom of choice we can find a selection o' : U -> E
such that o' (U) с v anda'(x) = a. Note that x \>n(a(x)) is auto-
matically bounded, as we have v (B) < 1 + \>n(a) f°r all g e V. Next,
choose a continuous function f : X -> [0,1] c 1R such that f (x) = 1
and f 1([0,1 ]) c u. We now define a selection a by
a (y)
f(y)(y)
0
if у e U
if у | U
On U we have a = f-a' + (1 - f) *0, whence a is e-n-continuous on U by
(3.9). On the open set V = X \ f \[0,1]) , the selection a agrees
with the continuous section 0. As U and V cover X and as a(x) =
= f(x)-a'(x) = 1*a = a, the proof is complete.)
3.13	Assume that X is compact or that v = v for all n,m e JN. Given
r —	n m
an e-n-continuous selection a:: X -* E and a point xq e X, there is a
e/2-(n+1)- continuous selection o' : X E such that
35
(i)	fln(o' - a) s |-e
(ii) a' (xQ) = a(xQ) .
(Let x e X. By (3.12) there is an e/2-(n+1)-continuous selection
such that ax(x) = a(x). Obviously, ax is also е/2-n-continuous.
Choose an open e-n-thin neighborhood v of a(x), an е/2-n-thin
neighborhood of ax(x) (= a(x)) and an open neighborhood U of x
such that o(U) c v and ax(U) c W. As V n W is an open set around
о(x) and as p is open, the set Ux := и n p(V n W) is an open neigh-
borhood of x. Moreover, if у is an element of U , then there is an
a e V n W such that у = p(a). Hence we obtain the inequality
~ °(У)) £	(У) - “) + v (a - a(y))
s e + e/2
4-
as a, a(у) e V and a/ax(y) e W.
Let U be any open neighborhood of x such that и с и . Replacing
_	о
each Ux ,xq / x e X, by Ux \ U, we obtain the following:
There is an open cover (UjJieI of X and e/2-(n+1)-continuous
selections (a.). _ such that
i iel
(i) xQ belongs to exactly one CL and for this index i we
о
have a. (x ) = a (x ) .
i о	о
о
(ii) For every i e I and every x e It we have the inequality
Vn(°i(x) " a(x)) ~ l’e-
As X is paracompact, we may find a partition of unity	sub-
ordinate to the cover (ир£е1- ВУ the property (i) above we may
conclude that f (xQ) = 1 and f-j_(xo) - 0 for i / i . We now define
о
our selection o' by
o' = .L f.’a.
1 el 1 i
36
Note that this sum is locally finite, i.e. each point has a neighbor-
hood U such that {i : f(U) / {0}} is finite. Whence (3.9), applied
to these neighborhoods shows that the restriction
U -> E
a'/U :
U :
p 1(U) of o' to U is e/2-(n+1)-continuous. Especially, o' will
satisfy the property (3.5(ii)), which is a local property.
Before we show the n+1-boundedness of o', we prove that o' has the
properties (i) and (ii) of (3.13):
As a'(xq) = 2 f±(x );а±(х ) = f (xQ).а£ (xq) = a(xQ), the property
i el	о	о
(ii)is satisfied.
Next, we check property (i): Let x e X be fixed and let i £ I be any
element. Then we have either x | th, in which case we conclude
f i (x) • \>n ((x) - a (x) ) = 0; or we have x e th, and then it is true
that f^ (x) • \>n (o^ (x) - a (x) ) < ^.f (x)»e. Whence for all iel and
all x e X we have f.(x)•v (a.(x) - a(x)) < 4 f.(x)• e. As the f.(x)
i n i	z i	i
sum up to 1, this yields for every x e X the inequality
vn(a'(x) - a(x)) = vn ( У ^(х)-а±(х) - ( У f ± (x) ) • a (x) )
iel	ieI
= v ( У fi(x).(ai(x) - a(x))
iel
£ У f (x) • v (a. (x) - a (x) )
iel
- Л f i ‘ 2 ’ e = 2 ’ £
iel
and therefore (i) holds.
Finally, we have to show the n+1-boundedness of o':
Firstly, assume that X is compact. As the family (f^)^ is locally
finite, an easy compactness argument shows that there is a finite
subset J с i such that f / 0 if and only if i e J. Therefore the
sum we used in the definition of o' is actually finite. Thus, o' is
е/2-n+l-continuous by (3.9)
Now assume that v = for all n,m e IN. Then the triangle in-
equality and the fact that 9n+^ (o' - a) = 9n(a' ~ a) yield that the
37
map x -> v (o'(x)) is bounded, i.e. the property (i) of (3.5) holds.
As we checked the property (ii) of (3.5) already, the proof is com-
plete .)
Applying (3.12) and (3.13), we obtain after an obvious recursion:
3.14	Assume that either X is compact or that v = vm for every
pair of natural numbers n,m. If a e E is given, then there exists
a sequence (q ) „of selections such that
n ngB
(i)	an is (-1) n-n-continuous.
(ii)	Van+i " an) - i’(i)n for a11 n e3N
(iii)	an(p(a)) = a for all n e JN
3.15	If (a )	„ is a sequence of selection which satisfies the
П ПбЮ}
properties (i), (ii) and (iii) of (3.14), then there is a selection
a such that lim $ (a - a) = 0 for all m e JN. This
m n
П->оо
fies in addition the equation a(p(a)) = a.
selection satis-
(Let e > 0 and let n e JN. Choose a natural number N such that
о
nQ < N and such that
V	3 Лп
у	2 (7) < e.
n=N
Then for all natural numbers m > n N we have the inequality
m-n-1
no m n no i=0	n+i+1	n+1'
m-n-1
r	<>n (an+i+1 " %+i’
1=0 о
m-n-1
- Д ^n+i^an+i+1 ~ °n+P
1=0
, 3, , 1, n+i n	/ 3. , 1.1
<	)	(7) • (7)	< У (y) • (y)	< e-
i=0	i=N
38
In particular, since for every x e X the sets {a e E : \>n (a) < e }, n e ®,
and e > 0, form a neighborhood base of 0 in the stalk Ex with the
induced topology, the sequence (ап(х))п^ is a Cauchy sequence. As
all the stalks are assumed to be semicomplete, lim a (x) exists in
n-*°°
E . Now define a : X -* E by a (x) := lim a (x) . It remains to show
n->°°
that lim 0 (an ~	= 0 f°r aH no e ]N‘
n->“ о
Thus, let e > 0 and let n e ®. As we have seen above, we can find
о
a natural number N such that (a - a ) < e/2 for all m,n > N.
no m n
Whence, for every x e X and all m,n > N we have v (a (x) - a (x)) <
n m	n
о
< e/2. Sending n to infinity, this yields v (a (x) - a(x)) < e/2 for
о
all m > N. Since this inequality holds for all x e X and sincee >0 was
arbitrary, this yields lim (a - a) = 0.
n-*00 о
Clearly, by the definition of a, we have a(p(a)) = lim v (p(a)) =
n->“
= lim a = a.)
n-*°°
Now (3.11), (3.14) and (3.15) allow us to conclude:
3.16 If X is compact or if v = v for all n,m e JN, then for
— n m
every a e E there is a continuous section a : X E such that
a (p (a) ) = a.
It is now easy to finish the proof of our theorem: Let a e E and
let x = p(a). Choose a paracompact (resp. compact) neighborhood и of
x (such that only countably many of the seminorms have value differ-
ent form 0 on p 1(U) ). Now (3.16) applied to (p 1(U),p,U) yields
a bounded and continuous section a : U -* E such that a(x) = a.
Now (2.3) and (2.4) together with the fact that every locally para-
compact space is completely regular show that (E,p,X) is a full
bundle.

4.	Stone - WeierstraB theorems for bundles
The classical theorem of Stone and WeierstraB has been generalized
in many ways (see [Bu 58], [Br 59], [Bi 61], [G1 63], [We 65],
[NMP 71 ], [Ho 75], [Gi 77], [Mo 78]). The results which will be
represented in this section are due to Machado, Nachbin and Prolla
([NMP 71]) and K.H.Hofmann ([Ho 75]).
4.1	Definition. Let p : E + X be a bundle. A family	c
c С(X) is called locally finite, if every point x e X has a neighbor-
hood U such that {i : f^(y) / 0 for some у e U} is finite.
A subspace F c Г(р) is called fully additive, if for every locally
finite family	c (X) and every family (с^)1е1 c F the
selection £ ^i‘ai belongs to F, provided that this selection is
bounded.
A subspace F c Г(р) is called stalkwise dense, if for each x e X
the set ex(F) is dense in the stalk E , where ex : Ftp) -* Ex denotes
the evaluation map. □
It is obvious that every fully additive subspace F of г(p) is also
а С, (X)-submodule.
b
We now turn to our Stone-WeierstraB theorem, which in this form is
due to Hofmann, Machado, Nachbin and Prolla:
4.2	Theorem. Let p : E -* X be a bundle and let F c Ftp) be a
fully additive and stalkwise dense subspace of Ftp). Then under
each of the following two conditions, F is dense in Ftp):
40
(i)	The base space X is compact.
(ii)	The base space X is paracompact and p : E -> X is a bundle
of normed spaces, i.e. |j| = 1.
Proof. Let a e Г (p)• Then we have to show that for every e > 0 and
every seminorm Vj belonging to the bundle there is a section p e F
with 0j(a - p) < e.
Firstly, fix an arbitrary point x £ X. As F is stalkwise dense,
there is a section px e F such that \м(рх(х) - a(x)) < e/2. By
(1.6. (iii)) we can find an open neighborhood Ux of x such that
Vj(px(y) “ a(y)) < e/2 for all у e Ux.
Now the open sets Ux, x e X,cover X. As X is at least paracompact,
we may choose a partition of unity
ver (U )	. Especially, the family
X X Сл
(f ) subordinate to the open co-
X x cX
(f ) с С, (X) is locally finite.
X X ^X D
We define
Then p : X -> E is a continuous selection. Moreover, p is a
bounded selection: Indeed, if X is compact, then the boundedness
of p follows from (1.6.(iv)). On the other hand, if p : X + E is a
bundle of normed spaces, then the family of seminorms (v.). T
1 J
consists of one element only, which is just the seminorm we
used above.	in this case we only have to show that the
mapping x -> Vj (p(x) ) : X +1R is bounded. As we shall see in a moment,
we have Vj(p(y) - p(y)) < e for all у e X. Thus, in this case the
boundedness of p follows from the triangle inequality and the
boundedness of a.
Thus, in both cases the selection p will belong to F. It remains
to show that (a - p) < e.
Let us start with a у e Y. Then we may compute:
Now we have either у e Ux and hence \м(р(у) - Px(y)) < e/2, or
we have у $ Ux, in which case fx(y) = 0. Thus, in both cases we may
conclude that fx(y)•vj(а(у) - px(y)) - ^х(У)’е/2. This implies the
inequality v.(a(y) - p(y)) s	f (y) = e/2, i.e. $.(a - p) <
e/2 < e. □
For convenience we state the version of (4.2) which we shall use
most often:
4.3	Corollary. Let p : E -* X be a bundle of Banaoh spaces over
a compact base space X and let F c r(p) be a stalkwise dense
С(X)-submodule of Г(p). Then F is dense in Г (p)	□
We conclude this section with an application of our Stone-WeierstraB
theorem. In (2.9) we have seen that for completely regular base
spaces and "locally" completely metrizable Г(p) the evaluation
maps ex : r(p) -> Ex are quotient maps.
Now suppose that A c X is any subset. Then we also have an evaluation
map e : p -> p . : Г (p) -* Г (p) . Again we ask for conditions under
A	/ A	A
which this map is a quotient map. The Stone-WeierstraB helps to
find an answer:
42
4.4	Theorem. Let p : E X be a bundle with a countable family
of seminorms and assume that X is normal and that all stalks are
complete. If А с X is compact, then every section : A-> E may be
extended to a global section. Moreover, the evaluation map
eA :	7-s a Quotient map.
Proof. As Ftp) and Гд(р) are complete and metrizable, using Banach's
homomorphism theorem it is enough to show that is a topological
homomorphism, i.e. ед is open onto its image, and that the image
of ед is dense in Гд(р).
The fact that ед is topological follows as in (2.5) and (2.6) using
the normality of X instead of the regularity in the proofs.
It remains to show that the image of ед is dense in гд(р):
As p : E -* X is a full bundle by (2.9), the image of ед is stalk-
wise dense. Moreover, the image of ед is a C(A)-submodule of гд(р):
Let a e Гд(р) be of the form ° = £a*'°'^ for a certain °' e Г(р) and
let f e C(A) be a Ж-valued continuous function on A. As X is normal,
we can find an extension f e С, (X) of f. Now we have f-c =
b
= f'/A-a'/A = (f'-a,)/A = £A(f'-a,)-
Now an application of the Stone-WeierstraB theorem (4.2) completes
the proof. □
The following corollary, which is analougos to (2.10), has been
proved and reproved by several authors : K.H.Hofmann credited this
result to M.Dupre, J.M.G.Fell has shown it in [Fe 77], and J.W.
Kitchen and D.A.Robbins proved an even stronger version for compact
base spaces in [KR 80]: Every section a : A -> E may be extended
under the preservation of norm to a global section, provided that
X is compact, A с x is closed and p : E -» X is a bundle of Banach
spaces.
43
4.5	Corollary. If p : E -* X is a bundle of Banach spaces over a
normal base space X and if А с X is compact, then every section
a : X -* E may be extended to a global section. Moreover, the
evaluation map ед : Г(p) -* Гд(р) is a quotient map of Banach
spaces. □

5.	An alternative description of spaces of sections: Function modules
There is an alternative way of describing spaces of section which
does not make use of the topology on the bundle space E. For
bundles of Banach spaces, this description is due to F.Cunningham
(see [Cu 67]), and for the general setting we refer to the
paper of Nachbin, Machado and Prolla (see [NMP 71]).
Suppose that we begin with a bundle p : E + X of fi-spaces of a cer-
tain type т : I -* JN and suppose that the base space X is compact.
Let E := r(p). Then we know from (1.6), (1.9) and (2.2) that E
has the following properties:
(FM1) For every x e X there is a topological vector space E^; the
topology of E is induced by a family of seminorms (\m).	.
X	J J €«J
(FM2) E is a closed linear subspace of П°° E , where
x£X
oo	X
П E = {a e П E : sup v.(o(x)) < « for all j e J},
xeX x	x£X X x£X 21
equipped with the topology induced by the seminorms
(^)
jeJ
given by
0. (a) = sup v2? (a (x) ) .
3	xeX 3
(FM3) The set {a(x) : a e E] is dense in Ex for every x e X.
(FM4) The mapping x v^(a(x)) : X+JRis upper semicontinuous
for every a e E and every j e J.
(FM5) E is a (X)-module relative to the multiplication given by
(f»o) (x) := f(x)«a(x) for all x e X, f e %(X) and a e E.
45
(FM6) Each of the is a topological fi-space of type т and E is
a topological fi-subspace of ц°° E , i.e. if i e I and if f.
x eX
is one of the additional operations, then for all , . . . ,
e E we have f , (, • • • , ат ) e E and the mapping
f± : Ет(1) -> E
is continuous, where f^(u^
а ,.. ) (x) = f (a (x),. . . ,
T (1)	11
а ,.. (x)) .
т (i)
5.1	. Definition. Let X be a topological space and let т : I+ К be
a type. If E is a topological vector space satisfying (FM1) - (FM5),
then E is called a function module with seminorms (\m).	. Moreover,
1 3
if the axiom (FM6) holds, then E is called an ft-function module of
type t. The space X is called the base space; the vector spaces
Ex, x e X, are called the stalks of the function module. □
We shall see that every JJ-function module is in fact (isomorphic to)
the space of all sections of a bundle of fl-spaces of the same type,
provided that the base space is compact.
Let us start with the so called "standard construction of bundles",
which is due to K.H.Hofmann (see [Ho 75]) in the case of Banach
bundles and has been generalized by H.Moller ([Mo 78]) to our
present situation:
5.2	Let (E ) be a family of vector spaces. If we set
x x
E := и (x}xEx and p : (x,a) -* x : E -* X, then (E,p,X) is a fibred
xeX
vector space. As we already remarked in (1.4), every element of the
cartesian product П E may be viewed as a selection of (E,p,X).
xeX
Further, let (v^)^£j be a directed family of seminorms on Ex generat
ing a Hausdorff topology . Then we may define a directed family
46
(v ) . of seminorms on g by v ((x,a)) = v (a) .
] ]EJ	J	J
Suppose now that X carries a topology and that E is a linear sub-
space of п E such that (FM3) and (FM4) are satisfied. Then we have
xeX X
5.3	The sets of the form T(U,a,e,j) := {a e g : p(a) e U and
Vj(a - a(p(a))) < e}, where U с X is open, a e E, e > 0 and j e J,
form a base for a topology on E.
(We have to show that for each a £ T(U^,	,j) n T (U2 , a2 > e2'2^
there are an open set U3 с X, an element £ E, an > О and an
j3 e J such that a e T(U3,a3,e3,j3) с T(U1,	,j1) n T(U2 . a2 , e2,j2) .
Thus, let us suppose that such an и is given. Let j3 e J be an index
such that j .j ,j2 < j3- Moreover, we define e3 by the formula
:= 4- min {e. - v. (a - a. (p(a))) : к = 1,2}
□ Z	К "1,	К
Jk
Further, use (FM3) to find an element a3 e E such that
v (a - a (p(a) ) ) < e .
j 3	э	о
Then for к = 1,2 we have
(at(P(a)) - a,(p(a))) s v. (av(p(a)) - a) + (a - a7(p(a)))
< v (a. (p(a)) - a) + v. (a - a,(p(a)))
3k K	33	4
<	(a. (p(a)) - a) + (2-e	- e )
< Vj (ak(p(a)) - a) + (eR -
v . (a - a. (p (a) ) ) - e, )
3k K	J
By (FM4) we now can pick an open neighborhood U3 c n U2 of p(a)
such that v. (aq(y) - a. (y)) < e, - e, for all у e U,. We claim that
J э	К	К э	Э
T(U3,a3,e3,j3) С т(U1,a1,e ,j ) n T(U2,a2,e2,j2): indeed, let
В e T (Uq , a, ,	, j q) . Then p(£3) e U, c IL, k=1,2, and therefore
J	J J	J	JX
47
( В ~ ak (P ( В)) ) < v. ( В " a-> (P (В)))	(о7 (p ( B)) - ak(p(B)))
3k K	3k	J	3k J	K
<	Vj (B _ ст3 (p (B) )) + (ek ~ e3)
<	e3 + (ek " e3> = ek-
From now on, the set E always carries this topology.
5.4	If a e E, then the mapping о : X + E is continuous.
(Let xQ e X and let a(xQ) e T(a’,U,e,j). Then v( (a - a') (xQ)) < e.
By (FM4) there is an open neighborhood V c U of xQ such that
Vj(a(y) - o' (y)) < e for all у e V. Clearly, this implies a(V) c
c T(U,a',e,j), i.e. a is continuous at xQ.)
5.5	The mappings add : EvE -> E and seal : IKxE -> E are continuous.
(Let (a,B) e EvE and let T(U,a,e,j) be a neighborhood of a + В • Let
6 = I (e - Vj(a + В " a(p(a))). Then there are elements o^,o2 e. E
such that v • (a. (p (a) ) - a) < 6 and v-(a, (p(B)) ~ B) <5- For these
elements we have
Vj(a(p(a)) - (a^(p(a)) + a2(p(B))) <
<	v.(a(p(a)) - (a + B) ) + (a - a^pla))) + v4(B - a9(P(B)))
J	J	1	J	z
<	e - 2-6.
Pick an open neighborhood V с U of p(a) such that
Vj(a(y) - a1(y) - a2 (y)) < e - 2-6 for all у e V.
Then the set T(V,a1,6,j)xT(V,a2,6,j) n EvE is a neighborhood of
(a,B) and for all (a',B') e T(V,a1,6,j)xT(V,a2,6,j) n EvE we obtain
v-(a' + B' - a(p(a'))) < \).(a' - a1(p(a'))) + v-(B‘ ~ a,(p(B')))
J	J	I	J
+ v j ( (+ a 2 ~ a)(p(ct')))
<6 + 6 + £~2“6
= £
48
This shows the continuity of add.
Now let (ro'“o)£ 1КХЕ and let T(U,a,e,j) be an open neighborhood of
r -a . In this case we choose 6 = e - v.(a(p(a )) - r -a ). Pick an
о о	j о о о
element a’ e E such that ro-Vj(a’(p(aQ)) - aQ) < 6/2. Then we obtain
(го-С (P(%) ) - a(p(ao))) < v. (ro-a' (P(%) ) - r/%) +
+ v (rQ-aQ - a(P(ao)))
< e - 6/2.
Hence there is an open neighborhood V U of P(aQ) such that
Vj(ro-a'(y) - a(y)) < e - 6/2 for all у e V.
Choose a real number 0 < e' such that
£'•(£’ +	+ |rol-e’ + (e - 5/2) < e.
If |r - r | < s' and if a e T(V,о',e',j), then we calculate:
v (r*a - a(p(a))) <	((r - rQ)-a) + Vj(rQ’a - a(p(a)))
S |r - rol*v (a) + v..(ro’(a - a'(p(a)))) +
+ v (ro-a'(p(a)) - a(p(a)))
<	|r - roHvj(“ - o’(p(“))) + v (a' (p (a)) ) ) +
+ !roI•£’ + (e - 5/2)
<	£’•(£’ + V (a-(p(a)))) + |ro|-e’ + (e - 5/2)
<	e.
This implies that the mapping seal is continuous, too.)
5.6	The mappings p : E + X and 0 : X + E are continuous.
(This follows immediatly from the definitions of the topology on E
and (5.4) . )
5.7	If U c X is an open set, if о : U + E is continuous, if e
0
49
and if j e J, then the set {a e E : p'a) e U and v j ( a " a(P(a)))< e}
is open in E.
(Indeed, the mapping T : a -> a + a(p(a)) : P 1 (U) -> p 1 (U) is
a
continuous, as add, a and p are continuous. Because T has continuous
a
inverse T_^, it is a homeomorphism. Now note the set in question
is the image under of the open set {a e E : p(a) e U and Vj (a) < e}
and therefore is relatively open in p (U). As p (U) is open itself,
the result follows.)
5.8	Proposition. Let (Ex)x x be a family of topological vector
spaces whose resp. topologies are induced by seminorms v* : E -> 1R
j e J. Let E be a subspace of Ц°° E and assume that the index set
xeX X
X carries a topology such that (FM3) and (FM4) are satisfied. Then
there is a bundle p :E -* X with stalks (isomorphic to) Ex, x e X,
such that E is (up to isomorphy) a subspace of r(p)-	□
5.9	Theorem. If X is a compact topological space, then there
is a one-to-one correspondence between the class of all bundles
with base space X and the class of all function modules with base
space X. More explicitly:
If p : E -* X is a bundle, then Г(p) is a function module with
stalks p 1(x), x e X.
Conversely, if E is a function module with base base x and stalks
E^, x e X, then the construction given in (5.2) yields a bundle
PE : EE -* X and these two operations are inverse to each other.
Especially, if E is a function module with a compact base space x,
then there is a bundle p : E_ -* X having the same stalks as E such
that E - Г(p) and this isomorphism preserves the С (X)-module
structure.
50
Proof. Let us start with a function module E with base space X.
Then the construction given in (5.2) yields a bundle p_ :	X
such that E may be viewed as a closed subspace of Г(p ). From (FM3)
hi
and (FM5) we know that E is stalkwise dense and а С(X)-submodule of
Г(pE). Now the Stone-WeierstraB theorem (4.2) shows that E is dense
in Г(p ). As E was already closed in Г(р„), we obtain E - Г(р„).
hi	hi	hi
Conversely, let us suppose that we are given a bundle p : E X and
let E := Г(p). Clearly, as we remarked at the beginning of this
section, E is a function module. It is obvious that we may identify
the sets E and EE and the projections p and pE> We only have to
show that this identification is a homeomorphism for the topologies
on E and Ee resp. Firstly, note that the topology on EE is certainly
coarser than the topology on E, as we used only global sections in
the definition of the topology on E,,. Whence, applying (5.7), it
remains to prove that every local section of p : E + X is continuous
when viewed as a selection of : E„ -> X.
K E E
Thus, let a : U + E be a local section and let x e U. We want to
о
show that the mapping a : U -> E„ is continuous at x . Pick
h	о
neighborhoods V,W of xQ such that V W	W	U and let	f :	X -> [0,1]
be a continuous function such that f(V) =	{1}	and f(X \	W)	= {o}.
Define a global section a : X •* E by a (x)	= 0	for x e X	\ W	and
a(x) = f(x)•a(x) if x e u. (This definition makes sense , as on
(X \ W) n и = U \ W we have f(x)*o(x) = 0-a(x) = 0.) Then о is
continuous as it is continuous on the open sets X \ W and U and
as these open sets cover X. Because p : E ? X and pE : EE + X have
the same global sections by the part of the theorem already verified,
о : X -> E„ is continuous,
hi
To finish the proof, we only have to remark that a and a agree on
the open neighborhood V of xq. □
51
We now turn our attention to ^-function modules:
Let т : I -* JN be a type and let E be an ^-function module with stalks
(E )	„ ,base space X and seminorms
X X Сл
(Vх). _. If (E,p,x) is the bundle
J J eJ
constructed in (5.2), then for every i £ I we may define a function
т (i)
E
f. : V E
f±((x,a1) ,... (x,aT(i))) := (x,f±(o^ ,...,aT(±)))
Of course, we hope that we obtain a bundle of fi-spaces in this
manner. I do not know an answer to this question in general, but I
can offer some partial solutions:
Firstly, we take a closer look at the proof of the continuity of
the mapping add : EvE + E in (5.5). Then we will recognize that the
key inequality looks as follows:
v (add (a1, a2) - add(B1,B2))s vj (“i “ B-j) +	(“2 ~ B2> •
This means that the continuity of the addition in topological vector
spaces is in some sense uniform for all vector spaces. This does
not have to be true for the additional operations	j a priori.
Whence, if we would attempt to modify the proof of the continuity
of add to show the continuity of the (f.). we would have to
postulate something like an "uniform continuity" for the	j*
In this case, we would obtain some very technical condition like
(*) For every iel, every j e J and every e > 0 there is an
j' e J and a 6 > 0 such that for all x e X and all elements
a1,B1,•••,aT(i),BT(i) e Ex the inequalities	, (a1 - B1),••
•••'V (aT(i) - ^(i)’ < 6 lmply Vfi(a1........aT (i) ’ "
fi(‘	(i)) ) < £‘
Let us agree that we call an fi-space E an uniform ii-function module
52
provided that (*) holds.
A straightforward modification of the proof of the continuity of
the mapping add now shows:
5.10 Proposition. If E is a uniform ^-function module with base
space X, then there is a bundle of ^-spaces p : E -> X such that E is
(topologically and algebraioally isomorphio to) a closed subspace of
Г(р).	□
There are certain cases for which the uniform continuity of the
additional operations follows automatically. For instance, every
function module of Banach lattices is uniform. This follows from the
inequality ||avb - cvd|| < j|a - c || + ||b - d || .
On the other hand, if we restrict the class of base spaces, then the
uniformity is not needed:
5.11 Proposition. If X is a completely regular topological space
and if E is an Pt-function module with base space X satisfying the
stronger axiom
(FFM3) For every x e X we have E^ = {a(x) : a e E}.
then there is a bundle of il-spaces p : E -> X such that E is (topolo-
gically and algebraically isomorphic to) a closed subspace of Г(p)•
Proof. Let p : E + X be the bundle constructed in (5.2),(5.3). By
(5.8) it remains to show that p : E -* X is a bundle of fi-spaces.
Firstly, note that we may use the proof of (2.5) to obtain from
(FFM3) the condition
(**) For every x e X and every a e E we have -jX(a) =
x	J
= inf {0.(a) : a(x) = a, a e E}.
53
Clearly, condition (**) and (FFM3) imply
(***) If a e E, a e E and if (a - a (x) ) < e, then there is
x	J
a a' e E with Vj (a - a') < e and a'(x) = a.
т (i)
We now want to show that the mappings f^: V E -> E, i e I, are
continuous.
т (i)
Let i e I, let (a^,...,) e V E and let 0 be an open neighbor-
hood of fi(a1,...,aT)• Further, let xQ := ptc^) = ... =
By (FFM3) and (1.6(vii)) we may assume that there is an element
a e E, an open neighborhood U of xq, an e >0 and an j e J such that
0 = T(U,a,e,j). Applying (FFM3) once again, we find elements ,...
...,a e E such that a, (x ) = av for all 1 < к < т (i). As
fi(a1.....aT(i))(xo) = fi(W..........aT(i)(xo)) = fi(a1.....ttT(i))£°’
there is an open neighborhood U' c U of xq, an e' >0 and an j' e J
such that T (U',f,...,) , e',j') c 0 (use (1.6(viii)).
We now apply the fact that E itself is a topological fi-space, i.e.
that the operations f^ : E7^ -> E are continuous. Whence it is
possible to choose 6 > 0 and j” e J such that the inequalities
%" (a1 " ai)......%•' (<5t(1) "^(ii’ " 6
imply
Oj' (fi(a1....ат(i)> " fi(aV‘--'ac(i))) < £'•
т (i)
We show that f : V E -> E maps the open neighborhood
т (i)
T(U',Ol,6,j") x...,xT (U',aT(i),6,j"I n v E
of (a^,•..,aT) into 0.
Indeed, let , . . . ,	) belong to the first of these sets and let
x := ptgp = ... = p(6T (!)) . By (***) there are elements s-j ,' ''' ат (i)
e E with gk = o^(x) and , (ak - ak) < 5 for all 1 < к < т (i) .
Whence we may conclude that
54
V (f.(Ol,...,aT(i)) -	....< e'
and especially
Vj. (f. (ai (x),...,aT(i) (x)) - f. (Br...,BT(i))) < £'•
This gives us finally the relation
fi(Br•••-BT(i)) e T(U',f.(ai,...,aT(i)),e',j') c 0.	□
5.12	Remark. Let us point out that theorem (2.9) also holds for
function modules, as in its proof we only used the properties
listed in (FM1) - (FM5). This means that every function module
with a completely regular base space, complete stalks and a locally
countable family of seminorms satisfies (FFM3).
The following theorem is analogous to (5.9):
5.13	Theorem. Let X be a compact topological space and let
т : I -> К be a type. If p : E -> X is a bundle of Ct-spaces of type r
then Г(p) is an ^-function module of type t.
Conversely, if E is an ii-function module of type r with base space X
which satisfies the stronger axiom (FFM3), then there is a bundle
Pp = Ep •* x °f ii-spaces such that E is (topologically and algebra-
ically isomorphic to) the Ct-space Ftp).
Moreover, these two operations set up a one-to-one correspondence
between the class of all ii-function modules satisfying (FFM3) and
the class of all full bundles. □
We conclude the section with a few examples:
5.14	Bundles over the circle. Let = {z e IE : | z | = 1 } be the
55
unit circle. Furthermore, let F be any Banach space with norm ||•ft
and let T : F -* F be a linear contraction. Moreover, define
ET := {a e С ( [О, 2tt ] ,F) : a(2K) = T(a(O))}.
Clearly, ET is a Banach space under pointwise addition und scalar
multiplication when equipped with the norm || • || given by
|| a || := sup { || a(x) || : 0 < x < 2л}.
Moreover, ET may be viewed as a function module over S : For
every z e s'! let Ez := F, equipped with the norm || • ||. Then we may
identify E with a closed subspace of 11°°. E by sending a e E to
„ I z	1
z cS
8, where for 0 < ф < 2л we define
8(е1ф) := а(ф).
If we do so, ET becomes a function module with base space ,
stalks Ez = F, and (semi-)norm || • || . The only axiom which requires
a little bit of work in verifying is the axiom (FM4):
Let a e et’ We have to show that the mapping z -> ||8(z) || :	-> 1R is
upper semicontinuous. The only problematic point of is z = 1. Thus
let us assume that || 8(1) || < e. This means || a(0) || < e and
11 о (2 л) 11 = ||T (a (0)) 11 < ||T || || a(0) || < 1 • e = e. Therefore we can
find a 6 > 0 such that ф e [0,6[ и ]2л - 6,2л] implies ||а(ф) || < е.
Whence for z е {е1ф : |ф| < 6} we obtain ||8(z) ||	< е.
By (5.9) we obtain a bundle PT : ET -* S1 such that Г(РТ) = ET.
If we choose F = 1R with the usual norm and T = -1 ( i.e. T is multi-
plication with -1), then PT : ET -* is the Moebius strip.
If F = IR^ with the Euclidean metric and if T = (^ _^), then
56
Рт : Ет -+ S1 is homotopy equivalent to Klein's bottle.
More generally, if F = ]Rn with the Eukledian metric and if T : 1Rn-> 1Rn
is a linear operator with det(T) = -1, then we obtain a higher
dimensional analogon of the Moebius strip and Klein's bottle, resp.
It is well known, that for F = ]Rn and for invertible T : ]Rn -* ЖП the
bundle pT : ET -* S1 is locally trivial (see section 16 for defini-
tions) . Moreover, in this case we only have two isomorphism classes
of locally trivial bundles over s\ depending on whether det(T) > 0,
in which case we obtain the trivial bundle, or det(T) < 0.
We shall see in section 17 that every bundle p : E -* whose
stalks are of some fixed dimension n < °° and whose bundle space E is
Hausdorff is in fact locally trivial and therefore isomorphic to
one of the bundles pT : ET -* s\ where T : 1Rn -> 1Rn is invertible.
5.15 Sequence spaces. Let E be a topological vector space with a
all О-sequences in E equipped with the topology induced by the
seminorms ($.). _ given by
1 J
^’j((un)neN) = SUp {vj(Un> : П £
It is well known that с (E) is a closed subspace of П E , where
°	ndN n
En = E for every n e JN. Moreover, if we equip IN with any topology
finer than the cofinite topology (i.e. with any -topology), then
с (E) is a function module, as multiplication with a bounded
Ж-valued function does not lead out of the class of О-sequences and
as for every e > 0, every j e J and every О-sequence (ип)п^} in E t*ie
set {n eJN : Vj(ur) > e} is finite and thus closed.
57
Thus, we may construct a bundle p : E -> JN such that cq(E) с Г(р) ,
where E = U*E and where p is the first projection. The topology on
E may be described as follows:
Let (n ,u ) e E =JNxE. If we define a О-sequence (u ) e с (E) by
о о	n ndN о
un = u and un = 0 for un = 0 for n / nQ, then this sequence gives a
section of p : E -> JN passing through (nQ,uQ). Hence by (1.6(vii))
the sets of the form {(n,u) e E : n = nQ and Vj(u - uq) < e or n / nQ,
n e U and v (u) < e}, where U is an open neighborhood of nQ, e > 0
and j e J, form a neighborhood base at (nQ,uQ)
Finally
we calculate Г(р)
in two special cases:
a)	И carries the discrete topology. Then E carries the product topo-
logy ofJNxE and therefore Г(р) consists of all bounded sequences with
values in E.
b)	И carries the cofinite topology. In this case we have Г(р) = cq(E):
Indeed, let a e Г(p). Then there is an u^ e E such that a(1) = (1,u^).
As for a given e > 0 and a given j e J the set
0 := {(n,u) e E : n = 1 and v(u - u^) < e or n / 1 and
v (u) < e}
is an open neighborhood of a(1), we can find an open neighborhood
U of 1 such that a(U) c 0. As U is open in the cofinite topology,
there is an n e TJ such that {n : n > n } c u. This implies
о	о
(o(n)) < e for all n > nQ, i.e. a e cq(E).
5.16 Example. Let X = [0,1] be the unit interval with its usual
topology and let E be the completion of C([0,1]) under the norm
III • HI given by
58
= max {|f(0) |, sup r.|f(r)|} .
0<r<1
It is easy to see that E is a C([0,1 ])-submodule of Л°° E ,
0<r<1 r
where Er is identical with 1R equipped with the norm
ria!	r = 0
1	r • | a |	r / 0
Furthermore, E is indeed a function module with stalks E and base
r
space X. Hence , by (5.9) there is a bundle p : E -> 0,1 such that
E is isometrically isomorphic with Г(р). We may identify the set E
with [0,1]x]R. if we do so, the mapping p becomes the first projection.
Let us try to describe the topology of E. The open sets are given
by tubes and by (1.6(vii)) we may use tubes around constant sections.
Thus, let (r,a) e [0,1]x]R be an element of the bundle space. If
r / 0, then a neighborhood base at (r,a) is given by sets of the
form
{(s,B) : |r - s| < e, s-|a -	< eh 0 < e < r/2
and it turns out that the subset ]0,1]x]R с E carries the usual
product topology.
A neighborhood base at (0,a) e [0,1]x]R is given by
{(0,6) : | a - £31 < e } и { (s, 6) : |s|<e, s • | a - 61 < e}
Hence every open neighborhood of (0,a) contains the elements of the
form (s,0), if we only choose s < =—?—г , and we conclude that the
2	 I a I
closure of the set [0,1]x{0} с E is equal to {0}xlR и [0,1]x{0} and
that the closure of {a e E :	< 1 } is equal to the set
{OjxlR и {a e E : |||a||| < 1}. This provides us with an example in
which the "unit ball" {a e E :	< 1 } is not closed.
59
We leave it as an exercise for the reader to verify that
Г(р) = {f : [0,13 •’•JR : lim r«f(r) = s«f(s)}
r->s
i.e. f belongs to Г(р) if and only if the mapping f : [0,1] -* JR is
continuous at every r / 0 and satisfies the equation lim r«f(r) = 0.
r-*0
Especially, the mapping X{0} defined by
r 0
X{°] = ( 1
if s / 0
if s = 0
is an element Г(р).
We conclude this section with the remark that also all weighted
vector valued function spaces in the sense of Bierstedt, Kleinstiick,
Machado, Nachbin, Prolla et al. fall under the notion of
function modules. For a precise definition and a treatment of
these examples we refer to the papers of the authors just mentioned.
б. Some algebraic aspects of ^-spaces
In this section we collect some properties of fi-spaces which will be
needed later on. Nothing new will be found here, all the results are
folklore. Therefore we can confidently leave all the proofs to the
reader.
6.1 Definition. Let E and F be two ^-spaces of type т : I -* U.
(i) An Q-morphism from E into F is a linear map ф : E -> F such that
for all i e I and all a.,... . ,a ... e E we have
1	т (1)
fi(*(a1>......Ф<ат(1)	’ ’ = *(fi(a1....aT(i)))-
(ii) A linear subspace N c e is called an Si-ideal, if for all i e I
and all sequences of ordered pairs (a^,b^)
. . . ,(a , .. ,b ,.. )
т(i) т (i)
the relations
f. (a.,...,a . ..)
i 1	т (i)
b.. e N for all 1 < j < T(i) imply
fi(b1......bT(i)> £ N‘ D
If E is a vector lattice, then the Q-ideals are exactly the ideals
in the usual sense; if E is an algebra over IK, then the fi-ideals
are also the usual algebra ideals.
6.2 Proposition.
E •> F be an Sl-morphism between two
Si-spaces E and F. Then ker ф := ф (0)
is an Sl-ideal.
The
next result states a kind a reverse of
(6.2):
e E xE
a .
1
Let ф :
□
6.3 Proposition. Let E be a (topological) Si-space of type т and
let N be a (closed) Sl-ideal of E. Then E/N is also a (topological)
J2 -space of type where the additional operations (f^) on E/N
61
are defined by
f. (a. + N, . . . , a . . , + N) :— f.(a.r...ra , ., ) + N •
11	т (i)	11	т (i)
Moreover, the canonical quotient map л : E -* E/N is a (continuous
and open) ii-morphism.
Conversely, if ф : E -* F is a (continuous and open) quotient map and
an ii-morphism and if N = ker ф, then the Oi-spaoes E/N and F are
(topologically) isomorphic. □
6.4	Proposition. Let ф : E -* F be a (continuous) Cl-morphism
between the Q-spaces E and F3 let N = ker ф and let л : E-* E/N be the
canonical quotient map. Then there exists an injective (and contin-
uous) ii-morphism ф : E/N -> F such that ф = ф°л. □
6.5	Proposition. Let E be a topological Q-space. Then the closure
of an Oi-ideal is again an Ct-ideal. □
6.6	Proposition. Let E be a (topological) ll-space and let
be a family of (closed) Л-ideals. Then n N, and E N. (resp.
		XeL A ХеЛ A
E N,J are again (closed) Л-ideals. Especially, the (closed)
XeK A
Л-ideals form a complete sublattice of the complete lattice of all
(closed) subspaces.	□
6.7	Notation. With Id^(E) we denote the complete lattice of all
closed Q-ideals of a topological fi-space E.
7. A third description of spaces of sections : С(X)-convex modules
We know from (1.6) that for every bundle p : E -> X the space of all
sections is a C(X)-module. Now suppose that somebody gave us a
C(X)-module E. Then we ask ourselves: Is it possible to "spread
E continuously across X", i.e. is there a bundle p : E * X such that
E and Г(р) are isomorphic? In this section we shall describe those
С(X)-modules for which this is possible.
If we are dealing with ^-spaces, we also have to worry about the
additional operations. Hence we note the following: If p : E -* X is
a bundle of ^-spaces over a quasicompact base space, then r(p) is
a topological fi-space. Moreover, if A c x is a subset, then the
set Nft := {a e Г(р) :	=0} is a closed fi-ideal. Especially,
if f e С(X) is a continuous Ж-valued function on X and if we let
A = f 1 ( IK \ {0}) , then f1 := Nft = {а e Г(р) : f-a = 0} is an
Q-ideal. This leads to the following definition:
7.1 Definition. A C^tX)-tl-module E is a topological fi-space which
is at the same time a'C, (X)-module such that
b
(i)	the multiplication (f,a) -* f-a : С^(Х) xE + E is continuous,
(ii) for every element f eC,(X) the set f1 := {a e E : f-a = 0}
b
is an 0,-ideal. □
Note that by the continuity of the multiplication the fi-ideal f1 is
automatically closed for every f e С^(Х).
For the following results see also the work of J.Varela (EVa 75]) in
the case of Banach spaces and the paper EM6 78] of H.Mttller.
63
As it is always convenient to work with compact Hausdorff spaces in-
stead of arbitrary topological spaces, we shall heavily make use of
the following fact:
For every topological space X there is a compact (Hausdorff) space
BX (the Stone-Cech-compactification of X) and a continuous mapping
i : X -> BX such that
(i)	i(X) is dense in BX.
(ii)	the mapping f + foi : C (BX)-* С^(Х) is a bijection preser-
ving the sup-norm and the algebraic structure.
This means that in the proofs of many results concerning C(X)-modules
we may assume w.l.o.g. that the space X is compact. This will be at
least be possible as long as we do not talk about points of X.
7.2 Proposition. Let E be a C-^(X)-il-module and let I be an ideal
ofC-foCX.). Then the closure of the complex product I.E is an Pi-ideal.
Moreover, if X is compact and if I is a closed ideal, then
I
we
{f '• f/A
let Ед :=
= 0} for some closed subset А с X. In this case, if
{f eCb(X) : f 1(0) is a neighborhood of A}, then we
have the following equalities:
I.E = Fa-E
=	-{-fl : f/A = 1 }
=	и {f1 : f/y = 1 for some open set U d A}
=	uTfl = f/A =
Moreover, in all these cases we may restrict ourselves to continuous
functions f : X -> [0,1] c IK.
Proof. By the continuity of the multiplication with elements in
C(X) we may assume that I is closed. Moreover, from (7.1(ii)) we
64
know that f1 is an fi-ideal and therefore by (6.6) the closure
of E {f1 : f , = 1} is an fi-ideal. Hence by the above remarks
/ a
concerning the Stone-Cech compactification it is enough to prove
the assertions for compact X in (7.2).
Hence
we assume that X is compact and that А с X is a closed sub-
set. As F, is dense in I
A
{f : f/ft = 0}, we obtain I-E = Fft>E.
Let f e Fft. then there is an open neighborhood U of A such that
continuous function g : X -* [0,1] such that g(V) = {1} and g(X \ U) =
= {0}. Then g.f = 0 and whence for all a e E we have g-(f.a) = 0, i.e.
f-E c g1. This yields the inclusion
F^'E c ( и {f1 : f/y = 1 for some open set U э A})
Obviously, we have
(u {f1 : f, = 1 for some open set U = A}) c ( e {f1 : f , = 1})
/U	/А
Further, let a e E {f1 : f, = 1}. Then there are elements
/ a
f.,..., f e С(X) and a.,...,a e E such that f .	= 1, f.-a. =0
1	n	1	n	i/A	i i
and a. + ... + a = a. Let g = ff . Then g ,, = 1 and g*a. =0
1	n	’	1 n	/А	’	1
for all 1 < i s n. Thus, we obtain g-a = 0, i.e. a e g1. Hence we
obtain the inclusion
( E {f1 : f/A = 1})- с ( и {f1 : f = 1})-.
Finally, let a e и {f1 : f. = 1}. Then f»a = 0 for some f e С(X)
/ A
with f/A = 1. Let g = 1 - f. Then g vanishes on A and therefore
belongs to the ideal I. Moreover, we have g-a = (1 -f)-a=a-f-a=
= a, i.e. a e I-E. As this implies
( и {f : f = 1 }) c i.e
65
our proof is complete. □
7.3 Corollary. Let E be а СЛХ)-^-module and let А с X be a subset.
b
If we let I = {f : f= 0}, then
I>E с и {f1 : f/y = 1 for some open set U d A and О < f < 1 }
Proof. This assertion follows for compact X immediatly from (7.2). If
X is not compact, consider that Stone-Cech compactification BX of X
and let i
X + BX be the canonical map.
Then for f e С. (X) we have
b
f (A)
{0} if and only if there is a f e C(BX) with f = f'oi and
f'(i(A)) = 0. Moreover, if f'= 1 for some f' e C(BX)
open set U c BX, then for f = f' »i we have f^-1	= "I
and some
and i 1(U)
is
also open. Now some straightforward arguments complete the
proof. □
7.4 Proposition. Let E be a	-^-module. Then we have:
(i)	0 e E has a neighborhood base consisting of closed, convex
and circled sets A с E such that f-A c A for all f e С^(Х)
(ii)
The topology on E is generated by a family
(Vj)
jeJ
of semi-
norms
satisfying
Vj(f-a)
for all j e J.
with 11 f 11 < 1.
llf IHj (a)
Proof. Let v be the gauge function of a closed, convex and circled
neighborhood A of 0 fullfiling f-A c A for all f e C^tX) with
||f|| < 1. Then it is easy to check that v(f>a) < ||f||>v(a). Whence
(i) implies (ii). It remains to check (i):
Let U be any closed convex and circled neighborhood of 0. Then, by
the continuity of multiplication by f e C^tX), we can find an e > 0
and a closed, convex and circled neighborhood V of 0 such that
||f||	< e and v e V imply f.v e U. Let V := e-V. Then V is still a
66
closed, convex,circled neighborhood of 0 and we have f »V c U for all
f e С, (X) with Ilf II <1. Let
b	iiii
W := U {f «V : 11 f II < 1 }.
From 1 -V = V we obtain V c w c U. Let A be the closed, convex,circled
hull of W. Then A is a closed, convex and circled neighborhood of 0
contained in U. Moreover, f »W c W for all f e C^tX) with ||f||	<1
and the continuity of multiplication imply f»A c A for all f e C^tX)
with ||f|| < 1. This completes the proof. □
The next lemma, due to J.Varela [Va 75] in the case of Banach spaces
and due to H.Moller in our setting, opens the door to a connection
between C(X)-modules and bundles:
7.5 Lemma (Varela). Let E be а С, (X)-module and let v be a oon-
b
tinuous seminorm on E satisfying v(f-a) < ||f||-v(a) for all a e E
and all f e C^tX) . If А с X is any subset and if I = {f e C^CX) :
f/A = 0}, then for all a e E we have
v(a + I»E) := inf {v(a + b) : b г I-E}
= inf {v(f*a) : f e C^tX), 0 < f < 1 and A c f ^ ( {1 }) ° } .
Proof. Let
1 := inf {v(a + b) : b e I.E}
r := inf {v(f.a) : f e С, (X), 0 < f < 1 and A c f 1 ( {1 })°}.
b
If f : X ->JK is a continuous and bounded function with constant value
1	on A, then 1 - f vanishes on A and thus belongs to I. Thus, we
know that (1 - f)-a belongs to I-E for every a e E. As we may write
f.a = a + (f - 1)-a, we obtain 1 < r.
Conversely, let e > 0. By (7.3) and the definition of 1 we can find
67
an f e C^tX) with 0 < f < 1 and A c f \{1 })° and an element b e f1
such that v(a + b) < 1 + e. This implies x < v(f’a) = v(f>a + f*b) =
= v(f«(a + b)) < 11 f11 • v(a + b) < 1 + e. As e > 0 was arbitrary, this
yields r < 1.	□
Before we proceed, let us introduce some notations:
Let E again be a (X)-^-module, let v be a continuous seminorm on
E, let a e E and let x e X. We define
Zx := {f £ Cb(X) : f(x) = 0}
N := I -E
x	x
E := E/N
x	'	x
ex	: E	-*	Ex	is the canonical projection
ax := ex(a)
vX(ax) := inf {v(a + b) : b e NxJ
N := n Nx
xeX
By (7.2) the subspace Nx is an fi-ideal. Hence, by (6.3), the
quotient space Ex is a topological fi-space, too, and the quotient
map ex is an fi-morphism.
Moreover, we have
7.6	ex(f«a) = f(x)*ax for all a e E and all f e C^X)•
(Indeed, f(x) = 0 implies f-a e Nx , whence ex(f-a) = 0. Now let
f e C^tX) be arbitrary. Define a continuous and bounded function
g e (X) by g = f - f(x)-1. Then g vanishes at the point x and we
may compute: ex(f-a) = e ((f - f(x)>1)»a + f(x)«a) = ex(g-a + f(x)-a)
= ex(g*a) + f(x)-ex(a) = f(x)-ax-)
7.7	If we fix an element a e E and if v(f-b) < ||f||>v(b) for all
f e C^tX) and all b e E, then the mapping
68
v (а_) : X -> Ж
х -> v (ах)
is upper semicontinuous.
(Let \>x(ax) < M. By (7.5) there is an open neighborhood U of x and a
continuous mapping f
X -> [0,1 ] сЖ such that f^
1 and such that
v(f-a) < M. Using (7.5) again
we conclude that \Л(а )
M for all
У e U.)
It is now obvious from (7.4) and (7.7) that the fi-space E/N may
algebraically be embedded into П°° E and that this embedding is
xeX x
continuous. Moreover, under this embedding, the closure of E/N
in П°° E is an fi-function module with base space X and stalks Ex-
xeX
Further, the stronger axiom (FFM3) is automatically satisfied.
Thus, we may state:
7.8	Proposition. Let E be a
bundle p : E -* X and a continuous
i : E/N -* Г(р). In addition, this
every a e E there is an element u
If E is a	-ii-module and if X
bundle p : E -* X is in fact a bundle
(X)-module. Then there is a full
and injeotive ii-morphism
bundle has the property that for
e E/N with i(u)(p(a)) = a.
is oompletely regular, then this
of Ti-spaoes. □
In order to present the whole space E as a space of sections, we
need two things : Firstly, the mapping i : E/N -> Г(р) given in (7.8)
should be open onto its image. Secondly, the subspace N с E should
be trivial. Unfortunatly, the following example shows that it may
happen that N = E, even if E is a Banach space and even if the base
space X is compact:
7.9 Example.
Let X
[0,1] be the unit interval with its usual
69
topology and let E be the completion of C([O,1]) in the norm given
by
|||f HI = f |f (x) |dx .
Then E is a C ( [0,1 ])-module and we have |||f >m||| < ||f|| - |||m||| for all
f e C([0,1]) and all m e E. In this case, = E for all x e [0,1].
Indeed, let xQ e [0,1], let m e C([0,1]) с E and let e > 0. If we
define 6 =	' we таУ find an element f e C([0,1 ]) such that
0 < f < 1, f(xq) - 0 and f(x) = 1 for all x with |x - x | > 6. Then
the element f-m will belong to Nx and we have
|||m - f-mm = |||(1 - f) .m|||
1
= J (1 - f(x)) •|m(x) | dx
0
S 2.M||m|||
This yields m e N . As C([0,1]) is dense in E, we conclude that
E = N .
Suppose that our C(X)-module E may be represented as a C(X)-module of
sections in a bundle p : E -* X. From (1.6.(x)) we then may deduce
that E is locally C(X)-convex in the sense of the following defini-
tion:
7.10	Definition. Let E be a (X)-module.
(i)	A subset A с E is called С(X)-convex, if for all m,n e A and
all f e C^tX) with 0 < f <, 1 we have f-m + (1 - f) -n e A.
(ii)	The (X)-module E is said to be locally С(X)-convex, provided
that 0 e E has a neighborhood base of С(X)-convex sets. □
70
Note that for every С(X)-convex subset A с E the convex and circled
closed hull of A is also С(X)-convex. Hence E is locally C(X)-convex
if and only if 0 has a neighborhood base of closed, convex and
circled C(X)-convex subsets. If we pass to the gauge functions given
by these sets, we obtain:
7.11	Proposition. A	-module E is C(X)-convex if and only if
the topology on E is induced by
a family of seminorms
(Vj)
satis-
j eJ
fying the following condition
If m,n e. E with Vj (m) , Vj (n) s 1 and if f e C^CX) with
0 s f < -| f then Vj (f ’Ш + (1 - f) >n) £ 1.	□
Hence
we are once again led to a closer look on seminorms in
С(X)-modules. Let us start with two lemmata:
7.12	Lemma. Let E be С^(Х)-module and let v be a seminorm on E sa-
tisfying v(f-m) < ||f ||-v(m) for all m e E and all f e C^tX). Then
|f | < |g| implies v(f’in) < v(g>m) for all m e E, f,g e С (X).
b
Proof. Suppose that g(x) / 0 for all x e X. Then we may compute:
v(f-m) = v(g-|>m) < ||||H (g-m) < v(g-m).
The proof of the general case is a modification of this idea:
Firstly, we may assume without loss of generality that X is compact.
Let e > 0 and let A := {x : |g(x)| > e}. Then A is closed in X.
For every x e A we define h(x)
h : A -* IK is continuous and we
h : X +Ж be an extension of h
f (x)	.
:= —clearly, the mapping
have |h(x)| < 1 for all x e A. Let
with |h| < 1. Then an easy calculation
shows that ||h-g - f|| < 2-e. This implies v(f-m) < v((f - g-h) -m) +
+ v(h-g-m) < 2>e-v(m) + v(g-m). As e > 0 was arbitrary, we obtain
v(f >m) < v(g-m).
□
71
The next lemma is due to Bohnenblust and Kakutani (see [BK 41 ]) :
7.13	Lemma.	Let v be a seminorm on C^tX) satisfying \>(f.g) <
||f|| -\>(g). If v(fvg) = max {v(f) ,v(g) } whenever fAg = 0, then we
have v(fvg) = max {v (f) , v (g) } for all 0 < f,g e. C^fX) •
Proof. Again, we may assume without loss of generality that X is
compact. Let I = {f e С (X) : -j(f) = Oj.Then I is an ideal of C(X)
and v induces a norm on C(X)/I. Moreover, C(X)/I is a vector lattice
and we have |f + l| = |f| + I for all f e C(X). From (7.12) we
conclude that v(f) = v(|f|). This implies the equation v(|f + I,) =
= v(|f|) = v(f) = v(f + I). Hence the space С(X)/I is a normed
vector lattice.
Let (f + I) л (g + I) = 0 = (fAg) + I. Then fAg belongs to the ideal
I. Substituting f and g by f - fAg and g - fAg resp., we may assume
that fAg = 0. Using our hypothesis, we may compute: v((f + I)v(g +1))
= v(fvg) = max {v(f + I),v(g + I)} . Let E be the completion of
C(X)/I in the norm v. Some standard arguments show that E still
satisfies v(avb) = max {v(a),v(b)} whenever алЬ = 0. Now we deduce
from [BK 41] the E is an abstract M-space, i.e. E satisfies the
equation v(avb) = max {v(a),v(b)} for all a,b a 0.
After these preparations it is easy to show (7.13): For all
f,g e C^fX) with f,g > 0 we have v(fvg) = v(fvg + I) =
= v((f + I)v(g + I)) = max {v(f + I) , v (g + I)} = max {v(f),v(g)}.	□
The following result is due to several authors: The equivalence of
(1), (2) and (5) may be found in [Ho 75]. R.A. Bowshell [Bo 75]
showed that (3) and (5) are equivalent and raised the question
whether (3) and (4) are the same. In the present form, the next
proposition is once again due to H.Moller:
72
7.14	Proposition. Let E be a C^tX)-module and let v be a seminorm
on E satisfying \>(f.m) < ||f|| .\>(m). Then the following conditions
are equivalent:
(1)	If f	e C^tX)	with О < f <	1 and if m,n e E	with v(m),v(n) <1
then	we have	also v(f*m +	(1 - f) -n) < 1.
(2)	If О	< f,g e.	C^fX) an^ if	m,n e E' t*ien we	have
v(f*m + g«n)	< ||f + g||-max {v (m) , v (n) } .
(3)	If f,g e C^fX) with f.g = О and if m e E, then
v( (f + g) .m) = max {v(f *m) ,v(g*m) }.
(4)	If О i f,g e. Cfc (X) and if m e E, then we have
v((fvg)>m) = max {-j(f-m),v(g-m)}.
(5)	For every m t E the following equation is true:
v(m) = sup {v(m + I.E) : I is a maximal closed ideal of
Cb(X)}.
If X is quasicompact, then these conditions are also equivalent to
(5’) If m e E, then v(m) = sup {vX(mx) : x e X}.
Proof. The implication (2) -* (1) is trivial.
(1) -* (3): Assume that f.g = 0. Then we have |f + g| = |f| + |g| >
i | f|. Thus (7.12) gives us the inequality v(f.m) < v((f + g) «m) ,
i.e. max {v(f»m),v(g«m) } < v((f + g) -m) .
Conversely, suppose that v(f*m),v(g*m) < 1. We have to show that
v((f + g)-m) < 1 + v(m)>e, where e > 0 is arbitrary.
Thus, let e > 0 and let A = {x e X : f(x) = 0} and В = {x e X :
| f (x) | > e}. Choose any continuous mapping h : X -> [0,1] with
h(A) = {0} and h(B) = {1}. (Here we again made the assumption that
X is compact, which is possible w.l.o.g.) Then we have
|| f - h f || < e and (1 - h) -g = g, because g(x) / 0 implies x e A
and hence (1 - h)(x) = 1. We now conclude
73
v ( (f + g) -m) = \)(h-f-m + (1 - h)-g«m + (f - h .f) .m)
<	'j(h-f-m + (1 - h) .g«m) + ||f - h-f || -\>(in)
<	1 + v (m) • e.
(3)	-* (4): Let m e E. Define a seminorm v on С, (X) by у (f) : =
m £>	m
v(f«m). Now apply (7.13)
(4)	-> (5),(5')s The maximal closed ideals of С, (X) correspond to the
b
maximal closed ideals of C(BX). Hence we may assume w.l.o.g. that
X is quasicompact. In this case the maximal ideals of C^tX) are of
the form I = {f e С(X) : f(x) = Oj.Thus, it is enough to prove
(51) .
Obviously, we have sup {yX(mx) : x e X} < v(m).
Conversely, let e > 0 and assume that sup {yX(mx) : x e X} < v(m) - e.
Applying (7.5), we find for every point x e X an open neighborhood
V and a continuous function f : X -* [0,1] with f (V ) = {1} such
X	X	X x
that v(fx«m) < v(m) - e. As X is quasicompact, there is a finite
number of point x.,...,x e X such that V и ... и V = X. As this
n	x1	xn
implies f v ... v f = 1, we obtain from (4) the inequality
X1	n
v(m) = v((fx v ... v fx )-m)
1	n
= max {v(f >m) : 1 < i < n}
< v(m) - e
a contradiction.
(5),(5’) -* (2): Again, we may assume that X is compact and hence
it is enough to show that (5')implies (2). Firstly, note that
(f«m)x = f(x)-m by (7.6). This yields yX((f-m)x + (g-n)x) <
< |f(x) + g(x)|-max {yX(mx),vX(nx)} whenever f(x), g(x) > 0. Now
an easy calculation using (5') shows (2).	□
74
If we combine (7.11) and (7.14), we obtain
7.15	Proposition.	Let X be a quasicompact space and let E be a
С(X)-module. If E is locally С (X) -convex, then N = n {Ix’E : x e
= 0.
Proof. The topology on E is induced by a family of seminorm (\>.) . _
which satisfy condition (1) of (7.14). Let m e N. Then mx = 0 for
all x e X. Hence for all j e J and all x e X we have 'jX(m ) =0.
From (7.14) we deduce that Vj(m) = 0 for all j e J, i.e. m = 0. □
We now come to a central result, which appears in different form
already in the work of Nachbin (see [Na 59]). In the present form
however, this theorem is due to K.H.Hofmann for Banach spaces and
to H.Mbller in the general case:
7.16	Theorem. Let X be a quasicompact space and let E be a
С(X)-module. Then E is locally С(X)-convex if and only if E is
(topologically and algebraically isomorphic to) а С(X)-submodule of
Г(р), where p : E -* X is a bundle.
Moreover, if a e E , then there is an element a in (the image of)
E such that a(p(a)) = a.
If X is compact, then E is dense in Г(р). Hence, if E is complete,
then E =: Г (p) .
Proof. Every С(X)-submodule E of Г(р) is locally C(X)-convex by
(1.6.(x)). The other direction follows from (7.8) and (7.15); the
last statement is a consequence of the Stone-WeierstraB theorem
(4.2).	□
For С(X)-fi-modules, we deduce from (7.8) and (7.16) the following
75
7.17	Complement. Let E be an Qrspace which is at the same time a
С(X)-module for a certain compact space X. Then E is a locally
С (X) -convex С (X) - il-module if and only if E is (topologically and
algebraically isomorphic to) a dense С (X) -submodule of the Q,-space
Г(р), where p : E -* X is a bundle of 0,-spaces. □
Let us state some corollaries which will cover the most important
cases:
7.18	Definition. A normed С(X)-module E is called locally
С(X)-convex, if for all f e С, (X) with О < f < 1 and all m,n e E
b
with ||m|| , ||n||	< 1 we have ||f-m+ (1 - f)-n||	< 1. □
7.19	Corollary. Let X be a compact space and let E be a Banach space
which is а С(X)-module. Then E is locally С(X)-convex if and only
if there is a bundle p : E -* X of Banach spaces such that E is
isometrically isomorphic to Г(p).	□
In section 14, notably (14.11), we shall see that (up to isomorphy)
the bundle p : E -* X given in (7.19) is also unique. On the other
hand, the space X may be	to "large"	for E in the sense that "many"
stalks of the	bundle p :	E	-* X are 0. This happens for instance, if
we define a multiplication on E with elements of C(X) by f-m =
= f(xQ)-m for	a fixed xq	e	X. It is	then easy to see that all the
stalks of the	bundle p :	E	-* X will	be equal to 0, except for the
stalk over xQ, which will be equal to E itself. Hence we may as
well choose the smaller space {xQ} for a base space of the bundle
p : E X without losing any information. This leads us to the
following observations:
7.20	If E is any C^(X)-module, we define
76
ЕХ:= {f е С^(Х) : f .а = О for all a e E}
It is clear that E1 is a closed ideal of С(X). If X is compact, then
there is a closed subset А с X such that E1 = {f e C(X) : f=0}
and С(X)/Е1 = C(A). Obviously, E is also a C(A)-module. Hence we
may in all cases replace the compact space X by the smaller set A.
7.21	Definition. (i) A (X)-module E is called reduced if
f-a = 0 for all a e E implies f = 0.
(ii) A bundle p : t -> X is called reduced if {x e X : p (x) / 0}
is dense in X. □
Applying (2.2) and (1.5.Ill) we obtain
7.22 Proposition. If X is completely regular and if p : E -* X
is any bundle, then the C^tX)-module r(p) Is reduced if and only if
the bundle p : E -* X is reduced.	□
7.23 Proposition. Let p : E -* X be a reduced bundle of Banach spaces
over a completely regular base space X, then
T_ : С (X) -> В ( Г (p) )
f -> Tf, Tf(a) = f-a
is an isometry of Banach algebras.
Proof. Applying (2.10) we obtain
l|Tf II = sup { || f-ст II : II ст II s 1 }
=	sup sup { | f (x) | -Ц a(x) ||	:	11 ст 11 < 1 }
xeX
=	sup sup { | f (x) | • 11 a 11 : || a || s 1, a e E }
xeX	,
=	sup {|f(x)| : x e X, p (x) / 0}
77
= Hfll
as {x e X : p 1(x) / 0} is dense in X. □
7.24	Corollary. If E is a Banach space Which is a reduced
locally C(X)-convex С(X)-module for a certain compact space X, then
the mapping f ->	: C(X) -> B(E) is an isometry of Banach algebras. □
We conclude this section with some examples:
7.25	Let E be a Banach algebra. Recall that the centroid Z^(E) °f
E is the set of all bounded continuous operators T : E -> E satis-
fying a«T(b) = T(a-b) = T(a) «b for all a,b e E. If T belongs to the
centroid, then T1 = {a e E : T(a) = 0} is always a closed ideal.
Now let p : E + X be a reduced bundle of Banach algebras. It is easy
to verify that for f e C^tX) the mapping a + f»a : Г(р) + Г(р) be-
longs to the centroid of Г(р). We shall abreviate this fact by
writing Cb(X) c Z (Г(p)).
Conversely, if the Banach algebra E is a reduced С^(X)-module,
then С, (X) c Z„(E) implies that E is а С, (X)-П-module.
bn	b
As in a C*-algebra every closed ideal is a *-ideal, we can state:
7.26	Corollary. Let X be a compact space and let E be a Banach
algebra (C*-algebra) which is at the same time a reduced С(X)-module.
Then the following statements are equivalent:
(i)	E is locally C(X)-convex and C(X) c Z (E) .
(ii)	There is a bundle p : E + X of Banach algebras (C*-algebras)
such that E is isometrically isomorphic to Г(р)	□
78
As a matter of fact, for C*-algebras the inclusion C(X) c Z^(E)
implies that E is locally C(X)-convex (see section 14).
7.27	Let E be a Banach lattice and let S,T : E + E be bounded
linear operators. We say that S < T if S(a) < T(a) for all 0 < a e E.
The center of E is defined to be the set
Z^(E) := {T e В (E) : -r>Id < T < r-Id for some r £ ]R} .
It is known that Z^(E) is as an ordered vector space and as an
algebra overJR isometrically isomorphic to C(Y), where Y is a
compact space (see [Wi 71],[FGK 78]). Moreover, for all T e Z^(E)
and all a e E we have |T(a)| - |т|(|а|) (this follows immediately
from theorem (2.2) in [FGK 78]). Especially, all positive elements
0 < T e Z^(E) are lattice homomorphisms. Hence the equivalences
T (a) = 0 iff | T (a) | = 0 iff | T | ( | a |) = 0 if f | |T | (a) | = 0 iff
|T|(a) = 0 show that ker T is an ideal of E for every T e Z^(E)•
Now let p : E + X be a reduced bundle of Banach lattices. Some
straightforward arguments show that in this case we have
Cb(X) c Z (Г(р) ) , i.e. the operator a + f-a : Г(р) -> Г(р) belongs
to Z (E) for every feC.(X).
U	D
Conversely, if E is a Banach lattice which is a reduced (X)-module,
then the above arguments show that С^(Х) c Z^(E) implies that
f1 is an ideal of E for every f e C^tX)• Thus, we have the following
analog to (7.26):
7.28	Corollary. Let X be a compact space and let E be a Banach
lattice which also is a reduced С(X)-module. The following statements
are equivalent:
(i)	E is locally C(X)-convex and C(X) c Z^(E).
79
(ii)	There is a bundle p : E + X of Banach lattices such that E
is isometrically isomorphic to the Banach lattice Ftp). □
Problem. Is there a general notion of "center" for ^-spaces in
general? If so, can this center be described in the form
C^tPrim E), where Prim E is a set of "primitive" Q-ideals of E
carrying the hull-kernel topology? (If E is a Banach algebra or a
Banach lattice, see [DH 68] and [FGK 78], for a "topological" version
of this problem see section 14.)
8. С(X)-submodules of Г(р)
Let us suppose that F is a closed submodule of r(p), where p : E + X
is a bundle of Banach spaces over a compact base space X. Then, of
course, F is a locally C(X)-convex C(X)-module, too, and therefore
F may be represented as the Banach space of all sections in a
bundle p' : E' -> X. We shall see that E' may be identified with a
certain subset of E and we shall give some characterizations of
the subsets of E obtained in this way.
In the beginning of this section we return again to bundles of
fi-spaces with certain families of seminorms:
8 .1 Definition. Let т : I + B be a type and let p : E -> X be a
bundle of fi-spaces of type т with family of seminorms (Vj)jeJ.
A subset F c E is called a ii-subbundle if
(i) p 1(x) n F is a (non-empty) fi-subspace of Ex for every
x е X
(ii) Given a e F, j e J and e > 0, there is a neighborhood U of
p(a) and a section a e Гу(р) such that a (x) e F for all
x e U and such that Vj(a(p(a)) - a) < e.
A subbundle F is called stalkwise closed, if p (x) n F is closed in
Ex for every x e X. □
A large part of the following proposition follows immediatly from
the definitions:
8 .2 Proposition. Let p : E -> X be a bundle of ^-spaces with
family of seminorms (v.) If E c E is a ti-subbundle, then the
81
restriction
P/F :
F -+X is a bundle of Qrspaoes itself having
ь /,_) £j as a family of seminorms, when we equip F with the
Qrstruoture and the topology inherit from E-
Especially, the restriotion of p to F is still open.
Proof. The only interesting point to prove is the following: If
0 c F is open in F, then 0 is a union of tubes, i.e. we have to
verify axiom (1.5. II).
Let us start with an open set 0 c F- Then we may find an open set
0'с £ such that 0=0’ nF. Pick any a e 0. Then we may find an open
neighborhood of p(a), an index j e j, a real number e > 0 and
a local section a :	+ E such that Vj(a(p(a)) ~ a) < e and such
that {В e E : Vj ( 6 - a(p( 6) ) ) < e and p(g) e U1 } c O'. Let
6 = -i ( e - v . ( a(p(a)) - a) ) .
*	J
By ( 8.1 (ii)) there is an open neighborhood of p(a) and a con-
tinuous section P :	E such that Pft^) c F and v (p(p(a)) - a) <
< 5. Obviously we have v (p(p(a)) - a(p(a))) < e - 6. Hence there
is an open neighborhood U c n Uj of p(a) such that
vj(P(x) - о(x)) < e - 6 for all x e U. Moreover, using the triangle
inequality we obtain v (B - a(p(B))) < e whenever p(B) e U and
v^(g - p (p ( B)) ) < 6. This yields
a e {g e F : p(B) e U and v (В - p (p (B) ) ) < 5} c 0-	□
The next result is a trivial remark following from the definitions:
8.3	Proposition. If p : E X is a bundle of il-spaoes and if
F c E is an it-subbundle, then и cl (E n F) is a stalkwise olosed
xeX x
£2 -subbundle, where cl(Ex n F) denotes the closure of Ex n F
in E •	□
x
82
We now discuss the connection between subbundles of p : E + X and
С(X)-submodules of r(p)- First of all, we should remark that a
considerations of the Q-structure makes only sense if r(p) is a
fi-space and this is only guaranteed if the base space is quasi-
compact. This explains the somehow technical postulates in the
following proposition:
8.4	Proposition. Let p : E + X be a bundle of ^-spaces and
assume furthermore that r(p) Is algebraically an 0,-subspaee of
the cartesian product п E
„ x
xeX
(i)
If F с E is an Qrsubbundle, then
and а С^(Х)-submodule of Г(р) •
r(p,r) an
Q-subspaee
(ii)
Converselyif F c Ftp) is an Ct-subspace, then и e (F)
xeX X
is an ft-subbundle of E. □
Of course, even if we restrict ourselves to С(X)-submodules, there
is no reason to believe that this last proposition sets up a
one-to-one correspondence between all fi-subbundles of E and all
fi-subspaces and (X)-submodules of F(p). For instance, all Cj.(X)-sub-
modules of F(p) of the form r(P/p), F с E a subbundle, are fully
additive in the sense of (4.1). However, the following example shows
that even for trivial bundles with discrete base space and stalk 1R
we can find a (X)-submodule of r(p) which is not fully additive:
8.5	Example. Let X be any infinite, non-countable set, equipped
with the discrete topology, let E - X xIR and let p : E + X be the
first projection. Then r(p) consists of all bounded mappings from
X into 1R. Moreover, if F c F(p) is a fully additive C^CX)-sub-
module, then
83
a 1 (0) .
F = {a
e Г(р) : a/M = 0}
where
M = n
aeF
Indeed, the
inclusion F c {a e
Conversely
F(p) :	=0} holds trivially.
let x e X \ M. Then there is
0 and
suppose that =
an element т e F such that т(x) / 0.
j. . .	a (x)	,	. .
uous function ——\ *x i where v (x) -
т (x) лх	лх
Multiplying т with the contin-
1 and x (y) = 0 for x / y, we
(x) = a(x) and тх(у) = 0 for
obtain an element т e F such that т
x	x
x / у. Clearly, the family (t )	is locally finite and
X X
a = l т . This proves that a
x£X\M x
e F, as F is fully additive.
Now let Fc := {a e Ftp) : a ( JR \ {0}) is countable}. Then Fc is
a closed С(X)-submodule of F(p).As n a (0) = 0 and as F / F(p)
aeF
Fc is not fully additive.
8.6
Theorem.
be a bundle of ii-spaces and
assume
that
one of
the
following two
conditions is satisfied:
or
Then
The
The
base space X
base space X
of Banach spaces
the following statements
is compact.
is paracompact, p : £ -> X
and F(p) is an fi-subspace
of
a bundle
xeX
are true:
Let p : E X
(a) If F c £ is a stalkwise
closed Q,-subbundle, then F(p, ) is a
fully additive closed ti-submodule of Г(p).
(b) If F с Г(p) is a fully additive closed ti-submodule of Г(p),
then и cl(e (F) ) =: E is a stalkwise closed ii-subbundle of
xeX X	F
E.
Moreover, the
all stalkwise
mapping F
-> Г (Рд) is a
closed il-subbundles and
bijection between the set of
the set of all fully additive
closed il-submodules of Г (p) . The inverse of this mapping is given
by F E .
£
84
Proof. From the Stone-WeierstraB theorem (4.2) (or (4.3) resp.) we
conclude that F is dense in p(p )• As F is closed in r(p), we
/eF
obtain equality.
Conversely, by the definitions we have	) c Ex n F, an^ tEie
smaller set is dense in the larger one. As by assumption p was stalk-
wise closed, we obtain equality in this case, too. □
8.7 Remarks, (i) If X is compact, then we do not have to postulate
that F is fully additive: In this case, every С(X)-submodule of
Г(р) is fully additive.
(ii) If all stalks of the bundle p : E + X are complete and if the
bundle has a countable family of seminorms (v.). ,, then e (F) is
J jeJ	x
automatically closed in Ex, provided that F is a closed and fully
additive (X)-submodule of Г(p). Hence, under these conditions, we
may set E = и e (F).
£	,, X
xeX
(Indeed, from
As the bundle
a e cl(ex(F))
( 8.6) we know that F = Г(p,	), where E_ = и
/EF	F xeX
cl(exF).
p/ep :
we may
E_ X is full by (2.9) , for a given
г
find a a e F such that a(p(a)) = a, i.e.
ex(F) = cl(ex(F)).	)
As in section 3 we may virtually "weaken" the notion of subbundles in
certain situations. We shall do this in the following proposition,
which is an immediate consequence of (3.2):
8.8 Proposition. Let p : E X be a bundle of El-spaoes and
suppose that one of the following two properties are satisfied:
(a)	p : E X is a bundle of Banach spaces and X is locally
paracompact.
(b)	p : E + X has a locally countable family of seminorms
(v.).	, all stalks are semicomplete and the base space
3 3
X is locally compact.
85
Then ? c E is a stalkwise closed il-subbundle if and only if
(i)
(ii)
F n Ex is a closed ft-subspace of for every x e X.
The
restriction
P/F =F
+ X
is
still open.
□
Thus, if we restrict ourselves to bundles of Banach spaces with a
compact base space X, we are lead to a study of those "distributions"
of closed subspaces (F ) of the stalks such that the restriction
X X
of the projection p : E + X to и F is still open. We shall
xeX x
return to a further discussion of this topic in section 15.
9. Quotients of bundles and С(X)-modules
In the same way we can form quotients of a single topological vector
space, we may form quotients of bundles of vector spaces. As one
might expect, these quotients a closely related to quotient maps
between the corresponding C(X)-modules of sections.
Let p : E + X be a fixed bundle of ^-spaces with seminorms (v.) 
1 J
and let F с E be a stalkwise closed subbundle. This time we do not
require that F n Ex is an fi-subspace of Ex, but we postulate that
F n Ex is an fi-ideal of Ex- Let us agree to call such a subbundle
a stalkwise Fl-ideal.
A straightforeward proof shows:
9.1 Proposition. If the subbundle F с E is a stalkwise Fl-ideal
and if Ftp) is an Fl-subspace of the cartesian product of the stalks,
then r(P/p) is an Fl-ideal of Г(р)-	□
Hence we may form the quotient Г(р)/Г(Р/р)- It is fairly easy to
see that Г(р)/Г(Р/р) is a topological fi-space and a locally C(X)-con-
vex C(X)-module if we define
f-(a + r(P/F)) := f«a + r(P/f) for а11 о e Г(р), f e Cb(X).
It is less obvious to see that Г(р)/Г(Р/р) is even a (X)-fi-module
in the sense of (7.1), and we shall for the moment accept this fact
without proof. Hence, applying (7.16) and (7.17), we are led to the
conclusion that, at least for compact base spaces X, the quotient
Г(р)/Г(р/р) may be represented as the fi-space of all sections in a
suitable bundle of fi-spaces q : E' + X. As this idea works only for
87
compact base spaces and does not tell very much about the relation-
ship between the bundles E, F and E', we shall turn our attention
to another aspect which will yield the above facts automatically.
9.2 Again, let p : E + X be a bundle of ^-spaces and let FEE
be a subbundle which is stalkwise a closed ideal. We define an
equivalence relation e on E by setting
(a, 6) e 0p iff p(a) = P(B) and a - 6 e F.
Let E/F :- E/g^ and let Яр : E + E/F be the quotient map. We equip
E/F with the quotient topology. There is more structure we can add
to E/F:
First of all, note that 0^. c ker p. Hence there is a mapping
Pp : E/F -> X such that p = р^оЛр, i.e. the diagram
KF
E -2 E/F
pl	4- pF
X	X
idx
commutes. By definition of the topology on E/F, the mapping p^ is
continuous.
As p./ (x) = E /(F n E ) for every x e X, the stalks of p_ : E/F + X
carry an unique Q-space structure so, that Яр : E -> E/F induces
stalkwise a homomorphism of fi-spaces (see (6.3)). Hence
PF : E/F -> X is a fibred fi-space.
Finally, we define a family (\Л)• , of seminorms on E/F by
3 led
Vj(a’) := inf (B) • В e (cc’ ) } for all a' e E/F,
i.e. v? is stalkwise the quotient seminorm of v- modulo F-
Of course, we now wish to show that p^ : E/F -> X is a bundle of
88
Q-spaces with seminorms
and that r(Pj-) contains г(р)/Г(Р/р)
as an Q-subspace (which then will yield a proof for the fact that
Г(р)/Г(Р/р) is an Cb (X) - Q-module) .
We shall split the proof into a number a small steps:
9.3	If и c X is open and if a : U + E is a local section, then
the mapping : a -> a + a(p(a)) : P 1 (U) + p 1 (U) is a homeomorphism.
(This observation is already contained in the proof of (5.7)) .
9.4	The mapping л^- : E + E/F is open.
(Let 0 с E be open. We have to show that Лр1(лр(0)) is open. We claim
that
л^.1 ( л^_ ((?) ) = {a: p ( a) = p ( 8) and a - 6 e F for some 3 e (7}
= {a: there is an open neighborhood V c p ((?) of p (a) and
a local section a : v + F such that
a e т (0 n p~1 (v) ) }.
Indeed, if a is contained in the latter set, then a = 8 + a(p(a))
where a e Гу(р^р). As 8 e 0 and as a - В = a(p(a)) e F, we obtain
a e Яр1 (тТр(О) ) .
Conversely, let a e ttjJ (л^. ((?) ) . We have to find an open neighborhood
V c p(0) of p(a) and a local section a : V -> E such that
a = a(p(a)) + 8 for a certain 8 e <?• Firstly, choose a 8' e 0 such
that p(a) = p(8') and a - 8' e F. Then select an element j e J and
an e > 0 such that {y : p(y) = p(a) and (y - 8') < e} c 0. By the
definition of subbundles there is an open neighborhood V c p((?) of
p(a) and a local section a : V + F such that Vj(a - 8' ~ a(p(a))) < e-
Let 8 = a - a(p(a)). Then 8 e as desired.
Now we conclude that
Лр1 (Лр (0) ) = и {Тст((7 n p~1 (v) ) : V c p(fl) open, a e Гу(Р/р)}
89
and this set is open by ( 9.3) . )
9.5	The mappings
add : (E/F) v(E/F> + Е/F
seal : IKx(E/F) + Е/ F
О : X - e/F
as well as the additional mappings
т (i)
f± •• V (E/F) + E/F iel
are continuous.
(As all the proofs are similar, we show only the continuity of the
т (i)
mappings f± : V (E/F) -> E/F.
Firstly, note that the mapping •’ E -> E/F induces a mapping
т(i)	т (i)	т (i)
V TTF :	V E -	V (E/F)
Ц,...,^!)) -> (KF(ai) ,...,TTF(aT(i))) •
r(i)	t(i)
By the definition of the topologies on V E and V (E/F) resp.,
which is essentially the product topology, and by (	.4), the
т (i)
mapping V л is surjective, continuous and open, whence a quotient
“	F	т (i)
map. As the mapping f : V E E is continuous, the assertion
now follows from the	commutativity of the diagram т (i)
т (i) V E	V^r	,	.. F	t(i) -	V	(E/F)
f +	1 f±
E	-	E/F	) KF
9 .6 Given a' e E/F, j e J and e > 0, there is an open neighbor-
hood U of pF(a') and a continuous section o' : U E/F such that
vj(o' (PF(a') - a' )) < e.
90
(Let а е к (а1) . Then we may find an open neighborhood U of p(a) =
= pp(a') and a continuous section a e F^tp) such that \л(а(р(а)) - а)
Define o' :=	Then o' has the desired properties. )
е.
As F is stalkwise closed, (1.5.IV) and (1.6(viii)) imply
9.7
If a e E/F, then а
= 0 if and only if v(a) =0 for all j e J
It remains to check axiom (1.5. II), i.e. we have to show that the
tubes form a base for the topology on E/F:
9.8	If O'	c Е/f is open and if a'	e (?' , then we	can	find an open
set U c X, a	continuous section	а' :	U E/F, a j	e J	and a real
number	e > 0	such that the tube	T(U,o',e,j) is open and	satisfies
a'	e {8'	e E/F : p (6') e U	and	(а' (p( B') -	6')	< e} c O' 
-1	-1
(Let ae Яр (а') с л ((?') =• 0. Then there is an open set U c x, a
section а e Г^(р) , a jej and an e > 0 such that
a e {g e E : p(8) e U and v (o(p(B)) - 8) < e} c 0.
Let o' := iTpoa. As usual, we abbreviate
T(U,a,e,j) = {g e E : p(g) e U and v (a(p(g)) - g) < e} and
T(u,a> , e, j) = { g ’ e E/F : p F ( g ’ ) e U and (о' (Р(_ ( g' ) ) - g' ) < e }
The proof of (9.8) will be complete if we can show that
к (T(U,a,e,j)) =T(U,a',e,j)
as then we can conclude that a' e T(U,a',e,j) c O'- Moreover, the
set T(U,a',e,j) will be open, as the mapping is open.
The inclusion Яр(т(u,o,e,j)) c T(U,a',e,j) is easy to see, as by
91
definition we have v^ (o'(pp (lip ( В) ) ) — iTp(g)) = Vj (тг F ( a (p ( g) ) -&)) <
<Vj(a(p(g)) - g) .
Conversely, let g' e T(U,a' ,e,j). Then we know that Vj(a'(Pp(g')) -
- g')< e. Hence there is an element у e Яр1(a'(Pp(g')) - g') such
that Vj(y) < e. Define g = a(p(y)) - y- Then g belongs to T(U,a,e,j)
and Kj-(g) = g', i.e. g' e Лр(T(u,a,e,j)) .	)
9.9 The mapping
Пр : Г(р) r(pF)
a +
is a continuous homomorphism of (X)-modules and ^-spaces. Moreover
ker nF = r(P/f)•
(We only have to prove the continuity of Пр. But this follows imme-
diately from (Пр(a)) < Oj(a).	)
9.10 If X is compact or if X is paracompact and if p : E + X
is a bundle of normed spaces, then we have
(Пр(а)) = inf (а + p) : p e Г(р^р)}.
Especially, in both cases, the mapping Пр is open onto its image.
(It is easy to check that we always have (Пр(а)) < inf {9j(а + p) :
p e r(p/F)}.
Conversely, let M = О^(Пр(а)) and let e > 0. Then for every x e X
we have и^(Пр(а)(х)) = v j (iTp (a (x) )) < M + e. Hence there is a
certain a e F n E such that v.(a(x) + a) < M + e. Let
6 := M + e - Vj(a(x) + a).
By the definition of subbundles, there is a local section px : U + F
such that v.(a - p (p(a))) < 6; using (2.2) we may assume that
92
р е Г(р/Г) is a global section. As v.(a(x) + p (x) ) < M + e, there
x	/ r	j	x
is an open neighborhood Ux of x such that \>^(р(у) + px(y)) < M + e
for all у e Ux. We now follow the path we have walked several times
before; By passing to a refinement if necessary, we may assume that
the covering (U ) v is locally finite. Take a partition of unity
(f ) v subordinate to (U )	. Now define
p := I fx’px
x eX
As the family (f «p ) is locally finite, p maps X into F and is
a continuous selection. Moreover, p is bounded: This follows trivial-
ly in the case where the base space X is compact. If p : E + X is a
bundle of normed spaces, then the family of seminorms consists of
one element only, namely . Hence we only have to show that the
mapping x + Vj(p(x)) : X + His bounded. But this follows easily
from the triangle inequality and the following
v.(a(y) + p(y)) < У	f v (y) v( a (y) + p_,(y))
3	xeX x 3	x
< У f (у) • (м + e)
x eX
= M + e,
as we have either у e Ux and then v..(p(y) + px(y)) < M + e or we
have у | Ux in which case fx(y) - 0.
Hence in both cases, p will be a continuous section p : X + F.
Moreover, the above argument shows that	+ p) < M + e. This
yields the inequality inf {*0^ (о + p) : p e Г(Р/р)} < 0^(Пр(а)) + e-
As e > 0 was arbitrary, the proof is complete. )
From the Stone-WeierstraB theorem (4.2) we conclude:
9.11 Under the hypothesis of ( 9 .10), the image of Up is dense in
93
Г(Р F) •
9.12 If the bundle p :
Е + X satisfies the assumptions of ( 9.10)
and if in addition all
stalks of the bundle are complete and if the
family of seminorms is
countable, then the mapping Пр : Г(р) -»• Г(Рр)
is surjective.
(We already know that П (r(p)) is dense in r(Pp)
is topologically and algebraically isomorphic to
and that Пр(Г(р))
Г(р)/Г(Р, J • More-
over, by assumption and (1.10), r(p) is complete
and metric. As the
quotient of a complete metric space is again complete, r(p)/r(p
and therefore Пр(Г(р)) are complete. This
yields Пр(г(р)) = Г(Рр)-)
We collect all these partial results in a
theorem:
9.13 Theorem. Let p : E + X be a bundle
of Ft-spaces with seminorms
tv.) .	. Moreover, let F E be a stalkwise closed subbundle, which
3 JeJ
is stalkwise an Pl-ideal. Then pp : E/F -> X is a bundle of Pl-spaces,
where E/F carries the quotient topology and the quotient structure
of
Pl-spaces.
If
Яр : E -> E/F is the quotient map, then
Пр : Г(р) - Г(рр)
о
Кр»а
is
a continuous
homomorphism with kernel r(p
If
X is oompact
then Up is
open onto its image and the image of Up
is
dense.
Finally, if all
stalks of E
are complete and if the family of semi-
norms is countable, then Пр
□
is surjective.
9.14 Theorem.
Let p : E -> X be a bundle of Banach spaces over a
94
paracompact base space and let F с E be a stalkwise closed subbundle.
If we equip the stalkwise quotient E/F with the quotient topology,
the quotient structure and the quotient norm, then we obtain a bundle
of Banach spaces p^. s E/F -> X. Moreover, the quotient space Г(р)/Г(р/р)
is canonically isomorphic and isometric to Г(рр). □

10. Morphisms between bundles.
Having discussed subobjects and quotients, we should also make some
remarks on morphisms between bundles in general. As everybody would
expect by now, these morphisms will be closely related with homo-
morphisms between the corresponding spaces of sections.
10.1 Definition. (i) Let E and F be fi-spaces which are at the
same time (X)-modules. A linear map T : E + F is called a
Cb (X) -il-morphism, if T is a homomorphism of fi-spaces also preserving
the С, (X)-module structure,
b
(ii)	Let p : E + X and q : F -> X be bundles of ^-spaces of the
same type т and with seminorms (v.) . _ and (ц, ), resp. A mapping
J J eJ	к к ex\
A : E + F is called a morphism of Ц-bundles, if
a)	X is continuous.
b)	p = qoX, i.e. X preserves stalks.
с)	X, -1 , . : p (x) + q (x) is a homomorphism of fi-spaces.
/Р Iх)
d)	For every k e К there are elements j e J and О < M e 1R
such that v-(a) < M implies pk(X(a)) < 1. □
The property d) in some sense says that the family (X , -1	) v has
/p kXJ X^A
to be "equicontinuous". We shall illustrate this statement in
example ( 10. 20) .
From the point of view of "equicontinuity" it is not suprising that
property d) holds automatically if X is compact:
10.2 Proposition. If p : E -> X and q : F -+ X are bundles with
oompaot base space X and seminorms (v.). _ and
J J
X : E + F is any map, then the properties (a) ,
(Uk'keK г<33Рч and
(b) and (c) of (10.1)
imply property (d).
96
Proof. Let к e К be any index. Then {g e F : Uj, (	< 1 } is open
in F- As X is continuous, the set () := X 1 ( {£3 e F :	( B) < 1 }) is open
in E and contains the 0 of Ex for every x e X. Hence, for each x e X
we may find an open neighborhood Ux of x, an ex >0 and an element
j e J such that {a e E : pfa) e U and v. (a) < ex} c 0. Now the
x	-^x
compactness of X yields finitely many points x^,...,xn e X such
that и и ... и U = X. Let M = min {e ,...,e } and let j £ J be
1	n	1	n
any element such that j < j for all 1 < i < n. Then we obtain
-V
{a e E : v.(a) < M} с X (()). This of course implies ц. (X (a) ) < 1
1	x
We now enter the discussion of the connection between С, (X)-fi-module
b
morphisms and morphisms between bundles of fi-spaces. The next propo-
sition is straightforward:
10.3 Proposition. Let p : E * X and q : F + X be bundles of
il-spaees and assume that Г(р) and T(q) are il-subspaees of the resp.
cartesian product of their stalks. If X : E * F is a morphism
between bundles of Q-spaoeSj then
T, : Г(р) + F(q)	defined by
Л
T,(a)(x) = X(a(x))	a e Г(р), x e X
Л
is a continuous С^(Х)-il-module homomorphism. □
The following example shows that a converse of (10.3) does not
always hold:
10 .4 Example. Let cq be the Banach space of all convergent se-
quences in IK with limit 0, equipped with the supremum norm. As we
know from (5.15), we may identify cQ with the space Г(р), where
97
р : Е +N is a bundle of Banach spaces whose stalks are all isomor-
phic to IK and where IN carries the cofinal topology. As every
continuous f : IN -> IK is constant, every bounded linear operator
T : cQ + cq is a Cb( В ) - module homomorphism. Especially, the
shift S : cq -> cq, S((un)neN) = s(<un+1)neN) is a cON)-module homo-
morphism, which is not induced by a bundle morphism.
This example shows that the operator T has at least to leave the
subspaces Nx invariant in order to be induced by a bundle morphism:
10.5 Proposition. Let p : E + X and q : F + X be bundles of
il-spaces and assume that r(p) and r(q) are ft-subspaces of the
direct produot of their stalks.
Moreover, assume that p : E -+ X is a full bundle and that for every
x e X the evaluation map ex : Г(р) -+ p (x) is a quotient map in the
sense that (a) = inf {Oj(a) : a e Г(р), a(p(a)) = a}, where
denotes any of the seminorms belonging to the bundle p : E + X.
If T : Ftp) -> Г (q) is a continuous morphism between ^-spaces such
that
T({a e Г(р) : a(x) = 0}) c {p e r(q) : p (x) = 0},
then there is a morphism of bundles of 0,-spaces \ : E -* F such that
T = T .
Л
Especially, T is a	-module homomorphism.
Proof. As r(p) is a full bundle, the evaluation map ex : r(p) ->-p\x)
is surjective. If we denote the evaluation map r(q) + q (x) with
e , too (and hope that this will confuse nobody), then the assumption
that T maps {a e r(p) : a(x) = 0} into {p e r(q) : p(x) = 0} is
equivalent to ker ex <= ker(ex°T). Applying (6.2), (6.3) and (6.4), we
98
find an fi-morphism Xx
-1	-1
p (x) + q (x) such that Xx»ex
ex
oT.
T
r(p)	r(g)
-1 , x
P (x)	+ q (x)
We now define A : E + F by X(a)
^p(a)
(a). Then, by construction,
(b) and (c) of (10.1 (ii)) are satisfied. Clearly, we have T(a)(x) =
= (e ° T) (a) = (X ° e ) (a) = X (a(x)) - X(a(x)), i.e. T = T. It re-
X	X X	X	Л
mains to check that X has the properties (a) and (d) of (10.1(ii)).
Let (v.) . _ and (u, ), .. be the seminorms of p : E + X and q : F + X,
J J € J	К KeK
respectively.
As T : Ftp) + F(q) is continuous, for every к e К there is an M > 0
and an j e J such that 0. (a) < M implies fi, (T(a) ) < 1. Now let
1 K
a e E be such that v . (a) < M. As the evaluation таре : Г (p) * p (p(a))
1	x
is a quotient map in the sense that Vj(B) = inf {0^ (a) : a(p(B)) =6}»
we can find a section a e Г(р) such that О^(а) < M and a(p(a)) = a.
By the choice of j and M this implies fi^(T(a) ) < 1 and especially
Uk(X(a)) = uk(T(a) (p(a))) < 1. This shows (d) of (10.1. (ii)).
Finally, we show that X : E + F is continuous: Let a e E and let
0 be any open neighborhood of X(a). As the bundle p : E -> X is full,
there is a section a e Г(р) such that a(p(a)) = a. Hence T(a) is а
continuous section of q : F -»• X passing through X (a) . Therefore,
by (1.6(vii)), there is an open neighborhood U of p(a) = q(X(a)), an
e > 0 and an к e К such that
{6 e F : q(B) e U and (T(a)(p(6)) - 6) < el c 0.
As in the proof of (d) , we pick M > 0 and j e J such that 0^ (p) < M
99
implies (T(p)) < 1. We claim that the open set
{В e E : p(g) e U and ( g - a(p(g))) < M-e}
is contained in X ^ ((?) and thus A is continuous at a:
Assume that p ( в) e U and Vj ( В - a(p(B))) < M»e. By our assumptions,
we can find a section т e Г(р) such that т(p(В)) = В ” a(p(g)) and
Vj (T) < M - e. This implies Ck(T (т + a) - T (a) ) = flk(T(t)) < 1 • e = e.
Especially, evaluating this inequality at p(B)» we obtain
Ук (A ( В) ” T(a)(p(B))) < e» i.e. X (a) c 0.	□
Let us recall from (2.5) that the evaluation maps ex : r(p) + p1(p(x))
are automatically quotient maps in the sense of (10 .5), if the base
space X is completely regular and if the bundle p : E + X is full.
Moreover, we may apply (2.9) to obtain the fullness of p : E + X in
certain cases. In these cases we would only have to check whether
T({a e Г(р) : a(x) = О}) c {a e T(q) : a(x) = 0}. But if
X is completely regular and if T is a C(X)-module homomorphism, this
is always true:
10 .6 Proposition. Let p : E -* X and q : F + X be bundles with
a completely regular base space X and let T : Г(р) -> F(q) be a
continuous С^(Х)-module homomorphism- If a e Г(р) is a section,
then a(x) = 0 implies T(a)(x) = 0.
Proof. By (2.11) it is enough to consider sections of the form
f>a, where f e С^(Х) and f(x) = 0. But in this case we have ob-
viuously T(f-a)(x) = (f»T(a)) (x) = f (x) • (T (a) (x)) = 0.	□
Hence, for bundles with a completely regular base space, we have
the following corollary:
100
10 .7 Corollary. Let p ;E + x and q : F + X be bundles of c—spaces
having a completely regular base space X. Suppose in addition that
Г(р) and F(q) are Qr-spaees (which holds automatically, if X is com-
pact). If T : Г(р) -+ F(q) is a continuous C^tX) - fl- module homomor-
phism, then each of the following conditions implies that T is of the
form T , where X :
Aip	T
do .5) .•
E + F the morphism of Qrbundles constructed in
(1) The bundle p : E + X is full.
(2) The bundle p : E + X has a locally countable family of semi-
norms and all stalks (p (x)) are complete.
X
Moreover, in these eases the
assignment X -> T is a big ection be-
A
tween all 0,-bundle morphisms from E into F and all continuous
С^(Х)-0,-module homomorphisms from r(p) into r(q) with inverse
T - xT.
Proof. It remains to show that
cases (1) and (2) the bundle p :
a e Г(р) such that a(p(a)) = a.
X = X: Let a e E. As in both
TX
E + X is full, there is a section
Then an easy calculations gives
A(TJ
Л
A(TJ
Л
(a) =
= X.
A jT j (a(p(a)))
□
= T (a)(p(a))
Л
(X’a)(p(a)) = X(a), i.e.
Applying ( 10.7) to isomorphisms T : r(p) -»• r(q) only, we get an
answer to the question to what extent the space of all sections
determines the bundle up to isomorphy:
10 .8 Definition. Two bundles p : E + X and q : F + X of fi-spaces
are called isomorphic, if there is a bijection x : E + F such that
X and X 1 are morphisms of fi-bundles. □
Clearly, every isomorphism of bundles if a homeomorphism. Conversely
101
for bundles with compact base space we have:
10.9	Proposition. Let p : E -F x and q : F + X be bundles of
il-spaces with compact base space X. Then a mapping \ ; E + F "i-8 an
isomorphism if Srbundles if and only if
(1)	X is a homeomorphism.
(2)	X preserves stalks and X is stalkwise a homomorphism of
Qrspaoes. Q
10.10	Proposition. Let p : E -»• X and q : F + X be bundles of
ii-spaces and assume that Ftp) and F(q) are Q-spaces, If the bundles
E and F are isomorphic, then so are the C^tX) -Prmodules p(p)
F(q). The converse holds, provided that p : E->-X and q : F -+X are full
bundles and provided that X is completely regular.
In particular, this is the case if Г(p) and Г(p) are complete metric
spaces and if X is completely regular.
Proof. It is only the converse which requires a proof.
Let us assume that Ftp) and r(q) are complete metric spaces. Then
all quotients of Ftp) and F(q) are complete, too. From (2.6) we
know that the evaluation maps ex : F(p) + p (x) and ex : r(q) +
q 1(x) are quotient maps onto their images and hence the images are
complete. As these images are also dense in the stalks, we conclude
that the evaluation maps are surjections. Thus, the bundles
p : E + x and q : F -> X are full. Now apply (10 .7) to complete the
proof. □
As a corollary we obtain the uniqueness of the bundle representing
locally C(X)-convex С(X)-modules constructed in section 7:
10.11	Corollary. Let E be a complete metrizable locally C^tX)-con-
102
convex С (X) -Q-module, where X is a compact space. Then, up to
isomorphy, there is an unique bundle p : E + X of Q-spaces such that
E is isomorphic to Г(p).	□
Of course, all these results apply to bundles of Banach spaces. But
dealing with Banach spaces, we always have to worry about the pre-
servation of the norms, and this is what we shall do in the follow-
ing remarks:
10.12	Definition. Let p : E + X and q : F + X be bundles of
Banach spaces and let X : E + F be a bundle morphism. We define
II A II = sup { || Ap-1 (x) || : x e X}.	□
Note that by definition the maps X
/Р 1
(x)
p 1 (x) -> q 1 (x) are
bounded linear maps and that by (10.1(ii), property (d)) the number
|| X || is finite.
10.13	Proposition. Let p : E + X and q : F -> X be bundles of
Banaoh spaces and let X : E + F be a bundle morphism. Then
l|T.||	||X||.
Л
If all the evaluation maps : Г (p) -> p \x) are quotient maps of
Banach spaces, then we have equality. This is especially the case if
the base space X is completely regular.
Proof. Let us compute: For all a e Г(р) we have
||ТЛ (a) || = sup { ||TA (a) (x) || : x e x}
= sup { || X (a (x) ) 11 : x e X}
< sup { || x/p-1 (x) II • 11 a (x) 11 : x e X}
103
s Цх|| • INI ,
whence ||TA|| < || A || -
Conversely, assume that the evaluation map ex : Г(р) -* p \x) is a
quotient map of Banach spaces. Then for every bounded linear map
S : p 1 (x) + F into an arbitrary Banach space F we have ||S || =
= ||S»e || . Applying this to the equation e oT = X , -1 , . оe , we
x	x/p^xjx
obtain the inequality
II A/p“1 (x) II II A/p“1 (x) °ex II
= l|ex°T||
* llexll'llTll
s l|T|| •
This yields ||x|| < ||T|| .	□
10.14 Definition. Let p : E + X and q : F + X be bundles of Banach
spaces and let X : E + F be a morphism of bundles. If
^/p-1 (X) s P 1 (x) -> q 1 (x) is an isometry for each x e X, then x is
called on isometry of bundles. If in addition X is a bijection,
then X is called an isometrical isomorphism of bundles. Q
In this definition we do not require an isometrical isomorphism
to be open. But using axiom (1.5.II) it is very easy to show that
this is always the case. Hence we have
10'. 15 Proposition. Let p ; E + X and q : F + X be bundles of
Banaoh spaces. Then every isometrioal isomorphism is an isomorphism
in the sense of (10 .8). If X : E + F is an isometry (isometrioal
isomorphism') of bundles, then T. : Г(р) -+ Г (q) is an isometry
(isometrioaI isomorphism) of Banaoh spaces. □
104
For bundles with arbitrary base spaces, this is all I can say about
norm preserving C(X)-module homomorphisms. To obtain better results,
we have to consider bundles with completely regular base spaces:
10 .16 Proposition. Let p : E + X and q : F * X be bundles of
Ba.na.oh spaces with a completely regular base space X and let
X : E -> F be a morphism of bundles. Then:
(i) The operator T, is an isometry if and only if X is an
isometry of bundles.
(ii) The operator is an isometrioal isomorphism of Banaoh
spaces if and only if X is an isometrioal isomorphism of
bundles.
Proof. (i): Suppose that T, : Г (p) -»• Г (q) is an isometry. We
Л
have to show that || A (a) || - ||a. || for every a e E.
Let a e E. From (10.13) we know that || A , -1 . ,	< || A || -
/Р (P \&) )
= ||T^|| - 1» whence || A (a) || < ||a|| . To verify the converse in-
equality, we recall from (2.10) that the bundle p : E + x is full.
Therefore we can find a section a e Г(р) such that a(p(a)) = a. Now
suppose that there is an e > 0 such that || A (a) || + e < ||a|| • Then
we also have ||т^(а)(р(а))|| = 11 A (a) 11 < || a || - e. As norm : F + TR
is upper semicontinuous, there is an open neighborhood U of p(a)
such that ||ta (a) (x) || < ||a|| - e for all x e U. As usual, we
take a continuous function f : X -> [0,1] such that f(p(a)) = 1
and f(X \ U) = {o}. Then we conclude that ||f «а|| = ||т^ (f *а) || =
= ||f *Т^ (а) || < 11 а 11 - e, which is impossible as 11 a 11 =
= || f • а (p (a) ) || < || f »a|| . Because e > 0 was arbitrary, we have
shown that 11 а 11 < ||A(a) || .
The other implication follows from (10.15)
(ii): One implication is again clear by (10.15). Thus, suppose
105
that T. is an isometrical isomorphism. Then X is an isometry of
bundles by (ii) and X is a bijection, as a straightforward proof
using (10.7) shows. Thus, X is an isometrical isomorphism. Q
We collect all these partial results:
10.18 Summary. Let p : E + X and q : F -+ X be bundles of Banach
spaces with a completely regular base space. Then the mapping
X
X'
T (a) = X °a
Л
is a bijection between the set of all bundle morphisms X : E + F
and the set of all bounded С^(Х) -module homomorphisms T : Г(р) ->-r(q).
The inverse of the mapping is given by
T + Xm
T
where *T/ ~1 (x) = P~1 (x) q
diagram
T
Г(р)
(x) is the unique map such that the
F(q)
-1 , .	-1 . .
p (x)	+ q (x)
commutes.
Moreover,
the mapping X + T^ preserves norms, sends isometries of
bundles onto isometries of Banach spaces and isometrical isomor-
phisms of bundles onto isometrical isomorphisms of Banach spaces. □
10 .19 Remarks.
a)	If X ; E + F is injective, then so is
106
Т : Г(р) + F(q). Example ( 10.20) shows that the converse is false.
Л
b)	From (10.7) and Banach's homomorphism theorem we may conclude
that for bundles of Banach spaces with a completely regular base
space, a bundle morphism x : E + F is bijective whenever the operator
T : Ftp) + F(q) is bijective. Example (Ю.24) shows that the con-
Л
verse does not even hold for bundles with compact base spaces.
c)	Let us again consider bundles with completely regular base spaces
and let X : E + F be a bundle morphism. If T : Ftp) + F(q) is a
Л
surjection, then we may see that X is onto, too. Conversely, if
X is onto, then the image of T has not even to be dense in F(q)
Л
(see example (10.25). However, using the Stone-WeierstraB theorem
(4.3),	T.(F(p)) is dense in F(q) whenever the base space X is com-
pact and X is surjective. Again, example (10.24) will show that this
is all we can expect.
10.20	Example. Let X be an arbitrary topological space and let
E and F be topological vector spaces. Then we consider the trivial
bundles p : XxE + X and q : XxF + X, where p and q are the first
projections. We know from (1.8) that Г(р) = С^(Х,Е) and F(q) =
= С^(Х,Е), where С^(Х,Е) (resp. C^tXjF)) denotes the topological
vector space of all E-valued (F-valued) bounded continuous functions,
equipped with the topology of uniform convergence on X.
We shall give a description of all bundle morphisms X : XxE + XxF.
If X is completely regular, this will yield a description of all
(X)-module homomorphisms from C^tXjE) + С^(Х,Р).
Let us start with a bundle morphism X : XxE + XxF. Then the restric-
tion of X to {x}xE is linear and continuous. Hence for every x e X
107
there is a continuous linear mapping xx
: E + F such that
(x,a) = (x,xx(a))
for every (x,a) e XxE
Thus we have a mapping X_ : X + Lg(E,F), where Ls(E,F) denotes the
space of all continuous linear mappings from E into F, equipped with
the topology of pointwise convergence.
The mapping X_ : X + Lg(E,F) is continuous: Indeed, for every a e E
the mapping x + Xx(a) : X + F is continuous, as this mapping is the
composition of x + (x,a) + X(x,a) = (x,X (a)) + Xx(a). As the
topology on Lg(E,F) is the topology of pointwise convergence, the
continuity of X follows.
Moreover, the set {Xx : x e X} is equicontinuous: Take any continuous
seminorm ш on F. We have to find an open neighborhood и с E of 0
such that w(X (a)) < 1 for all x e X and all a e U.
Firstly, define a seminorm ц : XxF + JR on the bundle q : XxF + X
by setting ц(х,Ь) = w(b). By (1.8(i)) we may think of ц as one of
the seminorms belonging to the bundle q : XxF + X. As X : XxE + XxF
is a morphism of bundles, there is a real number M > 0 and a semi-
norm v : XxE + ]R of the bundle p : XxE + X such that v(x,a) < M
implies w(Л (a)) = ц(x,Лх(a)) = ц(Л(x,a)) < 1. Again by (1 .8(i) ) ,
we can find a continuous seminorm к : E + JR such that v(x,a) = к(a)
for all a e E. Now let U = {a e E : к(a) < M}. Then U is an open
neighborhood of О e E. Furthermore, a e U implies w(Xx(a)) < 1 for
all x e X. This shows the equicontinuity of the set {Лх : x e X}.
Conversely, let X_ : X + Lg(E,F) be a continuous function such that
the image {Xx : x e X} is equicontinuous. We define a mapping
108
X : XxE -> XxF	by
(x,a) -» (x,xx(a)) .
Then X is continuous: It is enough to show that (x,a) + Xx(a) :
XxE + F is continuous: Let (x ,a ) £ XxE and let W be a neigh-
o о
borhood of Xx (aQ). Pick any neighborhood V of 0 e F such that
о
Xx (aQ) + V c W and let U be any neighborhood of 0 e F such that
о
U + U с v. As the mapping x + Xx(aQ) is continuous, we may find an
open neighborhood S of x such that X (a ) e X (a ) + U for all
о
x e S. Moreover, the equicontinuity of {Xx : x eX} yields an open
neighborhood T с E of 0 such that ^X(T) c U for all x e X. Thus, for
(x,a + t) e Sx(a + T) we have Av(a + t) = X (a ) + X (t) e
О	(J	X О	X	X
С X (a)+U+UcX (a ) + V c W.
X O	X о
о	о
It is now obvious that X satisfies the properties (a), (b) and (c) of
definition (10.1(ii)). We check property (d) :
Let ц : XxF + F be one of the seminorms of the bundle q : XxF + X.
By (1.8(i)) we may assume that ц(х,Ь) = w(b), where w is a certain
continuous seminorm on F. Again, we make use of the equicontinuity
of the set {Xx : x e X} to find a continuous seminorm к on E and a
number M > 0 such that к(a) < M implies w(X (a)) < 1 for all x e X.
Now the mapping v : XxE + ]R defined by v((x,a)) - к(a) is a contin-
uous seminorm on XxE and v((x,a)) < M implies ц(Х(х,а)) < 1.
Thus, X is a bundle morphism, and we have shown
10.21	Let X be a topological space and let E,F be topological
vector spaces. Then the mapping
( X : XxE + XxF)
-	( X_ : X 1S(E,F)	)
109
is a bijection between the set of all bundle morphisms from XxE into
XxF and the set of all continuous mappings from X into Ls(E,F) such
that the image is equicontinuous.
Of course, if we wish to consider an additional rf-structure on E and
F, then we have to replace L (E,F) by the subspace of all fi-homo-
morphisms.
In certain cases every bounded subset of Lg(E,F) is already equi-
continuous. This is for instance so, if E is a Baire space and
especially if E is a Banach space (see [Sch 71, theorem III.4.2]).
Thus, we can state:
10.22	Let X be a topological space and let E and F be topological
vector spaces such that E is a Baire space. Then the mapping
(A : XxE XxF) (X : X L (E,F) )
-	s
is a bijection between the set of all bundle morphisms and
Cb(X,LS(E,F)).
We may interpret (10.22) as a bundle representation of the set of
all bundle morphisms between XxE and XxF. We shall return to this
idea in a later section, when we discuss bundles of operators.
Combining (10.7) and (10.21) we obtain:
10	.23 Let X be a completely regular space and let E and F be
topological vector spaces (such that E is a Baire space). Then a
continuous operator T : Cb(X,E) -»• Cb(X,F) is a (X)-module homo-
morphism if and only if there is a continuous mapping X : X + Lg(E,F)
110
such that X(X) is equicontinuous (bounded) and T(a)(x) = X(x)(a(x))
for all x e X and all a e C, (X,E).
b
Concretely, we take X = [0,1] with its usual topology and E = F = JR.
Then the C(X)-module homomorphisms from C([O,1]) into C([0,1]) are
given by multiplication with continuous functions f e C([O,1]) (as
everybody knows) . If we take the mapping id : x + x e C([O,1 ]) , then
the C(X)-module homomorphism
T : C([0,1 ]) - C([0,1 ])
f - id • f
is injective, but the corresponding bundle homomorphis XT : X xJR +
+ X xJR is given by
XT : X xJR + X xJR
(x,r) + (x,r-x)
and thus is not injective on the fiber over x = 0.
10.24 Example. Let p : [0,1] xJR + [0,1] be the bundle constructed
in (5.16). Recall that [0,1] x]R does not carry the product topology
and that F(p) is the completion of C([0,1]) in the norm
HIf HI = max { | f(0) | , sup {x-|f(x)| : 0 < x < 1}}.
Recall also that the canonical injection T : C ([0,1 ]) -+ Ftp) is a
C([0,1])-module homomorphism which is not surjective. Nevertheless,
the corresponding bundle morphism XT : [0,1] x]R + [0,1] x]R from
the trivial bundle pr^ : [0,1 ] x]R + [0,1 ] into p : [0,1 ] xIR -> [0,1 ]
is the identity map and therefore a bijection.
10.25 ‘Example. This time we take as base space X the whole real
line and consider the trivial bundle pr^ : JRx JR + JR. In this case
111
Г(pr.) = С. ( JR) . As an operator T : C, ( ]R) + C, ( ]R) we take
lb	b	b
2
multiplication with the continuous function exp(-x ). Then T maps
Cfc( 3R) into the closed subspace C ( 3R) of all continuous functions
on ]R vanishing at infinity. Thus, T is not surjective.
In this case again, the corresponding bundle map XT : ]R x]R + 1R *]R is
even a homeomorphism.

11. Bundles of operators
In this section we shall study spaces of continuous operators into
the space of sections of a bundle. The basic ideas may be explained
with the following example:
Let E be a normed space and let X be a compact space. By K(E,C(X))
we denote the Banach space of all compact operators from E into
С(X). It is well-known that K(E,C(X)) is isometrically isomorphic
with the Banach space C(X,E') of all norm-continuous mappings from
X into E', equipped with the supremum norm. The canonical iso-
morphism
Ф :	K(E,C(X))	C(E,E')
is given by
Ф(и) (x) - e »u
where ex : С (X) +Ж is the usual evaluation map.
Hence, we have obtained a bundle representation of the space of all
compact operators. The stalks of this bundle are all identical with
E' = L^(E, Ж), i.e. they may be viewed as the set of all bounded
operators from E into the stalks of the trivial bundle pr^ : XxIK + X
equipped with the topology of uniform convergence on bounded sets.
Unfortunately, this example also shows that we can not expect such a
nice representation in general: Let us try to present the Banach
space L^tEjCtX)) of all bounded operators with the operator norm
as a space of sections in a bundle of Banach spaces. If X is infinite
then 1^(Е,С(Х)) is strictly larger than К(E,C(X)); hence L^tEjCtX))
113
cannot be represented in the form C(X,E'), where E' carries the norm
topology. However, it is known that L (E,C(X)) equipped with the
topology of pointwise convergence is topological isomorphic to
C(X,Eg), where E^ carries the a(E',E)-topology. This shows that we
have to choose an appropriate topology in order to obtain a ‘'nice" re-
presentation of L(E,C(X)) by sections in a bundle.
On the other hand, L^(E,C(X)), equipped with the operator norm,
may indeed be written as the space of all sections in a bundle of
Banach spaces with base space X. However, the stalks of this bundle
are not as nice as they are in the other bundle. We shall give a
rather technical description of them.
To start our discussion, we recall some facts concerning topologies
on the space L(E,F) of all continuous linear operators from E into
F (see [Sch 71 , III.3J):
Let E and F be topological vector spaces and let S be a family of
bounded subsets of E such that the linear hull of и S is dense in
E (a family with the second property is called total) . If we equip
L(E,F) with the topology of uniform convergence on all subset
S e S of E, then L(E,F) becomes a locally convex Hausdorff topological
vector space. A base of open neighborhoods of 0 for this topology
is given by sets of the form
U(S,U) := {T e L(E,F) : T(S) c U}
where S runs through all elements of S and U ranges over an open
neighborhood base of 0 e F.
If the topology of F is generated by a family of seminorms (v.). _,
J J
then the topology of uniform convergence on subsets in S is generated
114
by the family of seminorms (v .) c . _. given by
Ь fJ
V (T) = sup V.(T(u))
b,:] ueS 3
If the family (vj)jej is directed and we want to have the same pro-
perty for the family (v .)	.	, we have to require that the
Ь / J	/ J €J
family S is directed in the sense that for every pair S^,S2 e S there
is an element c $ such that и	S^. This is the case for
the following examples:
a)	The topology of pointwise convergence: S is the family of all
finite subsets of E and we denote by Ls(E,F) the space L(E,F)
equipped with this topology.
b)	The topology of compact convergence: S is the family of all com-
pact subsets of E; the corresponding space is denoted by L (E,F).
c)	The topology of compact, convex circled convergence, provided
that E is quasicomplete: S is the family of all compact, convex
circled subsets of E; the space of operators with this topology is
denoted by L (E,F).
cc
d)	The topology of precompact convergence: S consists of all pre-
compact subsets; the space is denoted by L
pc
(E,F)
e)	The topology of bounded convergence: S consists of all bounded
subsets; the space of operators is denoted by 1^(Е,Е).
f)	In general, we denote the space L(E,F) equipped with the topology
of uniform convergence on all subsets S e S by L (E,F) .
Note that L^tEjF) is a normed space provided that E and F are normed
spaces. In this case we may take S = {B^ (E) }. Moreover, the correspond-
ing seminorm is the operator norm.
115
Now let us suppose that F is a topological C^(X)-module. Then we may
define a multiplication with elements of С^(Х) on L(E,F) in the
following way:
(foT)(u) := fо(T(u)).
It is obvious that L(E,F) will be а С, (X)-module under this multi-
fa
plication. Moreover, we have:
11.1	Proposition. Let E be a topological vector space and let
F be a topological C^tX)-module.
(i)	If S c E is any subset and if и c F is a С^СХ.)-convex subset
then {T e L(E,F) : T(S) c U} is С, (X) -convex.
b
(ii)	If S is any family of bounded subsets of E and if F is a
locally С^(Х)-convex С^(Х)-module, then L^(E,F) is a locally
C^tX)-convex (X)-module, too. □
If E and F are Banach spaces and if F is a locally (X)-convex
C^(X)-module as a normed space (recall that for normed spaces the
С, (X)-convexity means that the closed unit ball is С, (X)-convex),
b	b
then we may apply (11.1(i)) with S =	(E) and U = B^ (F) to obtain:
11.2	Corollary. Let E and F be Banach spaces and let us assume
that E is a locally C^tX)-convex С^(Х)-module. Then L^tEjF),
equipped with the operator norm, is a locally С^(Х)-convex С^(Х)-mo-
dule, too. □
Hence, applying (7.16) we learn the following:
11.3	Corollary. If p : E + X is a bundle with a quasicompact base
зразе and if F is a topogical vector space, then there is a bundle
116
q s F + X such that L$(F,F(p)) is isomorphic to а С(X)-submodule of
F(q)j provided that S is total in F. □
Our first problem will be to identify the stalks of the bundle
q : F + X. If we look at the examples at the beginning, we would hope
that they are at least subspaces of L^(F,p (x)). If we recall the
construction of the stalks (see section 7), it seems to be reasonable
to restrict ourselves to completely regular base spaces X, as other-
wise in might happen that every Ж-valued continuous function is
constant.In this case the construction in section 7 leads to bundles
whose stalks are isomorphic to the whole space, which certainly is
no progress at all.
The second problem then will be to decide whether or not L (F,T(p)) is
not only dense in F(q) but even equal to F(q). For compact base
spaces, a first answer is
11.4	Proposition. Let p : E + x be a bundle with a compact base
space X such that all stalks are complete. If F is a bornological
space and a the family S of subsets of F contains the closure of
every Q—sequence then there is a bundle q : F -> X such that
L$(F,r(p)) is isomorphic to F(q).
Proof. We know from (1.10) that Г(р) is complete. Hence, we may
deduce from [Sch 71, p.117, exercise 8] that L^(F,F(p)) is complete.
The proposition is now an easy consequence of (7.16).	□
Before we get to work and identify the stalks of the bundle q : F + X,
we close our general discussion with a corollary:
115 Corollary. Let p : E -> X be a bundle of Banach spaces and let
117
F be a Banach space. Then the spaces L (F,r(p)) and Lb(F, p(p) ) таУ
both be represented as the space of all sections in a bundle
qc : Fc + X and qb : Fb -► X resp. □
From now on we shall pass to a slightly more general situation:
We shall always consider a bundle p : E + X such that the base space
X is at least completely regular. Moreover, L will always denote a
Cb(X)-submodule of L(F,r(p)). Finally, S will be a family of bounded
subsets such that F = <uS>and such that T(S) is precompact for
every T e L and every S e S. The space 1 will always carry the topo-
logy of uniform convergence on subsets S £ S, i.e. the relative
topology inherited from L^(F,F(p)).
11.6	Proposition. Under the above assumptions, the closure in L of
the set IX«L = {f-T : f e Cb(X), f(x) = 0, T e 1} is equal to
{T e L : T(F) c Nx}, where Nx = {a e Г(р) : a(x) = 0}.
Proof. Let T e L and let f e Cb(X) such that f(x) =0. Then for every
a e F we have (f»T)(a)(x) = f(x)•(T(a)(x)) = 0, i.e. f.T(a) e N .
This implies T(F) c N .
Moreover, the set {T : T(F) c NxJ is closed: Indeed, let (T^)
be any convergent net contained in {T e L : T(F) £ NXJ- Then this
net is also convergent in the topology of pointwise convergence. But
the set {T e L : T(F) c NxJ is obviously closed in the topology of
pointwise convergence. Hence {T e L : T(F) c NxJ is closed in L.
It remains to show that I -L is dense in {T e L : T(F) c N }. To prove
X	x
this, let T e L beany continuous operator such that T(F) c Nx, let
S e S be any element, let e > 0 and let be one of the seminorms
belonging to the bundle p : E + X. It suffices to find an function
118
g e С, (X) such that g(x) = 0 and sup g.((1 - g) .T (s) ) < e.
s£S 3
As T(S) is precompact, we can find a^,...,an e S such that for every
a e S there is an i £ {1,...,n} such that	-a)) < e/2. As
Т(а^)(x) = 0 for all 1 < i < n and as Vj г E +1R is upper semicontin-
uous, we can find an open neighborhood U of x such that -;^(Т(а^)(у))
< e/2 for all у e U and all 1 < i < n. Use the fact that X is com-
pletely regular to find a continuous mapping g : X + [0,1] satisfying
g(x) = 0 and g(X \ U) = {1 }. By standard arguments we obtain
$j((1 - g) (Ttap ) < e/2 for all 1 < i <. n. If a e S is arbitrary, then
there is a certain i e {1,...,n} such that g^((l-g) .T(a - a^))	<
< Oj(T(a - a^)) < e/2. Now the triangle inequality yields
Oj ((1 - g) -T(a)) < e for all a e S. Hence we have
sup 0. ( (1 - g) -T(s) ) < e,
s eS 3
as desired. □
In the following, we shall again make use of our convention to denote
the stalks of the bundle p : E + X by Ex, x e X. Further, we let
Nx = {a e Г(р) : a (x) =0} and ex : Ftp) + Ex be the evaluation
map.
If L is a subspace of L$(F,r(p)), we define
Nx = {T e L : T(F) c Nx}
and
Lx = i/Nx	equipped with the quotient topology.
In the following, we shall give a description of a family of semi-
norms generating the topology on L :
Let Vj be on of the seminorms of the bundle p : E -> X and let S e S.
We define:
119
хм (Т + n4 := inf {sup v. (T(a) + T' (а) ) : Т' е яЧ
S'D	Х	aeS 3	Х
Then the seminorms (\>х .)	. will generate the topology on I.
ь । J Ь , J	X
Nobody can work with such a formula, therefore we give an alternative
expression for these seminorms:
IL 7 Proposition.
Under the assumptions made in the remarks
preoeeding Q.1 .6), we have
(i) The mapping T +	+ exoT : Lx -+ L(F,Ex) is well defined,
linear and injective.
(ii) For every seminorm Vj : E -»• JR of the bundle p : E + X, every
S e S, every x e X and every T e L we have
M . (T + n4 = sup V. (e °T(a) )
x aeS J x
(Hi) In particular, the mapping T + N^ -+ e^oT •• L L(F,Ex) 7-s an
embedding.
Proof. The property (i) is a consequence of (11.6) and (iii)
follows immediately from (ii). Thus, it remains to check (ii):
First of all, for every T' e	we have X
sup 0 . (T(a) + T'(a)) = sup aeS -1	aeS	sup V.(T(a) (у) + T' (a) (y) ) угХ -1
> sup aeS	хм(T(a)(x) + T'(a)(x))
= sup aeS	Vj(T(a) (x))	(since T'(a) (x) = 0 )
= sup aeS	v.(ev’T(a)) , J x
x	L and therefore Vg j(T + N^) > < x \ (vs,j)S, j ‘	sup v. (e °T(a)) by the definition of the aeS 3 x
Conversely, suppose that there is a C > 0 such that
120
^s,j(T +
> C > sup v (ev"T(a)).
aeS 3 X
In this case, we let
e := y(C - sup \>.(e oT(a))) .
2 aeS 3 X
As the set T(S) c r(p) is precompact, we may find elements a^,...,an
e S such that for every a e S there is an index i e {1,...,n} with
0. (T(a) - T(a.)) < e and we conclude that v.< ( e °T (a.) ) < C - e for
ji	j x i
all 1 s i £ n.
Now the upper semicontinuity of the mappings у + \>^(Т(а^) (у)) : X + 1R
yields an open neighborhood U of x such that -;^(Т(а^)(у)) < С - e for
all у e U and all i e {1,...,n}. Choose a continuous function
g : X ->- [0,1] such that g(x) = 0 and g (X \ U) = {1}. Then we have
O.((1 - g)-T(a.))	= sup v.((1 - g(y))-T(a.)(y))
3	yeX 3
< C - e
for all 1 < i < n.
If a e S is arbitrary, then ^^.(Tta^) - T(a)) < e for a certain
i e {l,...,n}. Hence the triangle inequality yields
0 ((1 - g)-T(a))	< C
and therefore
sup 0,(T(a) - g-T(a)) < C.
aeS 3
From (11.6) we conclude that -g-T e N^- This leads to the contra-
diction
C <	«2,j(T + N^)
< sup v (T(a) - g-T(a))
aeS 3
C. □
121
11.8 Proposition. Under the same assumptions, we have
(i) The mapping x •+ sup v. (e oT(a)): X + ]R is upper semicon-
aeS 21 X
tinuous for every T e L, S e S and every seminorm Vj : E + 1R
of the bundle p : E + X.
(ii) sup O.(T(a)) = sup sup v.(e °T(a)).
aeS 3	xeX aeS 3 X
x	L
Proof. Using (7.7), we conclude that the mapping x + $s (T + Nx>
is upper semicontinuous. Thus, (i) follows from (11.7).
The proof of (ii) is an easy calculation:
sup O.(T(a))	= sup sup v.(T(a)(x))
aeS 3	aeS хеХ 3
= sup sup v.(e oT(a)).	□
xeX aeS 3
We are now in the position to prove a bundle representation of
С(X)-submodules L c E$(F,r(p)). Our first result is still rather
technical:
11.9 Proposition.
regular base space
gical vector space.
Let p : E -> x be a bundle with a completely
and seminorms (v.). , and let F be a topolo-
Further, let L c L^(F,F(p)) be a	-submo-
dule, where S is a directed family of bounded subsets of F such
that F = <uS>and such that T(S) is precompact in Г(р) for every
S e S and every T e L.
Then there is a full bundle q^
to а С^(Х)-submodule of F(q^).
:	X such that L is isomorphic
The stalk over x e X of this bundle
may be chosen to be a subspace of L^(F,Ex), where Ex is the
stalk over x of the bundle p : E * X. In this ease, the canonical
injection
Ф : L
r(qL)
122
is given by Ф(Т) (x) = ex°Tj where ex : r(p) -+ Ex is the canonical
eva Illation.
Proof.
For every x e X let Mx = {ex°T : T e £} c L (F,Ex). If
(v.). , is the
1 J
topology on Mx
family of seminorms of the bundle p : E + X, then the
is induced by the seminorms (ш
x
S, j
(S,j)eSxJ given ЬУ
We ц(а) = sup V.(a(u))	, a e M .
b,:l	u£S 3	X
Moreover, by (11.8(ii)), the space L may be identified with a sub-
space of П°° M . The embedding L + JI°° M is given by
xeX X	xeX X
T -> ф(Т)
Ф(Т)(x)	= ex°T
It is now easy to verify that L, viewed as a subspace of n°° M ,
xeX x
satisfies the axioms (FM3) and (FM4) of section 5. Therefore an
application of (5.8) completes the proof. □
In general, there is no reason to believe that L is isomorphic to
the space of all section of r(q^). For example, let X be compact
and let N c£ (F,C(X)) be the space of all nuclear operators from a
normed space F into С(X), equipped with the operator norm. As every
nuclear operator is compact, the above result applies to N and we
obtain a bundle q^ : F^ + X such that N may be identified with a
С(X)-submodule of Г(q^). The Stone-WeierstraB theorem implies that
N is dense in r(q^) and it turns out that F(q^) is isomorphic to the
space of all compact operators, i.e. N is strictly contained in
r(qN) .
Thus, it is of some interest to study the space of all sections
of r(q^). It turns out that every section of the bundle q^ : F^ + X
may be viewed as a linear operator from F into r(p), but these
operators will not be continuous in general.
123
11.10 Proposition.
Let p : E -> X, L c E$(F,r(p)) and S be as in
(11.9) .
(i)
If I e r(qL)
is a continuous section, then
T : F + r(p)
defined by
T (u) (x)	= E(x) (u)	for all x e X , all u e F
is a linear map between F and r(p)-
eontinuous:
(a)	S contains a neighborhood of 0 e F.
(b)	F is bornological, X is compact and S contains the clo-
sure of every О-sequence in F.
In these eases, the mapping E + T : F(q^) + E$(F,r(p))
is an embedding.
IL 11 Remarks (i) If we compare the case (b) of this proposition
with (11 .4), we see that we may drop the completeness of the
stalks in the hypothesis of (11.4).
(ii) We shall see in the following proof that T will be always
sequentially continuous, provided that X is compact and that S con-
tains the closure of every О-sequence in F.
Proof of (11.10).	(i) : Obviously, the mapping T will be linear.
Whence it is enough to show that T^, maps F into Г(р) •
Thus, let us start with uq e F. As the family of sets S generates F,
we may assume that u e S for a certain e S .
J	о о	о
Firstly, we show that T^,(uq) is bounded: Let be any of the semi-
norms of the bundle p : E + X. If the seminorms (ш- ) o . on the
Ь z J Ь t J
124
bundle q^ :
F^ * X are defined as in the proof of (11.9)
then we
may estimate:
0. (T (u ))	= sup v.(T (u )(x))
3 L ° xex 3 L °
= sup v.(E(x)(u ))
xeX 3
< sup sup v. ( E (x) (u) )
ueSQ xcX 3
= sup ш* (E(x))
XeX o'3
as E belongs to Ftq^) and
therefore is a bounded selection.
To show the continuity of the mapping T^,(uq) : x + E , we state the
following
(*) Let x e X and let T e L be such that e oT = E (x ) . Then for
о	x	о
о
every seminorm belonging to the bundle p : E + X, every
e > 0 and every S e S there is an open neighborhood W of xq
such that v (E(x)(u) - T(u)(x)) < e for all u e S and all
x e W.
Indeed, the property (*) follows immediatlely from the upper semi-
continuity of the mapping x + wX .(E(x) - e »T) = sup v.(E(x) (u) -
S'3	x u£S 3
- T(u)(x)).
Now (*) implies the continuity of T^(uq) at x°: Firstly, by the
definition of the stalks of
(see the proof of (11.9)), we can
pick
an operator T e L such
have
open
T (u )(x ) = T(u )(x )
Eoo	о о
neighborhood У of T^(uq)(xq) looks like
that г »T = г x . In this case we
t.x	о
о
and as T(uq) belongs to p(P)» a typical
У = {a e E : p(a) e W, Vj(a - T(uQ)(p(a))) < e}.
125
where W is an open set around x . Now use (*) to find an open
о
neighborhood W of xQ such that v.(E(x)(uq) - T(uq)(x)) < e for all
x e W. Then by definition the mapping T^ maps the neighborhood W n W'
of x into V.
о
(ii): Case a. Let U e S be a neighborhood of 0 e F. As E belongs to
Ttq^), it is a bounded selection. Hence for every j e J the number
sup 0 • (T (u) )	= sup sup v. (E(x)(u))
u<U J L	ueu xeX J
= sup w* . ( E(x))
xex ,J
is finite. Clearly, this implies the continuity of T^,.
Case b. By ESch 71, II.8.3] we have to show that (T<un))n
converges to 0 for every О-sequence (un)nelj in F-
Fix e > 0 and let v. : E + 1R be a seminorm of the bundle p : E + X.
3
If (u ) is a fixed О-sequence in F, we show :
n n eJN
(**) For every x e X there is a neighborhood U of x and a natural
number N eN such that for all n N and all у e U we have
sup v . (T (u ) (y) ) £ e.
yeu J L n
Once (**) is established, an easy compactness argument will finish
the proof.
To convince the reader of (**), we shall again use (*): Firstly,
choose again any T e L such that e,{°T = E(x) and let S = {0} и {un :
n eU}. Note that S belongs to S by our assumption. Thus (*)
yields an open neighborhood U of x such that
sup v . (T (u ) (y) - T(u )(y'l) < e/2
yeU 11 L n	n
for all n e'u. As the operator T : F + Г(р) is continuous, we con-
clude that lim T(u ) =0. Therefore there is an N e JN such that
n->-°°
126
sup V. (T (u ) (у) ) < e/2
yeX J
for all n e Using the triangle inequality, these two inequalities
together yield (**).
To show that the mapping e + T is an embedding, we have to recall
that the topology on L (F,r(p)) is induced by the seminorms .,
Ь	ьf j
j e J, S e S given by
.(T)	= sup sup v.(T(u)(x))
,J	ueS xeX J
and the topology on r(qp
is
given be the seminorms
j e J and
“S, j '
S e S defined by
fflg j(E) = sup sup v.(E(X)(U)).
,J	xeX ueS J
An easy computation shows that for e e г(д^) we have
as,j(E) = ^,з(те’
and thus the proof is complete.
□
11.12 Corollary. Let p : E + X be a bundle with a completely
regular base space and let F be a topological vector space. Then
there is a bundle q : F + X such that L (F,r(p)) equipped with the
pc
topology of precompact convergence is isomorphic to a C^(X)-submodule
of F(q). The stalk over xeX of this bundle may be choosen to be a
E ). where E is the stalk over x of the bundle
x' ’	X	J
p : E + X. Tn this case, the canonical injection ф : L (F,r(p)) ->
Pc
+ F(q) is given as in (11.9)
Moreover, in each of the following cases (a) and (b), the map ф is
surjective with inverse
subspace of L (F,
Y : F(q)
E
- Lnc(F'r(P))
pu
127
where T^(u)(x) = £(x)(u) for all u e F and all x e X:
a)	F is finite dimensional.
b)	F is bornological and X is compact.
Proof. Only the verification of the surjectivity of Ф is of some
interest. But this follows from (11.10), if we note that 0 e F has a
precompact neighborhood, provided that F is finite dimensional ,
whence case (a) of (ll.10(ii)) applies under these circumstances.
Moreover, the closure of every О-sequence is precompact and thus
case (b) of (ll.10(ii)) applies in case (b) of (11.12).	□
Our next corollary concerns spaces of compact operators. Recall that
an operator К : F + E between topological vector spaces is called
compact, if there is a neighborhood U of 0 e F such that K(U) is
relatively compact in E. By K(F,E) we denote the subspace of L^(F,E)
of all compact operators, equipped with the topology of bounded
convergence. If и <= F is a neighborhood of 0, we let
KytFjE) = {Ke K(E,F) : K(U) is relatively compact}
and we equip this space with the topology of uniform convergence on
U (which may be finer than the topology inherited from K(F,E)
If p : E + X is a bundle, then K(F,r(p)) and Ky(F,r(P)) are Cb(X)-sub-
modules of L(F,r(p)). Therefore, we can state:
1L13 Corollary. Let p : E + X be a bundle with a completely
regular base space, let F be a topological vector space and let
U c F be a neighborhood of 0. Then there is a bundle q : F ->- X such
that Ku(F,r(p))is isomorphic to a C^(X)-submodule of r(q). The stalk
over x e X of this bundle may be choosen to be a subspace of
128
K„(F,E ), where E = p (x) . In this ease, the canonical injection
U x	x r ' '
Ф : Ky(F,r(p)) -> F(q) is given as in (11.9). Moreover, we have a
(topologioal) embedding
V : r(q) LD(F,r(p))
E + T
where 1у(Р,Г(р)) denotes the space L(F,r(p)) equipped with the
topology of uniform convergence on U.
If X is compact and if all stalks of the bundle p : E + X are
quasicomplete, then ф is a bijection with inverse Ч-
Proof. We again apply (11.9) to establish the existence of such a
bundle. Note that the stalk F^ of the bundle q : F + X may be identified
with {exoK : K e Ky(F,г(p)) } and hence is contained in Ky(F,Ex) .
The fact that 4 : F(q) + LyCFjEtp)) is a topological embedding
follows from (ll.10(ii)), case (a).
Finally, if X is compact, then the image of Kg(F, r(p)) under Ф is
dense in r(q) by the Stone-WeierstraB theorem (4.2). As the restric-
tion of 4 to the image of Ф is the inverse of ф, this implies that
Kg(F,r(p)) is dense in the image of 4. Now we know from (1.10) that
Г(р) is quasicomplete whenever all the stalks are quasicomplete.
From the proof of (III.9.3) in [Sch 71] we conclude that Ky(F,r(p))
is closed in EyCF^Cp)). This shows that К (F,r(p)) is equal to
the image of 4 and ч is the inverse of Ф. □
Of course, we can apply (11.13)
space. If in addition p : E -> X
we obtain:
to K(F,r(p)), where F is a normed
is a bundle of Banach spaces, then
IL 14 Corollary. Let p : E + X be a bundle of Banach spaces
129
X completely regular, and let F be a normed spaoe. Then there is a
bundle q : F + X of Banaoh spaces such that the Banaoh spaoe
K(F,r(p)) of all compact operators equipped with the operator norm
is isometrically isomorphic to а С^(Х)-submodule of r(q)- The stalk
over x e X of this bundle may be ohoosen to be a closed subspace of
K(F,Ex)j equipped with the operator norm. In this case, the canonical
injection ф : K(F,r(p)) -»• r(q) is given as in (11.9).
If X is compact, then ф is bijective.
Proof. This result is a variation of (11.13); there are two things
which have to be checked:
(i) The mapping ф is an isometry: This follows immediately from the
definition of the operator norm, the definition of the stalks of the
bundle q : F + X as it was given in the proof of (11.9) and
(11.8(ii)).
(ii) The stalks, as they have been defined in the proof of (11.9),
are Banach spaces and thus closed subspaces of K(F,Ex), x e X
From (11.7) we may conclude that the stalks are isometrically
isomorphic to quotients of K(F,F(p)) and thus are complete, since
K(F,r(p)) is a Banach space. □
In these last three corollaries the stalks of the bundle q : E + X
were always subspaces of larger spaces: They were subspaces of
L (F,E ) in (11.12), subspaces of К (F,E ) in (11.13) and subspaces
p C X	и X
of K(F,r(p)) in (11.14). In which cases do we obtain the whole space
as stalk? It turns out that at least in the first and in the last
case the answers are the same: It suffices that all stalks of the
bundle p : E + X have the approximation property in the sense of
Grothendieck (see [Gr 55]). Alternatively, we could postulate that
the bundle p : E + X is locally trivial.
130
The problem we are dealing with in this context is the following:
Given a point x e X in the base space of the bundle p : E -+ X and
an operator t : F + Ex, can we find a "lifting" T : F -+ r(p) such
that ex°T = t?
11.15 Proposition. Let p : E + X be a bundle over a completely
regular base space, let F be a topological vector space and let S be
a directed and total family of bounded subsets of F. Then for every
x e X, the closure of {e oT : T e Lc(F,r(p)) and dim T(F) < co} in
X	э
LC(F,E ) contains all operators of finite rank.
О X
Proof. Let t g L^(F,Ex) be of finite rank, i.e.
n
t = У фс®а.
i=1	1	1
for certain elements а- e E and certain elements e F'. Given
I X	*1
S e S and an open, convex and circled neighborhood U c Ex of 0, we
have to find an element T e L$(F,r(p)) such that dim T(F) < co and
such that (t - ex°T)(S) c U.
Firstly, note that ф^(3) is bounded in Ж for every 1 < i < n. Thus,
we can find a constant M > 0 such that |ф^(з)| < M for all s e S
and all i e {1,...,n}. Moreover, by (1.5.Ill) and (2.2), the set
{а(х) : а e Г(р) } c Ex is dense in Ex> Hence we can find sections
a ,...,n e Ftp) such that a. - a.(x) e *U for all i e {1,...,n}.
in	ii	M«n
Now define
n
т := Уф.9o : F -> Г (p) .
i=1	1	1
Then, by definition, T is of finite rank and for all s e S we have
n
(t - e от)(s)	= ( l ф.в(а. - а (x)))(s)
x	±=1 i i i
n
= У Ф . (s) • (а. - a. (X) )
—-1	1	-1-
131
6 Л, *i(s) мЬги
1=1
S и ,
i.e. (t - exoT)(S) c u. □
It is now evident that we are lead to spaces with the approximation
property:
11-16 Definition. A locally convex topological vector space E has
the approximation property, provided that for every locally convex
topological vector space F the linear operators of finite rank from
E into F are dense in L (F,E).
pc'
□
A. Grothendieck ([Gr 55]) showed that for Banach spaces E this
definition is equivalent to the following statement:
For every normed space F the linear operators of finite rank from
E into F are dense in K(F,E).
We now can state:
11.17	Complement. (i) Let p : E + X be a bundle over a completely
regular base space X such that all the stalks have the approximation
property. Then the stalks of the bundle q : F + X in (11.12) may be
chosen
to be dense subspaces of
Lpc(F'Ex>' x £ X-
(ii) If in addition p
E -> X is a bundle of Banach spaces, then the
stalks of the bundle q
F -+ X in (18.14) may be chosen to be
K(F,Ex), x e X. □
IL 18 Remark.
Under the conditions of
(11.12) and (11.17) we can
choose the whole spaces L (F,E ), x e X,
pc x
as the stalks of the bundle
132
q :F + X. In this case however, it may happen that the bundle
q :F + X is no longer a full bundle, although I do not know of any
example to illustrate this.
With this new choice of the stalks even the second half of (11.12)
remains valid. To show this, we would have to generalize (11.10),
notabely the properties (*) and (**) in the proof of (11.10). As we
are not going to use these facts in the following, we leave the
details to the reader.
In the next theorem we apply the results obtained so far to the
approximation property of spaces of sections:
11.19	Theorem. Let p : E + X be a bundle over a oompact base space
X. Then the space of all sections r(P) has the approximation property,
provided that every stalk E , x e X, has the approximation property.
X
Proof. Let F be a topological vector space and let F'®r(p) be the
set of all linear operators from F into Г(р) of finite rank. We have
to show that F'®F(p) is dense in L „(F,T(p))
pc
Firstly, note that F'»r(p) is а С(X)-submodule of L (F,F(p)), since
pc
the multiplication with elements f e С(X) is linear. From (11.12) and
(11.17 (i) ) we know that there is a bundle q : F -> X with stalks
isomorphic to the dense subspaces {^"T • T e. LpC(F,F(p))} of
L (F,E ), x e X, such that L (F,F(p)) may be identified with a
pc X	pC
С(X)-submodule of Г(q). Under these identifications the set
{T(x) : T e F'®F(p)} is dense in Ux°T : T e lpC(F,F(p)} by (11.15).
Hence the Stone-WeierstraB theorem (4.2) yields that F'®F(p) is
dense in L (F,F(p)).	□
pc
For a more detailed discussion of the approximation property of
133
spaces of sections, we refer to [Gi 78], [Pr 79], and [Bi 80].
Another important case of С(X)-submodules of L(F,i(p)) was already
discussed in section 10 and we shall add some facts here:
Let us consider a second bundle p' : E' + X. Then the set of all
(X)-module homomorphisms from Г(р') into Г(р) form а С^(Х)-sub-
module of L( Г(р' ) , Г(р) ) . We shall assume that p' : E' + X is a full
bundle and that the base space X is completely regular. Under these
conditions we saw in (10.7) that every continuous (X)-module homo-
morphism T : Г(р') + Г(р) may be "decomposed" into a bundle morphism
XT : E' -> E and this "decomposition" may be indeed been thought of
as a section in the bundle constructed in (11.9) . To explain this,
let us start with a lemma:
11.20	Lemma. Let E and F be locally convex topological vector
space, let M be a closed subspace of F and let л : F + F/M be the
quotient map. If S is an updirected and total family of bounded
subsets of F, then the mapping
4	= LK(S)(F/M'E)
T
Ls(F,E)
To-n
is a topological embedding with range {T e L^(F,E) : T(M) = 0}.
Proof. Let T г L^(F,E) and assume that T(M) = 0. Then we have
T(S) с и if and only if T(S + M) <= u, where S e S and where U e E
is an open neigborhood of 0.	□
Let us apply ( IL 9) to the situation where F = Г(р') for a full
bundle p' : E' X, where S is a directed family of precompact
134
subsets of r(p') such that F is generated by и s and where
L = Mod ( Г (p' ) , Г (p) ) . Then we find a bundle q : F + X such that L
is (isomorphic to) a (X)-submodule of F(q), the stalks of this
bundle being{ex°T : T e Mod(Г(p') , Г(p)) } c 1$(Г(р'),Е ). As
p' : E' X is a full bundle, the evaluation map ex : Г(р') + p' (x)
is a (topological) quotient map by (2.7). Hence by (10.6) and (11.20)
the subspace{ex°T : T e Mod(Г (p' ) ,Г(p)) } c L (Г(р'),Ех) may be
identified with a subspace of L„, . (E',E ) , where E' = p' (x) and
r	s(x) xx	x
where S(x) = {ex(S) : S e S}. Under this identification, the operator
e »T : Г(р') + E corresponds to the unique operator T : E' -> E
XX	xxx
such that the diagram
T
Г(р')	- F(p)
is commutative. It is clear from the proof of Q.0 .5) and (10.7) that
Tx =	1' where : E' + E is the unique bundle morphism
such that T = T, . Let us agree that we write Am(x) instead of
Л	1
AT/p'-1(x)•
Applying (11.9) we obtain a bundle q : F X such that Mod(Г(p'),Г(p))
<= L$ ( Г (p' ) , Г (p) ) is isomorphic to a (X) -submodule of Г (q) . The
stalks of this bundle may be chosen to be subspaces of L , .(E',E )
and the canonical injection is given by A + AT. Furthermore, the
family of seminorms of the bundle q : F + X is defined by
“S,j = F *	®
Л + sup v.(A{s[q(A)]}),
ses 3
where S e S and where v.. : E +]R is one of the seminorms of the
135
bundle p : E -> X.
If X is compact and if Г(р') is bornological, then Mod(г(p'),Г(P))
and F(q) are isomorphic, provided that S contains the closure of
every O-sequence.
We state a special case of these observations as a theorem:
11.21 Theorem. Let p : E -»• X and p' : E' + X be bundles of Banaoh
spates over a compact base spaoe X. Then there is a bundle q : F + X
such that the C(,X)-module Mod ( Г (p') , Г (p) ) equipped with the topology
of oompaot eonvergenoe is topologically and
algebraioally isomorphic to F(q). The stalks of this bundle may
be chosen to be subspaoes of fc(E^,Ex)• this
case, the canonical isomorphism is given by

Mod(F(p' ) ,F(p) ) F(q)
T
T
Proof. Let S denote the family of all compact
subsets of Г(р'). If we can show that S(x) is the family of all
compact subsets of E^, the theorem will follow
from the discussions preceeding (11.21).
Thus, we are dealing with the following problem: Given a Banach
space E, a closed linear subspace F, a compact
subset A c e/F, is there a compact subset в с E
such that A = В + F?. But this is a well-known result from the
theory of Banach spaces. □
12.	Excursion: Continuous lattices and bundles
In the past years a certain type of lattices appeared in mathematics,
which seem to be a natural background of a large variety of order
theoretical properties of mathematical structures. These lattices
were called continuous lattices by D.Scott in [Sc 72]. In the follow-
ing years, K.H.Hofmann and A.Stralka discovered that this
type of lattices was already known to other mathematicians in differ-
ent areas. J.D.Lawson, for instance, called them compact topological
semilattices with small semilattices, moreover A.Day and 0.Wyler
found them as "algebras" of the filter monad in category theory.
Also in functional analysis the concept of continuous lattices seems
to be useful. In this section we shall collect a few results and
definitions which will be needed later on. With a few exceptions,
the proofs may be found in [Comp 80].
12.1	Let L be a complete lattice. A subset D <= L is said to be
directed, if every pair a,b e D has an upper bound in D.
If a,b e L are two elements, we say that a is way below b, if every
directed set d with sup D > b contains an element d e D such that
a < d.
We shall abbreviate the phrase "a is way below b" by writing a << b.
12.2	A complete lattice L is called continuous lattice, if for all
a e L we have a = sup {b : b << a}.
123 We add a couple of examples which will be of significance:
137
(i)	Let X be a locally compact topological space. By 0(X) we denote
the complete lattice of all open subsets of X, ordered by inclusion.
Then 0(X) is a continuous lattice. Moreover, we have U <<V if and
only if U is compact and contained in V (i.e. if U is relatively
compact in V in the topological sense).
(ii)	Let К be a compact convex subset of a locally convex topological
vector space and let Conv(K) be the complete lattice of all closed
convex subsets of К, ordered by dual inclusion (i.e. A < В iff
В c a). Then Conv(K) is a continuous lattice. Here, we have
A v в = A n В
А л в = conv(А и В) (where conv(M) denotes the closed
convex hull of M)
A << В iff В c a° (where ° is the topological kernel
operator)
12.4	Every continuous lattices carries two important topologies,
which we will use later on:
(i)	The Scott topology. Let L be a (continuous) lattice. A subset
U c l is said to be Scott-open if
(1)	u e U and u < v imply v e U
(2)	If D c L is directed and if sup D e U, then U n D / 0.
It is easy to verify that the Scott open sets form a topology on L
which will be called the Scott-topology.
In a continuous lattice the sets of the form V(a) : = {x e L : a << x}
form a base for the Scott-topology. A mapping f : L * V between two
complete lattices is Scott-continuous (i.e. is continuous with
respect to the Scott-topologies on L and V) if and only if for
138
every directed subset D c L we have f(sup D) = sup f(D).
(ii)	The topology generated by the Scott-topology together with all
sets of the form L \ +a, a e L, is called the Lawson-topology. On a
continuous lattice, the Lawson-topology is always compact and Haus-
dorff. Further, the mapping л : LxL + L is continuous and the Lawson
topology is uniquely determined by these properties.
An л-homomorphism f : L + V between continuous lattices L and V is
Lawson-continuous (i.e. continuous with respect to the Lawson topolo-
gies) if and only if f preserves suprema of directed sets and
arbitrary infima.
12.5	If X is locally compact and if А с X is a compact subset of X,
then {U e 0(X) : A c u} is a typical Scott-open set. Of course,
instead of using 0(X) we may consider the complete lattice C1(X)
of all closed subsets of X, ordered by dual inclusion. In this case,
{B e C1(X) : A n В / 0} is Scott-open for every compact subset А с X.
Let К be a compact convex subset of a locally convex topological
vector space. If и с к is relatively open in К, then {A e Conv(K) :
A c u} is Scott open in Conv(K).
12.6	Let X again be locally compact. Then the Lawson topology on
0(X) (or, equivalently, on C1(X) ) is the well-known Hausdorff
topology. We will see in a moment that the same is true for the
continuous lattice Conv(K);
12.7	Proposition. If К is a compact convex set in a locally convex
topological vector space, then the inclusion Conv(K) + C1(X) is con-
tinuous for the resp. Lawson topologies. Especially, Conv(K) is
closed in Cl(K).
139
Proof. Firstly, we show that Conv(K) is closed in C1(K). Let
A e C1(K) \ Conv(K) be a closed subset of К which is not convex.
We have to find an open neighborhood of A which does not intersect
Conv(K). Pick a X e [0,1 ] and elements a,b e A such that
X*a + (1 - X) *b =: c | A. Let W be an open set around c such that
W n A = 0. As the mapping (x,y) -+ x«x + (1 - X) «У : KxK + К is con-
tinuous, there are open sets U,V around a and b resp. such that
X*U + (1 - X)«V c W. Now the set
{C e C1(K) : C 4 К \ U, C 4 К \ V, C n W = 0}
is open in the Lawson topology of C1(K) and contains A. Moreover,
this open set is disjoint from Conv(K) : IfC4K\U, C4K\V
and C n W = 0, we may pick elements x e C n U and у e С n V. Then
the convex combination X«x + (1 - X) *y belongs to X-U + (1 - X)-V c
c w and therefore cannot belong to C as C n X = 0. Hence C is not
convex.
Next, we claim that the Lawson topology on Conv(K) is coarser than
the topology induced by the Lawson topology on C1(K). This will
finish the proof, as both topologies are compact.
Let U c Conv(K) be Scott open and let A e U. By (12.3) and (12.4) we
may find a В	e Conv(K) such	that A c B° and	such that	С	c	B° implies
C e U for all C e Conv(K).	The set S := {С	e C1(K) :	С	c	B°} is
open in C1(K)	and we have A	e S n Conv(K) c	U. Hence	U	is open in
the topology	induced by the	Lawson topology	on C1(K).
Finally, let A e Conv(K) and let U = {B e Conv(K) : В $ A} =
= {B e Conv(K) : A 4 B}. Then V = Conv(K) n {B e C1(K) : A 4 В} and
therefore V is open in the topology induced by C1(K), too. As these
two types of sets generate the Lawson topology on Conv(K), our proof
is complete. □
140
12.8	Let us return to continuous lattices in general. If L is a
continuous lattice and if a. is an ultrafilter on L, we know that
и has to converge in the Lawson topology on L. The limit of this
ultrafilter may by calculated as follows:
lim и = sup inf M
Men
Translated to converging nets (xpj in L, the formula reads as
lim x. = sup inf x.
iel 1 iel j >i 3
12.9	An element p e L of a lattice L is called prime, if а л b < p
implies a < p or b < p for all a,b e L.
Prime elements in a continuous lattice have a much stronger property,
as the following lemma shows:
1210 Lemma Let L be a continuous Lattice, Let V be a compLete
Lattice and Let f : L + V be a mapping such that f(sup D) = sup f(D)
for every directed set D <= L. If A c L is compact in the Lawson-topo-
Logy and if p e V is a prime eLement of V, then inf f(A) < p impLies
f(a) < p for some a e A. □
For a proof of (1210) we refer to [GK 77] or [Comp 80].
Let us now return to bundles. One connection between bundles and
continuous lattices comes out of the following considerations:
12.11	Suppose that p : E + X is a bundle with a compact base space.
Then 0(X) is a continuous lattice. Moreover, we have a canonical
mapping between 0(X) and the complete lattice С(Г(р)) of all closed
subspaces ,of Г(р) given by
141
i : 0(Х) С (Г(р) )
U * NX\U {° е Г(р) : °/Х и °} 
This mapping satisfies the hypothesis of (12.10) (see also [GK 77]):
12.12	Proposition. Let p : E + X be a bundle with a compact base
space. Then the mapping i : 0(X) + С(Г(р)) preserves directed supre-
me, i.e. if (U,), . is a directed family of open subsets of X, then
А A € A
i( и UJ = ( и i(U,))
ЛеЛ A ЛеЛ
Proof. The mapping i is monotone, whence we have the inclusion
i( и U.) = ( и i(Uj) .
ЛеЛ A ХсЛ A
Conversely, let a e i( и U.). We have to show: For every e > 0 and
ЛеЛ A
every seminorm v. belonging to the bundle there is an a' e и i(U )
3	ЛеЛ A
Oj(a - a') < e.
e > 0 and Vj be given. We define U := {x e X : Vj(a(x)) < e.
= 0, we obtain X \
compactness of X and the fact
such that
Thus, let
As al X\U
U U Л	П
ЛеЛ A	ЛеЛ
that (U,), . is directed yields a
A A eA
XQ e Л such that X \ c U. Choose any continuous function
о
f : X + [0,1] c JR such that f(X \ U ) = {0}and f(X \ U) = {1} and
Ao
set o' = f*a. Obviously, the section o' belongs to i(U ) c
Ao
с и i(U.). An easy calculation shows that O.(a - o') < e. □
ЛеЛ A	3
If we take (12.10) and (12.12) together
we have done most of the
proof of the following
12.13	Proposition. Let p : E + X be a bundle over a compact base
space and let V с С(Г(р)) be any complete lattice of closed sub-
spaces of Г(p) containing all subspaces of the form
142
= {a e Г(р) :	= 0} , A closed. If P / Г(р) is a prime element
of V, the there is a unique x e X such that Nx c p.
Proof. Suppose that we would have с P and с P, where x / y.
Let f : X +Ж be any continuous function which takes the value 0 at
x and the value 1 at y. If а e r(p), then we may write
а = f «а + (1 - f) •а
e N + N
x у
C P,
and hence P = Г(р), a contradiction. This shows that there is at
most one such x.
To ensure the existence of such x e X, we let
К := {X \ {x} : x e X} с О(X).
It is well known that К is compact in the Lawson-topology of 0(X)
(see [Comp 80]). Moreover, n i(K) = n N = {0}. Hence, if P is
xeX
prime in V, we can find an x e X such that с P by (1210) and
(12.12).	□
12.14	Let L be any complete lattice. By Spec(L) we abbreviate the
set of all prime elements of L which are different from the largest
element 1. The sets of the form
s(a) := {p e Spec(L) : a $ p} , a e L,
form a topology on Spec(L), the so called hull-kernel-topology. The
closed sets of this topology are exactly the sets of the form
h(a) := {p e Spec(L) : a < p} , a e L.
12.15	If p : E -> X is a bundle with a compact base space and if
143
V с С(Г(р)) is a complete lattice of closed subspaces containing
{Nft : A e C1(X)}, then (12.13) means that we have a mapping
fix : Spec (V) + X
which sends every prime element P e Spec(V) to the unique x e X
such that Nx с P. (In this case we say: P is fixed at x.)
12.16	Proposition. If p : E + X is our favorite bundle with a
compact base space and if V с С(Г(р)) is a complete lattioe of
olosed subspaces oontaining {N^ : A e C1(X)}, then the mapping
fix : Spec(V) -»• X
is continuous, where Spec(V) carries the hull-kernel-topology.
Proof. Let A с x be closed, with the notations of (12.14) we show
that fix 1(A) = h(N&).
Indeed, if fix(P) e A, then N, c N,. c p, i.e. P e h(N„).
A fix(P)	A
Conversely, assume that P e	but xQ := fix(P) | A. Then we
obtain N <= P and N c p. using Uryson's lemma, we show as in the
A	xo
proof of(12.13) that Г(р) = Nx + c p, contradicting P / Г(р). [
13.	M-structure and bundles
Let E be а С(X)-^-module. Again, we ask: Under which conditions is
it true that E is isomorphic to a space of sections in a bundle? We
saw in (7.21) and (7.23) that for the algebraic point of view the
Banach algebra С(X), viewed as a space of bounded operators on E,
should be contained in the "center" of the fi-space E. It is a bit
surprising that on the topological side there is also a notion of
center of E. We shall see that for Banach spaces E there is a
commutative, closed subalgebra Zt(E) c B(E) containing the identity
such that E is locally C(X)-convex if and only if С(X) c Z (E).
Hence, if we are interested in representations by sections in
a bundle, the intersection of the two "centers" Zt(E) and Z^(E) is
the right object to look at. We shall see furthermore, that the
intersection Zt(E) n Z^(E) is a commutative Banach algebra with unit
and thus of the form C(X) for a certain compact Hausdorff space X.
This space X is the maximal space such that the d'-space E may
be represented as the space of all sections in a bundle p : E + X
of fi-spaces. Unfortunately, it is difficult to get hold of the stalks of
this bundle. They are by no means "indecomposable" in the sense
that Z. (E ) n ZO(E ) is one-dimensional, or, in other words, that
every bundle representation of the stalk with a compact base space
leads to a one-point base space.
In the same sense as the topological center Zt(E) is the counter-
part of the algebraical center, we shall find a topological
analog of fi-ideals: The M-ideals. It is remarkable that, as in
the algebraical situation, the topological center Zt(E) may be
represented as the Banach algebra of all continuous IK-valued
145
functions on the space of all "primitive" M-ideals, equipped with
the hull-kernel topology.
In this section, we shall restrict ourselves to Banach spaces and
bundles of Banach spaces.
Let us start with a list of results from the theory of M-structure
in Banach spaces. The proofs may all be found in the lecture notes
of E.Behrends ( [Be 79]) or in the earlier paper of E.M.Alfsen and
E.G.Effros ([AE 72]).
13.1	Let E be a real or complex Banach space. A projection on E is
a continuous and linear map p : E + E such that pop = p. a pro-
jection p is called an L-projeetion, if we have in addition
l|m П = ||p(m) || + ||m - p(m) || for all m e E.
If p,q are two L-projections on the same Banach space E, then
poq = qop, i.e. L-projections commute. Moreover, the operators
рлд := poq
pvq := p + q - poq
p1 := Id - p
are L-projections, too. Hence the L-projections form a Boolean
algebra. In this Boolean algebra, we have p < q iff poq = p. More-
over, if (PjJi j is an increasing family of L-projections, then this
family is pointwise convergent to an L-projection. Thus, the
Boolean algebra of all L-projections is complete and we denote this
Boolean algebra by IPT (E).
13.2	The Banach algebra Cu(E) generated by IPT (E) in B(E) is called
L
the Cunning ham algebra of E. As L-projections commute, Cu(E) is a
146
commutative algebra with unit, which is of the form C(X), where X is a
compact space. Moreover, the space X may be identified with the
Stone dual IPT (E)* of the Boolean algebra IPT (E) and thus is an ex-
tremaly disconnected space. This implies that the Cunningham
algebra is an order complete Banach lattice, i.e. if M c Cu(E) is
order bounded, then sup M exists in Cu(E). If M is in addition
directed, then M converges to sup M in the strong operator topology,
i.e. for every a e E the net {T(a) : T e M} converges to (sup M) (a)
in the norm topology of E.
13.3	a subspace F с E is called an L-ideal, if it is the range of
an L-projection. This is equivalent	to the fact	that	F has	a	comple-
ment F1 in E such that ||m + n||	=	||m|| + ||n ||	for	all m	e	F	and
all n e F1. It follows that F1 is uniquely determined and that F1 is
also an L-ideal. Moreover, if F	and	G are two L-ideals, then	F	+ G	is
closed and F + G as well as F n	G are L-ideals.
13.4	The L-ideals in Banach spaces behave especially nice with
respect to the extreme points of the unit ball of E. Let B^(E) =
= {m e E :	||m||	< 1 } be the unit ball of E and let F be an L-ideal.
Then we have
extr (F n B1(E)) = F n extr B1 (E).
Moreover, if F and G are two L-ideals, then
(F + G) n В (E) = conv ((F n В (E)) и (G n В (E)).
13.5	Again, let F с E be a closed subspace. If its polar F° с E' is
an L-ideal in the topological dual E' of E equipped with its
canonical norm, then F is called an M-ideaZ. of E.
We record a few properties of M-ideals:
147
(i)	Finite intersections and arbitrary closed linear spans of
M-ideals are again M-ideals. Whence the M-ideals form a
sublattice of the lattice of all closed subspaces of E. This
lattice is complete, although arbitrary intersections in the
lattice of M-ideals and the lattice of all closed subspaces
do not agree.
By M(E) we denote the lattice of all M-ideals of E.
(ii)	The sum of two M-ideals is closed.
(iii)	A subspace F с E is an M-ideal if and only if it has the
following 3-ball property:
If В(пи,Г1) = {m e E : ||m - m || < r^} , i=1,2,3 are three
open balls such that B(m^,r^) n Btir^,^) n B(m3,r3) / 0 and
В(т±,г±) n F / 0, then	n B(m2,r2) n В (m ,r3) nF#
/ 0.
This 3-ball property may be used to show the following proposition
(see also [GK 77] and [Be 79] in the case of function modules):
13.6 Proposition. Let p : E + X be a bundle of Banaoh spaces with
a oompaot base spaoe X and let и X be an open subset. Then the
subspace i(U) := {a e Г(р) : alvir. = 0} is an м-ideal of Ftp).
I X\U
Proof. We let В : = {a e F(p) : Ila - p [[ < e } for к = 1,2,3
к	к к
Suppose that a e В. n Bo n В, and a e В n i(U) for к = 1,2,3.
1	к к
Choose e such that 0 < e < e - || pk - ak || and e < ek - ||pk - a||
for к = 1,2,3 and let
V := {x e X : ||ak(x) II < e/2 for к = 1,2,3}.
148
Then V is open. Choose a continuous function f : X + [0,1] such that
f(x) = 0 for x | U and f(x) = 1 for x e X \ V. Define a new section
o' e Ftp) by a' : = f-p. We show that a' e B^ n B2 n B3 n i(U) : Firstly
we have o' e i(U) as f(x) = 0 for all x e X \ U. In order to show
that o' e B^, consider ] | a' (x) - pk(x) ||.
If x | V, then a' (x) = a(x) and hence ||a' (x) - pk(x) ||	< ||a - pk||
< £ — £ .
" к
If x eV, then we compute
||a'(x) - p (x) || = ||f(x)-a(x) - f(x).p.(x) +
К	л
+ (f (X) - 1)’(pk(x) - ak(x) + ak(x) ||
<	| f (x) | • 11 a(x) - pk(x) || +
+ I 1 - f (x) I • ( II Pk(x) - ak(x) II + 11 a k(x) || )
<	f(x).(e - e) + (1 - f(x)).(e.- e + e/2)
К	л
-	e к “ e/2‘
Thus we have shown that II Pk” o' II £ e ~ e/2 < ek ,as desired. □
Hence, if p : E + X is a bundle of Banach spaces, we may
"rediscover" the open subsets of X in M(F(p)) via the mapping i.
The next result makes it even clearer what bundles should have to do
with M-structure:
13.7 Let F ,...,F be M-ideals of E such that F. + ... + F = E and
1	n	In
F. n ••• n F = {0}. Then E is isometrically isomorphic to the
n n
cartesian product П E/F., equipped with the supremum norm.
i=1	1
We may interprete (13.7.) as a representation of E by sections in a
bundle p : E + {1,...,n} , where the stalk over i is just the
quotient space E/F^.
149
13.8	- To generalize (13.7), we need again some notation . Let E be
again be a Banach space and let F be an M-ideal of E. If there is
an extreme point p e extr B^ (E') such that is F maximal among all
M-ideals contained in ker(p) (or equivalently, if F° is the smallest
a (E' ,E)-closed L-ideal containing p), then F is called primitive.
Every primitive M-ideal F is a prime element in the lattice of all
M-ideals, i.e. if G,H are two M-ideals, then G n H <= F implies G c F
or H c F. Moreover, every M-ideal G is the intersection of all
primitive M-ideals containing G.
Especially, the lattice of all M-ideals is distributive (every
lattice with the property that each element is a meet of prime ele-
ments is distributive.)
With Spec(E) we denote the set of all primitive M-ideals. If P is
a primitive M-ideal, then we let Ep := E/P and ap := a + P be
the equivalence class of a e E modulo P. Note that Ep is a Banach
space when equipped with the quotient norm and that we have a
linear map
* : E + iT {Ep : P e Spec(E)}
a + a where S(P) := ap.
13.9 Proposition.
The mapping * : E + H°° Ep is an isometry.
Proof. Obviously, * is a contraction. In order to show that
II a|| > II all , let a e E. Then we can find an extreme point
p e extr B^(E') such that ||a|| = p(a). Let P be the maximal M-ideal
contained in ker(p). By duality, we may identify (E/P)' with P°. Then
we obtain ||a|| = p(a) = p(ap) < ||p||-||ap|| = 11 ap 11 < ||a|| . □
13.10	We now return to Banach bundles. Let p : E + X be a bundle
of Banach spaces and let и с X be an open subset. By (13.6), the
150
set i(U) = N 0 = { a e Г(р) : a/X\u = °} is an M-ideal. As the
primitive M-ideals are prime, we may equip Spec Г(р) with the hull-
-kernel topology. Now (12.16) applies to the lattice of all M-ideals :
13.11	Proposition. Let p : E X be a bundle of Banaoh spates over
a oompaot base spaoe X. Then for every primitive Vi-ideal P there is
an unique element fix(P)
e X suoh that N,. . .
ГIX(r)
с P. Moreoverj the
mapping fix : Spec Г (p) -+ x is continuous, where Spec Г(p) carries the
hull-kernel topology. □
13.12	Let us consider again a locally C(X)-convex C(X)-module E,
where E is a Banach space and where x is compact. As in section 7, we
let Ix = {f e C(X) : f(x) = ol and Nx = Ix«E. From (7.6) we conclude
(*)	f«a - f(x) «a e Nx for all x e X, f e С(X), a e E.
Now we know from (7.19) that E is isometrically isomorphic to Г(р),
where p : E + X is a bundle of Banach spaces. Under this isomorphism,
Nx corresponds to {a e Г(р) : a(x) = 0}. Moreover, every continuous
function f e C(X) yields a continuous function <I>(f) e C^tSpec E) by
0(f)(P) := f(fix(P)). As fix(P) is the unique element of X with
Nx с P, the relation (*) implies
(**	) f*a - <I>(f) (P) «а e P for all P e Spec E
or
(	***) Ф (f) *a - 0(f) (P) *a e P for all P e Spec E.
In (***) of course, we defined 4>(f)«a := f>a.
The following important Dauns-Hofmann-Kaplansky multiplier theorem
shows that we may define f*a for every f e C^(Spec E) and every a e E
in such a manner that (***) remains valid. For a proof of this result
we refer to [AE 72], [EO 74] or to [Be 79]:
151
13.13	Theorem. Every Banach space E is a C^(Spec E)-moduZe such
that for every f e C^tSpec E) and every P e Spec E we have
f -a - f (P) -a e P. □
13.14	Corollary. Let E be a Banach space. Then
(i)	E is a reduced C^(Spec ^-locally convex	(Spec E.)-module.
(it) If X is any topological space, then E is a reduced C(X)-Zo-
cally convex C^CE.)-module if and only if there is an isometric
homomorphism of Banach algebras Ф : С^(Х) + C^(Spec E) such
that 4>(f) *a = f-a for all a e E.
Proof. Using (13.9) and the proof of (1.6.(x)) we see that E is
C^tSpec E)-locally convex. Moreover, the C^tSpec E)-module E is
reduced: Let 0 / f e. C^tSpec E) . Then f (P) / 0 for some P e Spec E.
Pick any a e E \ P. Then f(P) >a | P. As f«a - f(P)-a e P by (13.13),
we conclude that f«a / 0.
For a proof of (ii), assume that E is a reduced С(X)-locally convex
(X)-module. We may assume w.l.o.g. that X is compact. From
(7.19) and (7.22) we conclude that there is a reduced bundle
p : E + X such that E = Г(р) .
We claim that for reduced bundles the mapping fix : Spec Ftp) + X
has	dense image: Indeed,	let	x e X be	an element such	that E^	=
= p	1(x) / {0}. Then Nx	= {a	e F(p) :	a(x) = 0}	is an	M-ideal
which is different from F(p). By (13.8) we find a primitive M-ideal
P e	Spec F(p) such that	Nx с	P. Clearly, fix(P)	= x.
Now	the mapping Ф : f +	fofix : C^tX)	+ C^tSpec	F(p))	sends С^(Х)
isometrically onto a closed subalgebra of C^tSpec F(p)) = C^tSpec E).
From (**) and (13.13) we obtain
f*a - Ф (f) .a = f -a - Ф (f) (P) «а + Ф (f) (P) «а - Ф (f) «a
e P - P = P
152
for every P e Spec E. As the intersection of all primitive M-ideals
is 0, we obtain f«a = 0(f) «a.	- The converse follows from (i).	□
From (7.24) we know that we may identify C^tSpec E) with a closed
subalgebra of B(E) via the mapping f + T^, T^(a) = f«a. This
gives rise to the following definition:
13.1	5 Definition. Let E be a Banach space. The image of the mapping
f + Tf : C^tSpec E) + B(E) is called the (topological) center of E,
denoted by Zt(E). □
There are various other characterizations of Zt(E) which we shall
list. But firstly, we need a definition:
13.1	6 Definition. Let T : E + E be a bounded operator.
(i)	T is called M-bounded, if there is a real number r c IR such that
T(a) is contained in every ball of radius r which contains a.
(ii)	T is called a multiplier, if there is a mapping
aT : extr B1 (E' )	Ж
such that for every p e extr.B^(E') we have T'(p) = aT(p)«p, i.e.
every extreme point of the dual unit ball is an eigenvector for
the dual operator T' : E' + E' .
(iii)	If T and S are two multipliers on E, then S is called an
adjoint of T, if aT = ag
where denotes complex conjugation. □
From Eae 72] for the real case and [Be 79] for the arbitrary case we
draw the following conclusion:
13.1	7 Theorem. Let T : E + E be a bounded operator. Then
(i)	T is М-bounded if and only if T is a multiplier.
153
If the dual operator T' : E' + E' belongs to the Cunningham
algebra Cu(E') of E', then T is a multiplier. □
13.1	8 Theorem. Let T : E + E be a bounded operator. Then T belongs
to the oenter Zt(E) if and only if 1 is a multiplier which admits an
adjoint.
If E is a real Banaoh spaoe, then the following conditions are equi-
valent:
(i> T e Zt(E).
(ii)	T is M-bounded.
(iii)	T is a multiplier.
(iv)	T' e Cu(E').	□
13.19 Conclusion. Let E be a Banaoh spaoe, let X be a oompaot topolo-
gical spaoe and let us assume that E is а С(X)-module.
(i) E is locally C(X)-convex if and only if the operators
a -> f*a : E + E belong to г^(Е), f e С (X) .
(ii) E is a reduced locally C(X)-convex C(X)-module if and only if
X is a quotient of the Stone-Cech compactification В(Spec E)
of the spaoe of all primitive M-ideals, equipped with the
hull - kernel topology. □
As a consequence, the space В(Spec E) is the largest compact space
which can serve as a base space of a bundle in which we can have a
reduced representation of E by sections.
14.	An adequate M-theory for ^-spaces.
In this section we shall deal with a straightforward generalization
of the ideas in section 13.
14.1	Definition. Let E be a topological fi-Banach space. A closed
subspace M с E is called an M-fi-ideaZ, if it is an M-ideal and an
fi-ideal at the same time. By M^(E) we denote the set of all
M-Q-ideals. □
From (6.6) and (13.5.(i)) we obtain
14.2	Proposition. Finite intersections and arbitrary closed linear
spans of M-il-ideals leads again to M-Sl-ideals. Hence M^(E) is a
complete lattice, which is a sublattice of M(E) and therefore
distributive. □
14.3	Proposition. For every K-ideal ( 0,-ideal) F с E there is a
largest K-il-ideab kn(F) (k (F) ) contained in F. Whence we have two
kernel operators
kfi : M(E) - Mfi(E)
and
к : Idn(E) - Mn(E) □
Note that by (14.2) every closed subspace F с E contains a largest
M-Q-ideal. This leads to
14.4	Definition. An M-Q-ideal P с E is called primitive, if there
155
is an extreme point p e extr (E') of the dual unit ball such that
P is the largest M-fi-ideal of E contained in ker(p). By Spec^(E) we
denote the collection of all primitive M-Q-ideals. □
14.5	Proposition, The mapping maps Spec E onto Spec^ E.
(ii) Spec^ E consists of prime elements of M^(E) only.
(Hi) If we equip Spec E and Spec^ E
with their hull-kernel topolo-
gies,
then the restriction of k^ to Spec E
is continuous.
Proof. The first assertion is an obvious consequence of the defini-
tions .
(ii) Let P e Spec^ E and pick any Q e Spec E with P = k^(Q) c Q.
If M,N e M^(E) are M-fi-ideals with M n N cP, then we conclude
M n N c Q, hence w.l.o.g. M c Q as Q is prime. But this yields
kfi(M) = M c kfi(Q) = P.
(iii) Let A c Spec^ E be closed. Then there is an M-Q-ideal M of E
such that A = {P e Spec^ E : M cP}. An easy calculation shows that
k^ (A) = {P e Spec E : M с P }and therefore kJ(A) is closed in
Spec E. This means that k^ is continuous. □
The following result is a consequence of the Dauns-Hofmann-Kaplansky
multiplier theorem (13.13):
14 .6 Theorem. Let E be a topological (l-Banach space. Then E is
a C^tSpec^ E) -Cl-modu le
which is locally C^tSpec^ E)-convex and
reduced
Moreover,
for every P e Spec^ E,
every a e E and every
f e C^fSpec^ E) we have
f «a - f(P) «а e P
Proof. By (14 .5.(iii)), the Banach algebra C^tSpec^ E) may be
156
identified with a closed subalgebra of C^(Spec E) via the mapping
f + f°ko : C, (Speco E) + C, (Spec E) . Define an action of C,(Spec E)
о ь	XJ	о ь	XJ	LJ	о ь
on E by
f-a :=	*a.
By (13.14),the Banach space E becomes a reduced locally C^(Spec^ E)-
-convex C^(Spec^ E)-module in this way. Moreover, by (13.13) we have
f *a - f (kfi (Q) ) -a e Q
for every Q e Spec E.
Now let P e Spec^ E. From (13.8) we know that P = n {Q e Spec E :
P c Q}. If f e C^tSpec^ E) is given, then f is constant on the
closure of {P}. As the closure of {P} is the set {P' e Spec^ E :
P cP'}, we obtain f(P) = f(P') whenever P cP'. Thus, if
Q e Spec E is given and if P c Q, then P c k^(Q) and therefore
f(P) = f(k^(Q)). This yields
f -a - f(P) -a e Q
for all Q e Spec E with P c Q and thus
f-a - f(P)-a e P = n {Q e Spec E : P c Q}.
It remains to show that E is a C^tSpec^ E)-fi-module. So, let
f e C^tSpec^ E). Then for every a e E we have
a e f iff f «а = 0
iff f(P)-a e P for all P e Spec^ E
iff a e P for all P e f	\ {0})
iff a £ n f”1( Ж \ {0}),
i.e. f1 = n {P e Spec^ E : f(P) /0}. As an intersection of fi-ideals
yields again an ideal by (6.6), we conclude that f1 is an Q-ideal
for every f e C^tSpec^ E).	□
157
Substituting M(E)
by M^(E) and Spec E
by Spec^ E, the same proofs as
in (13.11) and (13.14) yield:
14.7 Proposition.
Let p ; E + X be a bundle of 0,-Banach spaces
over a compact base space X.
Then for every P e Spec^ Г(р) there is
an unique element fix(P)
mapping fix : Spec^ Г(р)
e X such that P c N,.	. .
f ix (p)
-> X is continuous.	□
Moreover, the
14.8	Proposition. If X is any compact topological space and if E
is an ti-Banach space, then E is a reduced and locally (X)-convex
(X) - il-space if and only if there is an isometric homomorphism
of Banach algebras Ф : C^(X) + C^tSpec^ E) such that for all f e C(X)
and all a e E we have Ф(f) «a = f-a. □
Again, (7.24) tells us that we may identify C^tSpec^ E) with a closed
subalgebra of B(E):
14.9	Definition. The image of the mapping f +	: C^tSpec^ E) ->-B(E)
is called the topological Q-center of E, denoted by ^(E). □
The following result is parallel to (13.19):
14.10	Theorem. Let E be a topological ii-Banach space, let x be a
compact space and let us assume that E is a C(X)-module.
(i)	E is a locally С (X) -convex С (X)-il-module if and only if the
operators a -+ f*a : E -> E belong to Z^ ^(E) for evei>y f e С (X) .
(ii) E is a reduced and locally C(X)-convex С (X)-il-module if and
only if X is a quotient of the Stone-Cech compactifioation
B(Spec^ E) via a mapping Ф : B(Spec^ E) -> X such that
f-a - f^(P))-a e P for all P e Spec^ E- □
158
Thus, as in (13.19), the space В(SpecE) is the largest compact base
space over which a representation of the fi-space E by all sections in
a bundle p : E + X is possible.
We conclude this sections with a few remarks concerning Banach
lattices, Banach algebras and C*-algebras. From (7.26) and (7.28)
we obtain:
14.11	Proposition. If E is a Banach algebra or a Banach lattice,
then Ztrfi(E) = Zt(E) n Zfi(E). □
If E is even a C*-algebra, then we conclude from [Be 79] and
[AE 72] that the M-ideals of E are exactly the closed two-sided
ideals of the algebra E. Moreover, an ideal is primitive in the
sense of M-ideals if and only if it is primitive as an ideal of
the C*-algebra E. Hence, in this case we have
Z(E) := Z.(E) = Z0(E) = Z. (E)
and
Spec E = Spec^ E
If we apply (14.10) to this situation, we obtain:
14.12	Corollary. (Dauns - Hofmann) Let E be a unital Z*-algebra
s/
and let X be the Stone-Cech compactification of Spec E, equipped with the
hull-kernel topology. Then there is a bundle p : E -> X of C*-algebras
such that E is isometrically isomorphic to the C*-algebra Г(p) of all
continuous sections of p. □

15. Duality
The material represented in the rest of this paper was developped in
order to give an useful representation of the dual space Г(р) ' of
the space of all sections in a bundle. Although I did not succeed to
my satisfaction, I believe thatmanyof the results discovered in this
untertaking are interesting in themselves.
An "optimal" representation of linear functionals on Г(р) would be
the following: Given a continuous linear form ф : Г(р) -»• Ж, where
p : E + X is a bundle with stalks (E )	, then find a family (ф ) v
of continuous linear functionals фх : Ex + Ж and a measure ц e M(X)
such that
Ф(о) = f ф (a(x)) dji(x)
x x
for all a e Г(p).
Of course, this requires that the mapping x + фх(а(х)) : X + Ж is
ц-integrable for every a e Г(р). As this is always the case if the map-
ping T(a) : X + Ж defined by T(a) (x) := фх(а(х)) is continuous and
as an easy calculation shows that T(f»a) = f>T(a) for all f e С(X)
and all аеГ(р), we are led to a study of the space of all (contin-
uous) C(X)-module homomorphisms T : Г(р) -»• C(X), denoted by
Mod(F(p)). It turns out that there is a close relation between the
"size" of Mod(r(p)) and the topology on E. We shall find out that
(with restrictions) the space Mod(r(p)) is "big" if and only if
E is Hausdorff and that Mod(r(p)) is "very big" if and only if the
mapping x + ||a(x) ||	: X + ]R is continuous for every a e Г(р) .
160
15 .1 We start with a list of notations which we shall use
frequently.
Let p : E + X be a full bundle. For every subset А с X we let
N = {a e Ftp) : а/л = 0}. Instead of N. ,we shall write N .
A	/ A	X j	X
Recall from (2.7) that for completely regular base spaces X the
quotient F(p)/Nx may be identified with the stalk Ex and whence the
dual space of Ex may be identified with the polar N° c r(p)' of
Nx- We shall always make this identification. Hence equations like
ф(а) = ф(а(х))
will make sense, provided that ф e Ex = N°.
Similarly, if X is normal and if А с X is compact, then Гд(р) may
be identified with Г(р)/Ыд, if p : E + X is a bundle of Banach
spaces. In this case we have N° = Гд(р)'.
Let p : E + X be a bundle of Banach spaces with a completely regular
base space X. For every x £ X let Bx с Г(р)' be the dual unit ball
of the stalk Ex- If A c x is a subset, we define
В = и В .
A , x
хеА
Note that Вд is not the dual unit ball of Гд(р) in general!
Finally, we let
Вд = BA \ {0}.
15 .2 Proposition. Let p : E + X be a full bundle with a com-
pletely regular base space.
(I)	If A <= x is closed and Е/ x e X \ A, then Nx + Ыд = Г (p) .
If p : E + X is a bundle of Banach spaces, then
161
(ii)	If x,y e X are distinct, then Bx n By = N° n N° = {0 }
(Hi) If к <= X is closed, then Вд = Bx n N°.
Proof, (i) Let f : X -> [0,1 ] be a continuous function such that
f(x) = 0 and f (A) = {1 }. If a e Г(р) is a section of the bundle p,
then f-a e Nx and (1 - f) *a e and thus a = f*a + (1 - f) «a belongs
to N + N_.
x A
(ii) follows immediatly from (i) by taking polars.
(iii)	If x is an element of А с X, then Nx contains Nft. Thus, using
the definitions, we obtain R c r n № с r n N°, i.e. в, <= Bv n N°.
Conversely, assume that 0 / ф e Bx n N°. Then we can find an x e X
such that ф e Bx n N°. We have to show that x belongs to A. Assume,
if possible, that x does not belong to A. Then (i) yields the contra-
diction фе Bv П N° C N° n N° = (N + N ) ° = Г (p) ° = {0 } .	□
X A X A X A
From now on
we shall always equip Bv with the weak-*-topology in-
X
duced by Г(p)'.
15.3 Proposition.
If X is completely regular and if A с x is com-
pact, then Вд is compact.
Proof. Let (Ф^)iel ^e a conver9ent net contained in Вд and let
Ф = lim Ф.. We have to show that ф e B,. Firstly, for every iel
iel 1	A
there is an x^ e A such that Ф^ e Bx . By the compactness of A
there is a convergent subnet (x.). T of (x.). T; let x := lim x..
j'jeJ	1'iel'	.	3
J e J
We show that ф e Bx-
Obviously, we have ||ф||	<1. Moreover, note that ф = lim ф..
jeJ 3
Now let a e Nx- We have to show that ф(а) = 0. Thus, let e > 0.
Then there is a neighborhood V of x such that || a(y) || < e for all
у e v. Whence we have eventually Цо(х^) [[ < e. This yields
162
| ф( a) ( = lim | ф. ( a) [
je J J
= lim | ф. ( a( x.) ) |
je J J J
<	liS ||ф. [[ • || o(x .) ||
je J J	J
<	Tim || a(x .) ||
je J -1
<	e
As e > 0 was arbitrary, we obtain ф( a) =0.	□
By (15.2) we have a mapping у :	+ X defined by у( ф) = x iff
ф e В , provided that X is completely regular. Since for every subset
-1	x
А с X we have у (A) = EL, (15.3) allows us to conclude:
15.4	Proposition. Let p : E + X be a bundle with a compact base
space X. Then B^. is compact. Moreover, the mapping у : gT + X is
continuous. □
Recall from section 12 that C1(X) denotes the complete lattice of
all closed subsets of X. If p : E + X is a bundle with a compact
base space, then we have a mapping
Y* : C1(X) Cl(By)
A
A + Вд
Note that y*(A) = Y 1(A) и {0}. This yields
15.5	Proposition. If p : E + X is a bundle
a compact base space x, then the mapping y* ;
ves arbitrary intersections and finite unions
of Banach spaces with
C1(X) C1(BV) preser-
Especially, y* is
continuous for the Seott-topologies on ci(X) and Cl (Вх)л resp. □
163
15.6	Proposition. Let p	:	f	+ X be a bundle of	Banaoh	spaces	with
a compact base space X. If	U	c	Bv Is open and if к c Bv is closed,
X	X
then the set {x e X : A n Bx	c	U} is open in X.
Proof. Let и = {C e C1(BX)	:	A	nC cU}. IfC e U	and if	С с C	is	a
closed subset of C, then C belongs also to U. Moreover, let
P c C1(BV) be down-directed (i.e. C. ,Cn e P implies the existence of
X	12
Cj e P such that c n C2> and assume that n P e U. By the
definition of U this means nP n A c U. Hence the compactness of
Bv allows us to find а С e P such that C n A c U, i.e. C e U- Thus,
X
U is open in the Scott-topology of C1(BV). As Y* is Scott-continuous,
(Y*) 1(U) = {C e C1(X) : A n Bc c U} is open in the Scott-topology of
C1(X) . Now recall from [Comp 80] that the mapping x + {x} : X + C1(X)
is even continuous for the Lawson-topology on C1(X). This implies
that the set {x e X : A n Bx c U} is open in X. □
In the following, let B^j be the unit ball of r(p) ' and let Conv
be the lattice of all closed convex subsets of B°. Recall that B° is
a continuous lattice when ordered by dual inclusion. In the next
proposition however, the lattice theoretical operations refer to the
normal set theoretical inclusion.
15.7	Proposition. Let p : E + X be a bundle of Banaoh spaces with
a compact base space X. Then
(i)	conv Вд = N° n B° for all closed subsets А с X.
(ii)	The mapping A + N° n B° : C1(X) + Conv B° preserves arbitrary
intersections and finite suprema. Especially, this mapping is
Scott-continuous.
Proof, (i) Let B1 be the unit ball of r(p). Recall that the mapping
164
ед : Г(Р) Гд(р)
°	* а/А
д
is a quotient map with kernel If is the unit ball of Гд(р),
then e (B^) =	+ B^ . Moreover, we have ||ед(а) II s 1 if and
only if ||o(x) || 4. 1 for all x e A. This implies that + B^ =
= {a e Г(р) :	|| a(x) || 4 1 for all x e a). As Вд contains 0, we
obtain
conv B_ =	B°°
A A
= ( и В )°°
xeX
= ( n B°)°
xeX
= {о e Г(р) :	|| a(x) || s 1 for all a e A}
= (Ыд + Bl)°
= (na + B1>°
= na n Bi>‘
(ii) If A,A' c X are closed, then
NAUA' " B1 =	BAuA'
= conv (В, и В, , )
A A
= conv (conv	BA U	conv Вд1)
= conv((N° n		и (N°, n B°))
= (N° П B°) V		n B°) ,
Hence finite suprema are preserved.
Moreover, N, is an M-ideal of Г(p) and therefore N? is an L-ideal of
T(p) ' . Thus, from (13.4) we conclude that
extr (N° n N ° n B°) = N° n N° n extr B°
/4/4	1	/4/4	I
= extr (N° n B°) n extr (N°, n B°) .
Using the Krein-Milman theorem
we obtain from (i) firstly the in-
clusion
165
extr.(N° n N°, n В?) с В, n B,,
А А	I	A A
BAnA'
and then
(as B. ,, is closed)
A nA
N° n N° n B° c conv(B, . ,)
A A 1	A nA
Conversely, A nA' c A,A' implies N ,N , c N , and therefore
А А	А ПА
N° ,, c n? n N?. ;• This shows that finite intersections are preserved.
A nA A A
Finally, let (A^)j be a down-directed family of closed subsets of
X. From (12.12) we conclude that N ,	= ( и N ) . Taking polars, we
n i	i
obtain n (N° n B°) = N° n B°.	□
Aj_	I	fiA^	|
15.8	Corollary. If p : E -+ X is a bundle with a compact base space
X and if A and В are closed subsets of X, then N^ + Ng = N^^g.
Proof. Obviously, we have N + N„ c N „.
AB AnB
Conversely, note that Nft and Ng are M-ideals. Therefore the sum
Nft + Ng is a closed subspace of Г(р) by (13.5 (ii)). it remains to
show that N, + N„ is dense in N, „. But this follows immediatly
А В	AnB
from (15.7(ii)).	□
These results may be interpreted that the semicontinuity of the
norm in a bundle p : E + X is somehow reflected in the semicontinuity
of the mapping A + N° n B° : C1(X) + Conv В ° or, if we wish, in the
semicontinuity of the mapping x -> В : X -> C1(BV) . Therefore, we
might expect that the points of continuity of the mapping norm : E IR
are "rediscovered" in points of continuity of these maps:
166
15.9	Proposition. Опое again, let p : E + X be a bundle of Banaoh
spaces with a oompaot base space X and let xq e X be a point. The
following oonditions are equivalent:
(i)	The mapping x + В : X + C1(BV) is continuous at x for the
X	X	о
Lawson-topology on C1(BV)-
A
(ii)	If W с в is open and if В n W / 0, then the set
X	(ii) * * * * * * * xo
{x e X : Bx n W / 0} is a neighborhood of xq.
(iii)	The mapping x + || a(x) || j X + JR is continuous at xq for
every a e Г(р).
(iii') The mapping norm : E + Ж : a -> j | a 11 is continuous at every
a with p(a) = x .
(iv)	If M с X is a subset of X and if xQ belongs to the closure
of M, then В <= B...
x M
о
Proof, (i) + (ii) : If W c Bv is open, then {A e C1(BV) : A n W / 0}
X	X
= {A e C1(BV) : A 4 B„ \ W} is open in the Lawson-topology. Thus (i)
X	X
implies (ii).
(ii) + (iii) : Let a e Г(р) , let e > 0 and let U = {x e X :
||a(x) || > || a(xo)||- e}. We have to show that U is a neighborhood of
x .
о
Let W = Bx n {ф e Г (p) ' : | ф (a) | > || a(xQ) II ” e}- Then W is open
in Bx- Moreover, we may find an ф in the dual unit ball Bx of Ex
о о
such that ф(а(х )) = ф(а) >	||a(x ) || - e. Hence the set W n В
о
is not empty and by (ii) the set V = {x e X : Bx n W / 0} is a neigh-
borhood of xq. We complete the proof of (iii) by showing that
V c U:
Indeed, if x e V, then Bx n W / 0. Hence there is an ф e Bx such
that | ф (a) [ > ||a(x )|| - e. As ||ф|| < 1 , we may conclude that
|| o(xQ) || - e < | ф(а) [ = | ф (o(x) ) |	< ||ф|| • ||a(x) ||	<	||a(x) [[ , i.e.
167
x e U.
(in) -+ (iv) : Assume that x e M but В <t B„. Let v = R, \ B„. Then
о	x т M	л M
о
V is open in В and v n В / 0. Moreover, we have
X	X
о
(1)	г«ф e V for all ф e V and all 1 < |r | < -|^ ц .
Indeed, as ||г«ф|| < 1 , we have г«ф e Bv. Now assume that г»ф e В .
X	M
Then |1| < 1 implies ф e r“1= (r“1«BM)- c B^.
Now pick any ф e V n Bx and let A = ф (1) . Then A is a closed
о
hyperplane of Г(р). Further, we have
(2)	n {ф e Bv :	|ф(а)|>1-е}сУ
UeA	X
e?0
To show this inclusion, let ф belong the the left hand side. Then
|ф(а)[ > 1 for all a e A and whence ker ф n A = 0. This means that
ker ф and A = ф \l) are parallel hyperplanes and thus ker ф = ker ф.
Therefore, we can find an element r eJK such that ф = г«ф. Pick
any a e A. Then ф(a) = г*ф(а) - r, whence |r| >1. Moreover,we have
|[ф[[ = |r [ • [|ф[[ < 1, i.e. |r[ <	11 ф । p . This implies г*ф = ф e V,
as we started with an ф eV.
Note that all the sets {ф e Bv : |ф(a) | > 1 - e} are closed. An easy
compactness argument shows:
(3)	There are sections а^,...,а e Г(р) and e > 0 with
n
ф(а.) = 1 and n {феВ„:[ф(а.)[>1-е}<=У.
1	i=1	X	1
Let C be the convex hull of а.,...,а . Then
1 n
ф(а) = 1 for all а e C.
Moreover, as C is compact, we can find elements p^,...,p e C such
that for every а e C there is an j £ {1,...,m} with
Ha - PjH < e/3.
168
Now
1 - e/3	<	1
=	*(pj)
=	*(Pj(xo))
* И Ф Ii * ti P-j (xo) Ii
s i|pj(xo)i!
and the assumption (iii) imply that there is an open neighborhood U
of xQ such that 1 - e/3 < ||	(x) || for all x e U. If a e C, then
||a - pj || < e/3 for a certain j. Therefore for all x e U we have
fl Pj (X) ||	<	11 О(X) - Pj (X) ||	+ II a(x) ||
<	e/3 + 11 о(x) ||
i. e.
1 - e/3	<	e/3 + || o(x) ||
As xQ belongs to the closure of M, we can pick an element x^ e M n U.
Thus we have shown:
2
(4) There is an x^ e M such that 1 - ye < ||a(x^) || for all aeC.
From now on, we work entirely in the stalk E . Let В с E be the
2	Xl	Xl
ball of radius 1 - ye and with center 0 and let Cx := {a(xp :
a e C}Then it is clear that В n C = 0 and hence О | С + B.
X1	X1
By the Hahn - Banach theorem we can find a continuous linear
functional ф : E + IK such that ||ф|| =1 and ker ф n (С + B) =
X1	X1
= 0.
Suppose that ||ф(а) || < 1 - e for a certain a e C. As ||ф|| = 1 , we
can find an element a e В such that |ф(a)| > 1 - e. If we multiply
a with an appropriate г e Ж with |r| < 1 we obtain the existence of
a' e В such that ф(а') = ф(а). Hence ф maps а(х ) - а' e С + В
1	Х1
onto О, a contradiction. Thus, we may conclude:
169
(5) There is an element ф e Bx such that |ф(а) | > 1 - e for
all a e C.
As we have | ф(a ) | > 1 - e for all i e {1 ,.. . ,n}, this ф belongs to
V. On the other hand, we have ф e Bx <= BM (Contradicting the fact
that V n BM = (Bx x B^) n BM = 0.
(iv) + (i): Let a be an ultrafilter on X converging to xq. We
have to show that lim В = В
и. x x
о
Firstly, note that
{x }	= n M
° Men
and
lim Bv = n ( и В )—	(see (8.8) )
Meu. xeM
i.e.
lim В = n B~ .
U x	M
Meu.
As (iv) implies В c lim В , it remains to check the other in-
xo u x
elusion. Let A be any closed neighborhood of xQ. Then A e и and
therefore
n BM <= n {вд : A is a closed neighborhood of xQ }
Meu.
- n {Вд : A is a closed neighborhood of xq }
= Bx	by (11.5) .
о
(iii') + (iii) is trivial.
(iii) + (iii1): Let a e E belong to the stalk E over xq and
о
choose any section a e Ftp) with a(xq) = a. Further, let e > 0. We
have to find an open neighborhood U of a such that
| I! - H«И I < e for а11 В e U.
170
An easy application of the triangle inequality shows that
U := {B :	В _ a(p(B)) || < e/2 and p ( B) eV} has the required
property, where V is any open set around xQ such that
| Ha(x) H " HaH i < £/2 for а11 X e V. □
1'5.10 Definition. We say that a bundle of Banach spaces p : E + X
has eontinuous norm, if the mapping norm : a -> 11 a 11 : E +1R is
continuous. □
In the following proposition we show that the continuity of the norm
may be expressed by the continuity of various other maps:
15.11 Theorem.	Let p : E -+ X be a bundle of Banach spaces
over a compact base space X. Then the following statements are
equivalent;
(i)	E has continuous norm.
(ii)	The mapping x + ||a(x) || : X +1R is continuous for every
a e Г(p) .
(iii)	If W с В is open, then the set {x e X : W n E' / 0} is
open in X .
(iv)	For every subset M c x we have B„ = B—.
MM
(v)	The mapping x + В : X + C1(BV) is continuous for the Lawson
X	X
topology.
(vi)	The mapping x +	: X -> C1(B°) is continuous for the Lawson
topology.
(vii)	The mapping x -> B^ : X + Conv B° is continuous for the Lawson
topology.
(viii)	The mapping A -> N° n B° : C1(X) -+ Conv B° is continuous for
the Lawson topology.
(ix)	The mapping A + Вд : C1(X) + C1(BX) is continuous for the
171
Lawson topology.
Proof. The equivalences of (i) , (ii) , (iii) ,(iv) and (v) follow
immediately from (15.9)
(v)	+ (vi) : As the embedding A + A : C1(BV) + C1(B°) preserves
X	I
arbitrary infima and suprema, it is Lawson continuous. Hence (v)
implies (vi).
(vi)	+ (vii) follows from (12,7).
(vii)	+ (vi) : The embeddings x + {x} : X + C1(X) and A + A
Conv В °
+ Cl(B^) are continuous, hence (vii) implies (vi) .
(vi) + (v) : The image of the mapping x + В is contained in C1(BV).
X	X
As the embedding A -> A : C1(BV) + C1(B°) is also a topological em-
X	1
bedding, (v) follows.
(iv) -» (viii) : We show that the mapping A + n B^ preserves
arbitrary suprema: Indeed, (15.7(i)) yields n B° = conv(B— ) =
ljAj_	I	UAj-
conv(BuA )	(by (iv)) = conv( и Вд ) = conv( и conv Вд ) =
= conv( и (N° n B°)), i.e. N° д n B° = sup(N° n B°).
I	|	rlj.	j
As this mapping always preserves arbitrary infima, it is Lawson-
-continuous by(12.4(ii) ) .
For (viii) -» (ix) and (ix) -» (v) use the arguments given in (vii) +
(vi) (v) .	□
We now develop a duality between "stalkwise convex" subsets of E and
"stalkwise convex" subsets of В . In the remainder of this section,
X — --------------------------------------
172
Е
->• X will always be a bundle of
Banach spaces with a compact
base space X.
15.12 Lemma. Let Kc В be closed and let e > 0. Define
X
Ke := {a e E : иеф(а) i e for all ф e Kn В , . }.
р(а)
Then the restriction p ; K£ X of the projection p : E -> x is
still open.
£
Proof. Let а e К and let U be an open neighborhood of a. We have
£
to show that p(U n К ) is a neighborhood of p(a).
First of all, we may assume that U has the form
U = {g e E : II т (p(B) ) - B|| < r and p(B) e W} ,
where т e Г(p) is a section with т(р(а)) = a and where W is an open
neighborhood of p(a).
£
If т =0, then the 0 of Ex is contained in U n К for every x e W
£
and whence p(U n К ) = W
From now on, we assume that т / 0. By passing to a smaller r if
necessary, we obtain ||t|| - r > 0. Choose a real number 5 such
that e < 6 < e • ц T ||'— and let 0 = {ф e Bx : йеф(а) < 6}. Then
0 is open and so is the set {x e X : KnBxc(?J (see (15.6) ).
Since for all ф e ®р(а) n K the lnecIuality Re <На) s e < 6 holds,
p(a) belongs to the open set Wn {x e X : KnBxc(?J-We claim that
Wn{xeX:KnBxc()}c p(u n K£) :
Let x eWn{xeX:KnB c (} }. Then the element 4*T(x ) belongs
о	x	о о
in fact to K£: Indeed, let ф be any element of К n Bx • Then ф be-
o e
longs to 0 , whence Re ф(а) = Re ф(т (xQ) ) < 6 , i.e. Re ф (-^• т (xQ) ) < e.
Moreover, we have ||-§-t(xo) - t(xq) || = (1 - |) • ||t(xq) || <
< (1 -	T|| т [|—‘ llTll = r' i-e. |-т(хо) e U. We finally conclude
173
j-T(xo) e U n Ke and therefore xQ e p(U n Ke) •	□
15.13 Proposition. Let К c E be a subset such that К n Ex is
closed, convex and non-empty for every x e X. If the restriction
p : К ->• X of the projection p : £ + X is still open, then for every
a e К there is a section a e Г(р) with a(p(a)) = a and a(x) e К for
every x e X.
Proof. Step 1 If e > 0 and if aQ e К are given, then there is a
section a e Г(р) such that a(p(aQ)) = aQ and such that for every
x e X there is an a с К n satisfying ||a(x) - a 11 < e.
(Proof of step 1: Let aQ e К and let x £ X be arbitrary. Then we
may find a section ax e r(p) such that ax(p(aQ)) = aQ and
ax(x) e К n Ex. Let
Ux := {В e E: ||ax(p(B) ) - B|| < eb
Then, by assumption, Vx := p(Ux n K) is an open neighborhood of x.
As X is compact and as the V , x e X, cover X, we can find finitely
many elements x.,...,x e X such that X = V и ... и V . Let
in	X1	xn
(fpi=1 be a partition of unity subordinate to the open cover
V ,...,V .We define a section a e Г(р) by
X1	n
Then
a(p(a ))	= У f.(p(a ))-a (p(a ))
<4	X
1 = 1	1
= у
1=1
174
Moreover, if x e X is given, let M := {i : f (x) / 0}. Then we have
x e V for every i e M, as f. vanishes outside V . Hence for
Xi	1	xi
every i e M we can find a e К n such that 11	(x) - f? || < e
Define
В := I f±(x)-Bi
ieM
As К n E is convex and as V f.(x) = 1, the element в belongs to
X	. Zmr 1
К n Ex- Finally, we have
||a(x) - Bii = 1П f±(x)	(x) - g||
ieM	i
S I fi(x) • Ila (X) - В, II
ieM 1	xi	1
< I f . (x) • e
ieM
= e
Step 2 Let e > 0 and let aQ e K. Assume that there is a section
т e Г(p) with t(p(ao)) = and assume that for every x e X there is
an a e К n E^ such that || т (x) - a|| < e. Then we can find a section
t' e Ftp) such that
(i)	|| т' - т || < e.
(	ii) т' (p (aQ) ) = a .
(iii)	For every x e X there is an a e К n Ex satisfying
|| a - т ' (p (a) ) ||	< e/2 .
(Proof of step 2:	Let xq = p(aQ) and let x e X be arbitrary. Then
there is an ax e Ex n К with ||t(x) - a || < e. We may assume that
a„ = a if x = x . Let т be any continuous section such that
x о	о	x J
Tx(x) = ax and such that for every у e X there is an a' e E^ n К
satisfying ||тх(у) - a'|| < e/2. As ||тх(х) - t(x)|| = ||ax - r(x) ||
< e, we can find an open neighborhood Ux of x such that
||тх(У) - т(у) || < e for all у e Ux. We may assume that xQ | Ux if
175
х / х . Let (f ) be a partition of unity subordinate to the open
О	XX
cover (U )	,, of x. Then f (x ) = 1 and f (x ) =0 for x / x .
x xeX	x о	x о	о
о
Define
Then т' (xq) = aQ and || т' - т|| s e by some standard arguments we
already used in the proof of the Stone-WeierstraB theorem (4.2).
Moreover, as in the proof of step 1, we see that for every x e X
there is an a e К n Ex such that ||t’ (x) - a|| < e/2.)
Step 3 For every aQ e К there is a a e Г(р) with a(p(aQ)) = aQ and
a(x) e К for all x e X.
(By induction, using step 1 and step 2, we can find a sequence
Tn e Г(р) such that
(i)	Tn(p(aQ)) = aQ for all n e]N.
(ii)	H тп - Tn+1 || s фП for all n e ]N.
(iii)	For every n e IN and every x e X there is an a e E n К such
that ||тп(х) - an,x|| < (|)n.
We compute that
Han,x-“n+1,xH ~ Han>x - Tn(x)|| + ||Tn(x) - Tn+1(x) ||
+ iiTn+1(x) - “n+1,xH
< з-фп.
Hence (Tn)n£]N and (“n x’neiN are CauchV sequences. Let a := lim tr.
'	П-*-"
Then for each n e ]N we have Tn(p(ao)) = ao and therefore o(p(aQ)) =
= a . Finally, if x e X, then a(x) = lim т (x) = lim a e К n E ,
П-*"00 n	n->.oo '
as К n E is closed.) □
x
15.14	Proposition. Let К c В be closed. Then for every x e X we
X
have
176
(К n Вх)° = {а е Г(р) : а(х) = т(х) for some т е К°}
= К° + N
х
= conv (К° и N )
Especially, К° + Nx is closed for every x e X
Proof. Firstly, we have
{а e Г(p) ; a(x)
т(x) for some т e K°}
= K° + N
x
c conv (K° U Nx)
С (K° и Nx)°°
=	(K°° n N°)°
=	(K°° n Ep °
=	(K°° n Bx)°
с (К П Bx)°.
Thus, it remains to show that (К n Bx)° с {ас Г(р) : а(x) = т(x) for
some т e K°}.
Let о e (K n В )°. Then
а(х) e {а e E : Re ф(а) < 1 for all ф e К n В ^1.
Hence, using (15.12) and (15.13), we can find a section т e Г(р)
such that
а(х) = t(x)
and
т(у) e {а e E : Re Ф(а) < 1 for all ф e К n В ,	}
P k uj
for all у e X.
Now let ф e K. Then ф belongs to К n B^ for a certain у e X and
whence Re ф(т) = Re ф(т(у)) < 1. But this implies т e K°. □
177
15.15	Corollary. If К с В is closed and if 0 e K, then
X
(i)	(conv K) n Bx = conv (K n Bx) for every x e X
(ii)	и conv (К n В ) is closed.
xeX	X
Proof, (i) From (15.14) we conclude that
conv (K n Bx) = (K n Bx)°°
= (conv (K° и N ) ) °
= (K° и Nx)°°°
= (K° и Nx)°
= K°° n
= conv К n Bx.
(ii) follows from (i), as
и conv (К n В ) = и (conv К) n В
хеХ	X хеХ	X
- В n conv К. □
X
We now go back to the discussion of subbundle as it was begun in
section 8. We shall apply the results obtained in the present
section in order to give a description of subbundles which uses
duality.
Firstly, recall from (8.8) that a subbundle F £ E is completely
determined by a "distribution" of closed subsoaces (F ) of the
X X€X
stalks such that the restriction of the projection p:E->-X to
и F is still open.
xeX X
178
The next lemma is certainly well-known to everyone working in
functional analysis:
15.16	Lemma. Let E be a Banach space and bet К с E' be a subset
such that
(a)	К is a(E',E)-compact, convex and circled.
(b)	||ф|| s 1 for all ф e K.
(c)	If 0 / ф e K, then ф/ ||ф|| e К.
If a e E and e > 0 are given such that |ф(а)| < e for abb ф e K, then
there is an ebement b e E such that ||a - b|| < e and ф (b) = О for
abb ф e К.
Moreover, 3K*K is the a(E',E)-cbosed subspace generated by K.
Proof. Let F с E' be the subspace generated by K. Then F = IK-K, as
К is convex and circled. Moreover, F n {ф e E' : ||ф|| < 1} = К by
the assumptions (b) and (c). From the Krein-Smulian theorem and (a)
we conclude that F is a(E',E)-closed.
Thus, E/F° is a Banach space and (E/F°)' is isometrically isomorphic
with F°° = F. The dual unit ball of E/F° may be identified with K.
Now let л : E + E/F° be the canonical projection. If |ф(а)| < e for
all ф e K, then || к (a) || < e. As || к (a) || = inf {||a - b|| : b e F°},
179
there is a b e F° = {u e E : ф(и) = 0 for all ф e K} such that
|[a - b ||	< e. □
We now return to our bundle p : E + X of Banach spaces with a compact
base space X.
15.17	Proposition. Let К c Bv be a closed set such that
(a)	Kx := К n Bx is convex and circled for every x e X
(b)	If 0 / ф e K, then ф/ ||ф[| e К.
Then EIZ = {a e E : ф(а) = 0 for all ф e К n В , . } is a subbundle of
К	T	r	p (a)
E.
Proof. Obviously, EK n Ex = {a e Ex : ф(а) = 0 for allф eK n Bx} is
a closed linear subspace of Ex- It remains to show that the
restriction p/r :	» X is open.
/EK K
Thus, let a e E„ and let U с E be an open set around a. We have to
К
show that p(U n E„) is a neighborhood of p(a). Firstly, we may assume
x\
without loss of generality that U has the form
U = {В e E : ||a(p(B)) “ B|| < e and p(g) e W},
where a e Ftp) is a section passing through a, where e > 0 and where
W is an open neighborhood of p(a). As in ( 15.12) we define
p / 3
Kb/ = {В e E : Re ф(В) * e/3 for all ф < К n B ( }}.
As К is circled, we may write
г /з	,	,
К '	= {В e E : ф(В) < e/3 for all ф e К n B^,.,}.
p p;
e/3
As а e К t using (15.12) and (15.13), we can find a section
£ /3
p € Г(р) such that p(p(a)) = a and p(x) e К ' for all x e X. As
p(p(cc) ) - о(р(а)) = а - а = 0 and as the mapping norm : E + JR is
180
upper semicontinuous, the set
W' := {x e W :	|| p(x) - p(x) ||	< e/3}
is an open neighborhood of p(a)• We claim that W' c p(U n EK):
Indeed, if x e W', then
|ф(p(x)) | < e/3 < e/2 for all ф e К n Bx
and
||a(x) - p(x) ||	< e/3
whence for ф e К p Bx we have
|ф(а(х))| < |ф(а(х) - p(x)) | + |ф(р(х))|
< || ф || • || a(x) - p(x) [| + e/2 < e
Applying (15.16) to the Banach space Ex and the compact set К n Bx,
we obtain an element a e Ex such that [| a - p(x) || < e and ф(а) = 0
for all ф e К n Bx- As p(a) = x and as a e U n EK» the proof is
complete. □
15.18 Theorem.
Let p : E + X be a bundle of Banaoh spaces with a
oompaot base spaoe X. Then the mapping
F + F° n В =: K„
X r
(resp. F - и (F n E )° n Bv =: Kc)
xeX	XXF
is a bisection between the set of all closed submodules of Г(p)
(resp. subbundles of E) onto the set of all closed subsets К с В
X
such that
(a)	К n Bx is convex and circled for every x e X.
(b)	If О / ф e K, then ф/ ||ф[| e К.
The inverse of this mapping is given by
К F := {p £ Г(р)
Jt\
ф(а(х)) = 0 for all x e X, ф e К n Bx
(resp. К + EK := {а e E : ф (а) = 0 for all ф e К n Bp } ) •
181
Moreover, the bijections given here and the bijections given in
( 8.6) commute when composed in the right order.
Proof. First of all, note that we may (and do) identify (F n Ex)°
with (Г(р/С) + N )° = Г(р/С)° n №• Hence we obtain
/г	X	/г	X
xeX
(F n Ex)
xeX
r(P/F)°
Kr(p/F).
xuuo xу f we 11a vc г	J / r ) • nex ex vjx сз ? x t x о	n
K /ek
the mappings F + Kr and К -> E are inverse to each other:
г	К
Let F с E	be	a subbundle. Then
EK	=	{a	e	E	:	ф(а) =	0	for	all	ф	e	Kp	n	®p(a)J
=	{a	e	E	:	ф(а) =	0	for	all	ф	e	(F	n	Ep(a))°J
=	{a	e	E	:	a e (F	n	Ep(a))	}
= F
Conversely, if К c Bv satisfies (a) and (b), then
X
Kr = и (E n E )° П В
Jt\	X	A
x£X
= и {а г E : ф(а) = 0 for all ф e К n В }° n B„
xeX	x	x л
= B n и Ж-(К n Bv)
A	v	X
x£X
where we used (15.16) to establish the equality Ж.(К n в ) =
= {а eEx : ф(а) =0 for all ф e К n Bx}°.	□
182
there
Г(Р) •
that
to
15.19 Corollary. Let p : E + X be a bundle of Banaoh spaces with
a compact base space X. If (F ) is a family of closed linear
X x ^x
subspaces of (E ) then for every x e X and every a £ F
X X ^X	О	Xq
is a section a e Г(р) such that
a(x ) = a
and	a(x) e Fx for all x e X
if and only if и F° n B„ is closed. □
X X
XcX
We conclude this section with a remark concerning M-ideals of
Firstly, let M be a primitive M-ideal. Then we recall from
that there is an element x e X such that Nx с M. This implies
M is a submodule of Г(р): Indeed, let f e C(X) and let a e M.
Then f-a - f(x)-a e Nx с M and f(x)-a e. M, hence f-a belongs
M, too.
As every M-ideal is the intersection of primitive M-ideals, we
obtain:
15.20 Proposition. Every Vi-ideal M с Г(р) is а С(X)-submodule of
Г(р).	□
Using the 3-ball property of M-ideals, we can show the following
(see [Be 79, p.86 ]) :
15.21 Theorem. Let p : E + X be a bundle of Banach spaces over a
compact base space X. A closed linear subspace M c Ftp) is an
Vi-ideal of Ftp) if and only if there is a subbundle E^ с E such that
(i) M = F(p, )
' M
(ii) E., is "stalkwise" an Vi-ideal of E. i.e. the linear
M	4	>
sub space E., n E is an Vi-ideal of E .	□
r M x	J x
.16. The closure of the "unit ball" of a bundle and separation axioms
As we already noticed in example (5.16), not all bundle spaces have
to be Hausdorff. The same example shows that the closure of the
О-section may contain a whole line in some stalk and the same is
true for the closure of the "unit ball" {a e E : ||a!i s 1 }. We shall
see in this section that example (5.16) is no exception and that
the structure of the closure of the "unit ball" of a bundle deter-
mines the strength of topological separation in the bundle space.
This section will contain a lot of rather technical results and I
can only hope that the readers will not loose their patience before
they reach the applications of the material presented here in
sections 17 and 19.
16.1 Lemma. Let p
E + X be a bundle with seminorms
Define
(1)
vj (a) := sup { r e 1R
Vj 1 ( ]r,°°[)
is a neighborhood of a}.
Then vj satisfies \ij(X-a) = |l|«Vj(a) for all \ e Ж.
Moreover, if a : и + E is a section where и с X is open, then for
every x e U we have
(2)	v.(a(x))
sup {r e 1R : {y : r < v-(a(y))} is a neighborhood of x
Furthermore
(3)
v4(a) = sup {inf v . (U )
J	J
: U is a neighborhood of a}
v . : E * TR by
j j£J
184
Proof. Firstly, let us check the last equality: As
inf Vj (( Jr,«>[) ) > r, the right hand side of this equation dominates
Vj(a). Conversely, let a e E and let U be a neighborhood of a. Then
for every r < inf Vj(U) the set
a neighborhood of a- This shows
is
Vj (Jr,oo[) contains U
equality.
Now suppose that \ij(a)
e >0. By the last equation there is a
neighborhood U of a such that inf v•(U)
> e/2
Thus, U and
{В e E •• Vj (B)
respectively.
e/2} are disjoint neighborhoods of
a and 0 e E ,
P1 a
Conversely, suppose that U and I/ are disjoint neighborhoods of a and
0 e E
p(a)
We may assume w.o.l.g. that (/={BeE:p(B) eW an^
16 .2 Lemma. Let p : E + X be a bundle with seminorms (v.)  T •
] ]eJ
(i) The mappings (v-) . , are lower semicontinuous.
J J
(ii) Let x e X. Then the restriction of w . to E is uniformly
3 xo
continuous for every j e J.
Proof. It remains to check (ii). Let e > 0 and choose 6 = e/3. Now
assume that Vj(a - B) <5- We show that |vj(a) - Vj(B)| < e;
Let U = {у e E : Vj(y) < 6} and let V
a contained in {у e E : Vj(y) > Vj (a)
be any open neighborhood of
 6}. As в ” a e U and as
the mapping add : EvE + E is open by (1.6.(xi)), the set W = add(U,U)
= {Y1 + У2 : P(y-j) = р(у2> i y-j e U, У2 e U} is an open neighborhood
of g. Moreover, if у e W, then у = у^ + У2 f°r certain y^ e U,
y9 e (/. Thus the triangle inequality yields v-(y) г v-(y9) - vJy.)
> ^j(a) - 6 - 6 - Vj(a) - 2-6. Thus, the set
{у e E : Vj(y) > Vj(a) - 2*6}
is an open neighborhood of B- This yields the inequality
185
v . ( В) 2 V. (а) - 2 *6, i.e. v. (а) - v. ( В)
J	J	3	J
Vj ( B) - vj ( a) < e and therefore | vj (a)
< e. By symmetry we get
v( В) I < e- □
The following example shows that the mappings (v•). , are in general
1 J
not continuous:
16.3 Example. Let X = [-1,1] c ]R, equipped with its usual topology.
We consider the following weight function w : X -+1R:
y 1 , r 0
W(r) = 4
’ r, r > 0
and equip C([-1,1]) with the weighted norm ||| • ||| given by
v(f) = III f III = sup {w(r) • |f (r) I : -1 £ r < 1 }
Let E be the completion of C([-1,1]) in the norm Ill-Ill . As in (5.16)
we see that there is a bundle p : E -> [-1,1 ] such that E is isometri-
cally isomorphic to Г(p). It turns out that
r III =i|li ' Р(а) / 0
v(a) = 4
I 0	, p(a) = 0
Now assume that v is continuous. Then for all а e Г(p) the composition
v°o is continuous. But this is impossible, as by construction the
constant mapping with value 1 belongs to r(p).
A later example will show that; the	do not have to be seminorms,
which in our situation means that they need not be sublinear.
16.4 Proposition. Let p : E -+ X be a bundle and assume that X is
Hausdorff. Then the following conditions are equivalent:
(i)	E is Hausdorff
186
(ii)	{ox e Ex : x e X} c t is closed.
(iii)	For every 0 / a e E there is a seminorm Vj : E -»• JR
of the bundle p : E -> X such that Vj (a) >0.
If X is in addition normal and second countable, if all stalks are
complete and if the bundle has a. countable family of seminorms (espe-
cially if p : E + X is a bundle of Banaoh spaces with a normal and
second countable base space), then conditions (i) - (iii) are also
equivalent to
(iv)	For all a e Г(р) the set {x e X : a(x) = 0} is
closed.
Proof, (i) + (ii) : Let a belong to the closure of {0x e Ex : x e X}.
we conclude that p(a) = lim p(0	) =
iel Xi
continuous, we may write 0 , . = lim
P(a) ieI
space, limits of nets are unique, if
limx. and as the О-section is
iel 1
0	. Because E is a Hausdorff
xi
they exist. Hence we conclude
that a = 0 , , .
P (a)
(ii) + (iii) : Let 0 / a e E. As the set {0x e Ex • x e X} is closed,
we can find an open neighborhood U of a such that g / 0 for all
6 e U. We may assume without loss of generality that
U = {g e E : p(g) e U, Vj(a(p(g)) - g) < e}
for a certain open neighborhood U of p(a), a certain seminorm
Vj : E +1R of our bundle and a certain local section a : U + E.
Now let
5 = i-(e - v.(a(p(a)) - a)).
*	J
Then the smaller set
V = {g e E : p(g) e U, v^(a(p(g)) - g) < e - 6}
187
is still an open neighborhood of a. Moreover, for all В e I/ we have
Vj (B) > 6, as v ( g) <6 for an element Bel/ would imply
Vj (a (p ( B))
= v.(a(p(B)) - 0p(6))
£ Vj ( a(p ( В) ) ~ B) + Vj ( B)
< e ~ 5 + 6
e .
Thus
°P( B)
e U contradicting
the choice of U. We now conclude that
V j ( a) - 6-
(iii) + (i) : Let a, в e E be two distinct elements of E. We have to
show that they have disjoint neighborhoods. This is obvious if
P(a) / p(B), as in this case we may take disjoint open neighborhoods
-1	-1
U and V of p (a) and p( B) resp. Then p (U) and p (V) are disjoint
open neighborhoods of a and В respectively.
Now suppose that p( a) = p(B). In this case a / В implies a - В / 0.
Using (iii) we can find a seminorm Vj : E -> ]R such that \Л(а - В) > О.
Let г := l«v.(a - В). Then there is an open neighborhood U Of а - В
such that Vj(y) > r for all у e U. As the mapping
E vE	E
(Y1'Y2> * y1 - Y2
is continuous, there are open neighborhoods I/ and W of a and В resp.
such that Y-j ” ^2 e for Y1 e Y2e " T^ese sets У an^ W
are disjoint: Indeed, if у e V nW, then 0 = у - у e U, contradicting
Vj(y') > r for all y' e U.
Finally, suppose that p : E -> X satisfies the additional properties
listed in (16.4). Then obviously (i) implies (iv). Conversely, we
shall show that (iv) implies (ii):
188
Let a e E belong to the closure of {0x e Ex : x e X}. Assume, if
possible, that 0 / a.
By (2.9) there is a section a e r(p) such that a(p(a)) = a and by
(1.6.(vii)) the sets of the form
T(u,a,e,j) ;= {В e E : P<B) e U, Vj (a(p(B) ) - B) < e}
form a neighborhood base of a, where U runs through all open neighbor
hoods of p(a), \>j runs through all seminorms of the bundle and where
e ranges over all positive numbers. Obviously, we may restrict our-
selves to a countable neighborhood base of p(a) and to real numbers
of the form 1, n e IN. As the family of seminorms was countable, too,
we conclude that a has a countable neighborhood base (^n)n£]N and
we may assume that	c for all n e IN. Moreover, the singleton
{0 , . } is closed and a / 0 , .. Therefore we may assume that
P ( a)	p (a)
0 , . i V for all n e IN.
p(a) n
As a belongs to the closure of {о eE : x e X}, for every n e IN we
x x
can pick an element x e X such that 0 el/. Obviously, x p(a)
n	x n	n
n
for all n e JN, lim 0	= a and thus lim xn = p(ct) . Hence the set
П->оо П	Пн-co
A : =	{p(a) } и {xn : n e IN}
is compact and the selection p : A -+ E defined by
,0	if x = x for some n e IN
, . I x	n
p(x) = 1 n
a	if x = p(a)
is continuous. By (4.4) we can find an extension p e Г(р) of p. For
this section, the set {x : p(x) = 0} is not closed, as p(xn) ~ p(xn)
= 0x , but p(lim x ) = ”p(p(a)) = a / 0.	□
n	n
16.5 Corollary. Let p : E + X be a bundle of finite dimensional
spaces over a Hausdorff spaoe X. Then E is Hausdorff if and only if
for every x e X there is a seminorm v- : E -> TR and a real number
-'x
189
Мх > ° 8иа^ that for all О / а е Ех the set
{В е Е : V. (В) > М «V. (а) }
Зх	х Зх
is a neighborhood of а- Moreover, in this ease the seminorm
\> 	: E + ]R may be chosen to be a norm when restricted to the stalk
-’x
Proof. Assume that E is Hausdorff. As E is finite dimensional, the
x
topology of Ex is induced by a norm || • || . Because the sets of the
form {u Ex : Vj(a)< e} form a (directed) neighborhood base at 0 eEx,
where v  : E -»• 3R runs through all seminorms of the bundle and where
e ranges over all positive numbers
we can find an index j e J
and an e > 0 such that {а e E : v- (a) < e} c {a e Ex :	||a|| < 1}.
•J
Hence the set {a e E : v- (a) < e}
x Do
of E and therefore v• is a norm on
x	3o
. is a norm on E whenever v 	£ v •
3	x	jo 3
Vj implies that Vj < Vj, whenever Vj
contains no non-trivial subspace
Ex- This also implies that
Moreover, the definition of the
Now we note that В = {a e Ex : ||a|| = 1} is compact. As E is Haus-
dorff, we can use (16.4) to find for a given element а e В a semi-
norm v- : E +1R of the bundle such that v- (a) > 0. By the above
Da	Da
remark we may assume that v- < v- . Now the sets of the form
3o -'a
Ua = {B e Ex : Vj (B) > 0}
J a
are open by the lower semicontinuity of the Vj- Since these sets cover
the compact set B, we can pick elements а^,...,ап e В such that
Вс и и ... и U . Choose any index j such that
“1
v. < v.	for all i e {1,...,n} .
3„ 3X
(JC j
Then we conclude that Vj
0 for all a e B.
Moreover,
norm on E as v 
x 3o
x
v. .As the restriction of v. to
jx	->x
Ex
V . is a
Jx
is contin-
190
uous by (16.2), the set v. (В) cjr is compact and does not contain 0.
-’x
Hence we can find an Lx >0 such that v  (B) c JL ,«>[. Now assume
-’x
that 0 / a e Ex- Then the element a/ ||a|| belongs to В and therefore
we have v. (a) > Lx • ||a|| .
Jx
As every two norms on a finite dimensional space are equivalent,
we can find a constant M >0 such that ||a|| -L > M »v- (a) for
x	x -’x
all a e Ex . This implies v. (a) > M -v- (a) for all a e E which are
^x	x -’x	~
different from 0. In particular, by the def inition of v . we conclude
-,x
that the set {£3 e E : v  (£3) > M -v- (a)-} is a neighborhood of a for
Dx	x Dx
every 0 / a e Ex- □
We now turn our attention to a different description of the v-:
16.6 Definition
Let p
E + X be a bundle
with seminorms
j'jeJ
For every j
e J and every M e IR we define
Cj,M := e E :	~	'
where denotes the topological closure in E.
If p : E + X is a bundle of Banach spaces, we let
CM := {a e E :	|| a £ M}“.	□
16.7 Proposition.
For every bundle p : E + X we have-.
(i)
(ii)
(iii)
(iv)
a e C. „
j ,M
and only if Vj(a)
Vj(a) = inf {M : a e Cj M}
£ M, provided that M/0.
If a : U + E is a local section, then Vj(a(x)) s M if and
only if x e n {y : v-(a(y)) £ M + e} .
e>0	-1
If p : E + X is a bundle of Banach spaces, then (i) holds
also for M = 0.
Proof, (i)
As is lower semicontinuous, the set	is
191
closed. As Vj (a) M implies Vj (a) < M, we obtain {a e E : \м(а) < M}
c v . 1 ( (-00,M]) and thus C . ,, c	).
j	J	j ,M j	'
Conversely, suppose that Vj(a) s M / 0. Then for every e > 0 we have
Vj (а) < M + e. From the last equation in 0.6 .1) we conclude that
every neighborhood U of a contains an element g such that Vj(3) <
< M + e and from the definition of C. ,, we deduce that а e C.	for
j,M	j,M+e
every e > 0. Since multiplication with scalars different from 0 is a
M
homeomorphism, this yields + e
C j ,M
for every e
> 0. Letting e
go to 0, we obtain а e C.
(ii): If v (a) / 0, then (ii) follows immediately from (i). On the
other hand, if v . (ot) = o, then using the same arguments as above,
we may conclude that a belongs to C .
for every e
> 0.
(iii): Assume that x belongs to {y : Vj(o(y)) < M + e} for every
e > 0. Then for every e > 0 we know that
а(х) £ a({y : Vj(a(y)) < M + e} )
c a({y : Vj(a(y)) <. M + e})
c Cj,M+e
and thus Vj(a(x)) s M + e by (i). As e > 0 was arbitrary, we obtain
vj ( a (x) ) < M.
Conversely, assume that хЛ(а(х)) s M. Using (i) again, we conclude
that a(x) e n {а e E : v.(a) < M + e} . Suppose that the element
e>0	21	_
x belongs to the open set V := U \ {y : Vj(a(y)) M + e} for some
e > 0. Then for all у e V we have Vj(a(y)) > M + e and therefore
\ij(a(x)) > M + e by (16.1), which is impossible. This concludes the
proof of (iii).
(iv): Finally, let us suppose that p : E + X is a bundle of Banach
192
spaces. We have to show that a belongs to the closure of
{0x e Ex : x e X} if and only if v(a) =0, where v : E +1R is given
by v( a) = || a || .
Assume that	5(a) = 0. If {g e E : ||g||	>	0}	were	a	neighborhood
of a, we	could find a local section a		U +	E and	an	e	> 0 such that
a e {8 e	E	: p(B) e U, || a(p(B) ) ~ B|| < e}	c {g e	E	:	В / 0}. Let
6 =	-	||a(p(a) ) - a|| )• Then as in	the	proof	of	(16.4, (ii) + (iii) )
we would see that v(a) > 6 > 0, a contradiction.
This other implication holds trivially. □
The following example shows that the mapping Vj : E -> 1R .need not
be seminbrms. To verify this, we shall construct a bundle of
Banach spaces p : E + X such that the closure of the unit ball
{a e E :	11 a 11 s 1 } is not stalkwise convex and then apply (16.7. (i)).
16 .8 Example. Let X = [0,1 ] be the unit interval with the usual
2
topology, let E = [0,1] x]R , equipped with the product topology and
let p : E + X be the first projection. We define a norm on E by
|| (r, (a,b) ) ||
max {|a|/2, | b | }
max { | a | , |b | }
max {|a|,|b|/2 }
r < 1/2
r = 1/2
r > 1/2
Using (3.2) , it is easy to check that p : E + X equipped with this
norm is a bundle of Banach spaces.
Moreover, we have
C1 n Р“1ф = {a e E : ||a|| < 1 }“ n p“1 (|)
= {(-l,a,b) : (|a| < 1 and |b|	2) or
( | a | <. 2 and | b | <. 1) }
and this set is not convex.
193
In the remainder of this section we restrict our attention to bundles
of Banach spaces. We shall continue the discussion of duality already
begun in section 15.
Let us recall some notations ( p : E + X is a bundle of Banach
spaces with a compact base space ):
Bx denotes the dual unit ball of the stalk Ex , identified
with a subset of r(p)' via the natural embedding
- Г(Р) ' •
Вд = и Bx , where A is a subset of X
xeA
The following result is a generalization of
(15.9) :
16.9 Proposition. Let p : E + X be a bundle of Banach spaces over
a compact base space X. For every x e X let Wx c Bx be a closed and
convex subset. If А с X is a subset, then we define W := и W .
A x£A X
Then the following conditions are equivalent:
(i)	If	А	с	X	is any subset, then x e	A implies Wx	с
(ii)	If	А	с	X	is any subset, then x e	A implies Wx	c conv
(iii)	If	u.	is an ultrafilter on X with	lim u. = x, then	Wx	c	lim	W^.
(iv)	If	U	с	В	is (relatively) open, then {x e X :	W	n	U	/	0}
is open in X.
(v)	If A c B„ is closed, then {x e X : W c A} is closed in X.
X	X
Here, the limit lim W is taken in the Lawson topology of Cl B„.
u. у	X
Proof, (i) + (ii)
is trivial.
(ii) + (iii): As all the sets Wx , x e X, are closed and convex and
194
as the embedding Conv Bv + Cl B„ is continuous by (12.7), we have
X	X
lim Wy = n {conv Вд : A e u.} by (12.8). As A £ a implies x e A, the
property (iii) follows from (ii).
(iii) + (v): Let A c Bv be closed and let a be an ultrafilter
X
containing {x e X : Wx c A}. We have to show that lim u. e {x : WxcA}
i.e. W..	e A. But this is true, since we only have to note that
lim a
{B e Cl Bv : В c A}is closed in the Lawson topology of Cl Bv and
X	X
as {x £ X : Wx c A} e u. Therefore, using (iii) we may conclude that
W. . c lim W c A.
lim и и у
(iv) -» (v) and (v) + (iv) are trivial.
(v) + (i) : Let x e A. Then by (v) the set {y :	с Щд} is closed.
As this set contains A, it contains the closure of A, too. This
yields Wx с	□
The next result allows us to use duality in order to identify all
stalkwise convex and closed subsets of E containing the unit ball
{a e E :	|| a ||	< 1 }:
16.10. Proposition. Let p : E -> X be a bundle of Banaoh spaces
with a oompaot base spaoe X. Then a subset С с E containing
{a e E : 11 a11 £ 1} such that C n Ex is convex for every x e X is
closed if and only if the family Wx = (C n Ex)° c ^x satisfies the
equivalent conditions of (16.9).
Proof. Let С с E be given and let us assume that C is closed,
stalkwise convex and contains {a e E : 11 a11 < 1}. Moreover, let
A be a subset of X and let xQ belong to the closure of A. We have to
195
show that W c conv W = (U~ . Let us compute:
xo	A A
W° =	( и W )°
x eA
о {с e Г(р) : a(x) e C }
xeA
=	{a e	Г(р)	: a(A) 	с C}
=	{a e	Г(р)	: a(A) 	- C}	(since C is closed)
c	{a e	Г(р)	: a(xQ)	c C}.
Now the result follows immediately by taking polars.
Conversely, assume that the family Wx = (С n Ex)°, x e X, satisfies
condition (v) of (16.9). We have to show that C is closed, or,
equivalently, that E \ C is open. Thus, let us start with an element
a e E \ C, let xq = p(a) and choose a e Г(р) such that p(xq) = a.
For each e > 0 we define
A = {ф e Bv : Re ф(а) < 1 + e}.
ь	л
Then all the sets A£ are closed and
A0 =	\ Ae •
e>0
If Wx = (c n E )° were contained in A£ for every e > 0, then
°	о	o
it would be contained in AQ, too. But for each x e X we have (C n Ex)
= Wx c Aq = {o}° if and only if а e (C n Ex)°° = {p e Г(p) :
P(x) e C}. This means that Wx c AQ if and only if а(x) e C. As
а = a (x ) was not in C, we conclude that W <t A„.
о	'	x T 0
о
Let e > о be a positive real number such that Wx Ф A^ and let
U := {x e X : IV n (B \A ) / 0}.
X	Л £
196
Then xq e U and U is open by (16-9.(v)). We define
0 := {В e E : p(B) e U and ||o(p(B) ) - B|| < e}-
Then 0 is an open neighborhood of a- Moreover, () n C = 0: Indeed,
for В e C n 0 we would conclude that р(в) e U, ||a(p(B)) ” B|| < e
and Re ф(B) £ 1 for all ф e W , .
P 1 B)
this would imply the inequality
Re ф (a(p( B))) = Re ( ф ( В)
= Re ф ( В)
s Re ф ( В)
< Re ф(В)
< 1 + е
and thus Wpjgj c A£ contradicting
= {x : W n \ AJ / 0}-
X	A	£
= (С n Ep(6)) . For all Ф e Wp(6)
+ (ф (a(p ( B)) ) “ Ф ( B)))
+ Re (ф (a(p ( B)) ) “ Ф ( B))
+ | ф (a (p ( B))) “ Ф (В) |
+ IIфИ - Иo(p(в)) - b||
the fact that p(B) e U =
□
In a later section, when we shall talk about the "internal" dual of
all Ftp) consisting of all C(X)-module homomorphisms from r(p) into
С(X), we shall need the largest family of subsets Wx c Bx such that
the properties of (16-9) are satisfied. By (16.10) we know that this
family is determined by the smallest stalkwise convex, closed subset
C of E containing {a :	||a|| s 1}. Hence it seems to be desirable
to have an explicit description of this set. I conjecture that this
set can be obtained by taking stalkwise the closed convex hull of
the topological closure of {a e E : 11 a11 £ 1}» but I do not have
any proof hereof. All I am able to do is to identify the stalkwise
polars of this set:
16.11	Notation. Let p : E + x be a bundle of Banach spaces, X
compact. For every x e X we define a subset <x c Bx by
197
Кх := п <Вм : M c X and х е МЬ
16.12	Proposition. Let р : £ X be a bundle of Banaoh spaces with
a oompaot base spaoe X. Then Kx is circled and
(H <x c Bx
(г г J	Kx = n (B°° : x e M} = n {conv BM : x e M} = n {Ijm B^ :
a. is an ultrafilter on X converging to x}
where lim В is calculated in the Lawson topology on Cl Bv-
U. X	X
x ls an
and therefore
Proof. As B,, is always circled, so is B„. Since
M	M
of sets of this form, к is circled, too.
x
(i)	: From x e (x} , we have К с В .
X X
(ii)	: For every subset M с X we have B„ c
MM
Kx c n {BM° : X e c n <conv BM : x e “}.
Moreover, let a. be an ultrafilter on X converging to x and let Вц
be the ultrafilter generated by the image of u. under the mapping
x B„ : X ->- Cl В (r(p)') (recall that В (p(p)') denotes the unit
X	J	J
ball of r(P)')• Then lim В = lim В and В is an ultrafilter having
и и x	и
a base whose elements consist of closed convex sets only. Hence,
by (12.7), the limits of Вц in Cl (г(p)') and Conv (г(p)') agree.
We may now calculate
Um В = n — О
Meu yeM y
As lim и = x implies x e M for all M e u, we conclude that
BM° c	Bu
xeM
for every ultrafilter a with lim и = x. We obtain the inequality
198
n {BM :xeM}cn {lim Bx : u is an ultrafilter on X converging to
x}.
It remains to show that the last set is contained in < . To do so, we
have to prove that for every M с X with x e M there is an ultrafilter
u on X converging to x such that lim c BM- But this is easy, as
x e M implies the existence of an ultrafilter u. on X with M e a and
lim и = x. For any such ultrafilter, we have
lim B„
u У
n ( и В )
Neu. yeM У
n
Neu
M
□

16.13
Proposition. Let p : E + X be a bundle of Banaoh spaces with a
compact base space X. Then for every x e X we have
Proof. Let ф e (C^	n Ex)°. We	have to show that ф belongs	to
B°° for every subset	И c x with	x	e M. Firstly, note that a	belongs
to B° if and only if	|| a(y) || <	1	for all у e M. Hence, x e	M and
a e B° imply a(x) e	a(M) c a(M)	c	C1, i.e. a(x) e C1 n Ex	' Thus'
we have shown that (C^ n Ex)° c B°° whenever x e M, i.e.
(Ci n Ex)° c Kx.
Conversely, assume that Ф e Kx and let a e n Ex- We have to
show that Re ф(а) <1. AsC^ n Ex is circled, Re ф(а) < 1 holds
if and only if |ф(а)| < 1.
Let а e Г(р) be a section such that а(х) = a and let e > 0. Using
( 16.7. (iii) ) , we conclude that x belongs to M, where M = {у e X :
||a(y) || < 1 + e}, and therefore ф belongs to BM- As for all ф e BM
we have ф(а) = ф(а(у)) for a certain у e M, we may estimate
199
I Ф (о) [
= i Ф(о(у)) |
£ НфН  Н о (у) Н
<	1 • (1 + е) .
As ф belongs to the closure of BM, this implies |ф(а)| < 1 + e. Since
e > 0 was arbitrary, we conclude that |ф(а)| < 1.	□

17 . Locally trivial bundles: A definition
In this section we shall introduce locally trivial bundles, a
classical concept which has been used in.differential geometrie and alge-
braical topology since a long time. We should note however, that
our definition of locally trivial bundles will differ slightly from
the usual one, the reason being that homoeomorphisms between
bundle spaces commuting with the projections are in general not what
we call isomorphisms of bundles (see ( 10.1 (i)d) and example (10.25).
Nevertheless, if the base space is locally compact, our notion of
locally trivial bundles will agree with the usual one.
17 .1 Definition. (i) Let p : E -+ X and q : F + X be two bundles
of fi-spaces having the same base space X. We say that p and q are
locally isomorphic, if every point x e X has a neighborhood Ux с X
-1	-1
such that the bundles p : p (Ux) + Ux and q : q (Ux) + Ux are
isomorphic.
(ii)	A bundle p : E + X is said to be locally trivial if it is
locally isomorphic to a trivial bundle pr^ : XxE + X, where E is
a topological fi-space (see example (1.8.(i)). The fi-space E is
called the stalk of the bundle p : E •* X. □
From (10.10) and (1.10) we conclude:
17.2 Proposition. Let p : E + X be a bundle with a family of
seminorms (v.). T.
(i) If p is locally trivial, then every point x e X has a neigh-
borhood Ux such that Гу (p) and Cj;)(Ux,E) are isomorphic
(as С^(их) -modules and as ii-spaces, if required), where E is
the stalk of the bundle p.
201
(ii) If X is completely regular and if the bundle p : E + X is
full, then the converse also holds. □
It is obvious that we have to insist on full bundles in order to get
the converse of (17.2.(i)): By (2.3) fullness is a "local" property
for bundles with a completely regular base space, and local properties
are preserved under local isomorphy. As every trivial bundle is
locally full, so is every locally trivial bundle. Hence every
locally trivial bundle over a completely regular base space has to
be full.
18. Local linear independence
Let us start with a locally trivial bundle p : E + X and let us
suppose that we are given linear independent elements ,...,an e Ex
с E. Then it is easy to see that we may find an open neighborhood U
of x and sections ,... , an e [^(p) such that (y),...,an(y) are
linearly independent for every у e U and, moreover, а^(х) = for
every 1 < i < n.
Unfortunately, as example (5.16) and the section defined there
show, this property does not characterize locally trivial bundles.
However, by adding some separations axioms to both E and X, this
property gives us the right idea for such a characterization.
We use again the notations of section 16:
Ifp : E + X is a bundle
Vj(a) := sup {r e ]R : {g : v j ( B)
> r} is a neighborhood of a}.
Recall that Vj(a)
> 0 if and only if a
and 0 ,
p (a.
have disjoint
neighborhoods.
The results of this sections are known for bundles of Banach spaces
with continuous norm (see [Go 49]).
18.1 Proposition.
Let p
E + X be a bundle with seminorms
(v?
jeJ
let x e X and let V
о
be an open neighborhood of xq. Assume that
a1'.'.'an e Гу(р) aVe 3iven su°h that (xQ),...,an(xQ) are linearly
independent. If for eaoh 0 / a e <a^(xq),...,an(xq)> in the linear
span of the a^(xQ) there is an index j e J suoh that Vj(a) > 0, then
there is a neighborhood U с V of x suoh that for every x e U the set
{a^(x),...,an(x)} is linearly independent.
203
Proof. Assume, if possible, that (18.1) is false. Then there is a
net (xi)ieI converging to xq and numbers (r1 j.) ieI' • • • ' (rn id such
that for every i e I we have
m. := max {Ir. . I,..., I r . I } >0
i	1 1 ,i1	1 n,i1 J
and
n
У r .-a (x.) = 0.
k=1	'
By dividing all the r^ by iru,
we may assume that |r, .I S 1 for
1 к, i1
all i e I and all к e {1,
,n). By multiplying with a unimodular num-
ber if necessary
we may assume
that one of the r,
к
is equal to 1
for all i e I. Furthermore, there is an index к e {1,...,п} such
о
that I = {i e I : r, .=1}is cofinal in I; without loss of gene-
о	к , i	3
о
rality we may assume that к = 1. Hence, by substituting I by IQ,
we may assume that r^ = 1 for all i e I. Finally, by selecting a
suitable subnet
we may assume that
exists for all 1 < к < n.
Now r,'0,(x ) is a limit point of the net (r, .-a. (x.)). as
К К О	л,1	-К.	1	161
the scalar multiplication is continuous. As the addition is contin-
n
uous, too, we conclude that £ rv*ov(x ) is a limit point of
n	k=1 k k °
( I rk /°k!xJ!^T =	Clearly, (3 is also a limit point
к=1 rl 1 le± xi 1£±	xo
of the latter net. Now suppose that
n
“	:= I rk’ak(xr>) 0
к=1 к К °
Then we could find an j e J such that v (a) > o, which means that
a and 0 have disjoint neighborhoods. As they are both a limit point
xo
of the same net, this is impossible. Hence a = 0, contradicting the
fact that the adx ) , 1 < к < n, are linearly independent and that
К о
Г1 = 1 . □
204
From the last proposition and (15.4) we conclude:
18.2	Theorem. Let	p :	E	+	X be a bundle	and assume	that	E	is
Hausdorff. Moreover,	let	, .	. . , an e Г(р)-	Then the set
{x e x :	(x),...,an(x)	are	linearly independent} is	open.	□
18.3	Theorem. Let	p :	E	+	X be a bundle	and assume	that	E	is
Hausdorff. Then the mapping
dim : X + ]R
x + dim E
x
is lower semioontinuous.
Proof. By definition (1.5) the set {a(x) : a e Гу(р) f°r some open
neighborhood U of x} is dense in Ex> Thus, if dim Ex > n, then there
are open neighborhoods U^,...,U of x and sections e Гу (p) ,
1 s i < n, such that the set {a^(x),...,5n(x)} is lineraly indepen-
dent. Let V := U. n ... n U and let a. := c.,,,. Then (18.4) and
I	n	i i/V
(18.1)	yield an open neighborhood U с V of x such that for all у e U
the set {a^(y),...,a (y)} is linearly independent. Especially, we
have dim E^ > n for all у e U. □
We continue with a result which may be thought of as an improvement
of (18.1) :
18.4	Proposition. Let p ; E + X be a bundle with seminorms (Vj)jgJ,
let xq e X be a point and let V be a neighborhood of xq. Further-
more, let ,...,an e Г^(р) be suoh that (xq) ,. . . ,an(xQ) are
linearly independent and assume that for every d‘. / a e <a^ (x ) , . ..
...,a (x )> с E there is an j e J suoh that v-(a) > 0. Then we
о	-1
oan find an open neighborhood W с V of x suoh that eaoh neighborhood
205
U с W of х has the following properties:
(i) The C^(U)-submodule of Гу(р) generated by /ц' ’ ‘ ‘ ' an/U
e Гу(р) is topologically and algebraioally isomorphic to
cb(u, Kn).
(ii) The C^tU) -submodule of Гу(р) generated by	‘ ‘
e Гу(р) is complete and hence closed in Гу(р).
Proof. Let А сЖп be defined by
A := {(rlf...,rn) e ЖП : max {|r1 ],..., | rn| } = 1}
Then A is compact and therefore the set
n
A != ri’ai(Xo) ' (rv-^n) « A>
is compact in E .As the set {a(x ),...,a (x )} is linearly inde-
5C	J О	n о
о
pendent and as (O,...,O) < A, we conclude that 0 < A- Thus, for every
a e A there is an index j e J such that
a
0 < e 8= v. (a) .
a J ~
Now by (16-2) the sets 0	:= {g e E :
a	xQ
cover A- As A is compact, we can find
Vj (6)
> ea/2 } are open and
•••'“n e A such that
A c 0 и ... и 0 .As the family of
a1	“n
directed, we can find an index j e J
j e {j r...rj }• Now define
a1	“n
seminorms of a bundle is always
such that
v . for all
-’o

6 : = 4-*min { e , . .. , e }.
2	“1
Then it is easy to check that
v. (a) > 6
Jo
for all a e A.
Now let (гЪ1......rbn),(r2,i.....r2,n).....(rmJ......rm,n) e Abe such
that for each (r^,...,r ) e A there is a certain 1 e {1,...,m} with
206
r. -r11|+...+|r - r. |	where
1 1	1,1'	1 n 1 ,n1	3-M
M := max {sup (a (y) ) ,..
У eV Jo
Then we obtain the inequality:
n
sup v ( I (r. -a. (y) - r »a (y) )
у eV 3o i=1	11	J-ri i
,sup V. (a (y))}.
У eV Jo
n
<	I |r - r |-sup v (a (y))
i=1	' у V Jo
n
<	У |r. - r. . [ *M
" i=1	1
S fi/3.
Then the triangle inequality yields for all у € V the relation
n	n
( У Г -a (y) ) s-v (У Г -a (y)) - 6/3.
Jo i=1	1	Jo i=1	1
Now use (16.1) and (18.1) to find a neighborhood W с V of xq such
that
(1) if у e W, then the set {a (y) ,...,	(y) } is linearly inde-
pendent .
n
(2)	( У r^ .’a.(y)) > 5 for all у e W and all 1 e {l,...,n}
3q i=1	'1 1
and let и c w be any neighborhood of xq. Then we conclude that
(1’) if у e U, then the set {(y),...,an(y)} is linearly inde-
pendent .
n ?
(2’) v. ( } r.-a(y) ) > ч-'б for all (r ,...,r ) e A and all
Jo i=1 1 1	J	1	n
у e U.
n
Let C, (U) be equipped with the norm ||ff || := У ||f.|| and
°	l	n	i=1 i
let Ггт(р) (as usual) be equipped with the family of seminorms (0.). _
и	3 3 eJ
given by 0.(a) = sup v.(a(y)). We define an operator
3 yeU °
T : Cb(U)n -» Гу(р)
(f1....fn> - X fi-i
As Cb(U)n = Сь(и,ЖП) , the proof will be complete if we can show that
207
T is a continuous and injective (U)-module homomorphism which is
open onto its image.
Obviously, T is a (U)-module homomorphism.
To show the injectivity, let T(f^,...,fn) = 0. Then for all у e U we
n
have У f.(у)-а.(у) = 0 and thus f. (у) = 0 for all у e U and all
i=1	1	1	1
i e {1,...,n} by (1 ') . This implies (f^,...,fn) = (0,.. . ,0) .
Furthermore, T is continuous, as for all (f^,...,fn) and all j e J
we have
VT(fi.......fn>> = v.l fi-°ilu)
1=1
< I IlfiH .(a.)
i=1	J
s II (f-!....fn) || • max^Cap.....Oj<an)}
It remains to show that T is open onto its image:
Let (f.j,...,f ) e C^tUpand let e > 0. Then there is а у e U such
that
3.C-
max { ||f.| || ,..., ||fn|| }	< max {If^y) |,...,|fn(y) |} +
If we abbreviate m = max {|f(y) |,.., | f (y) |}, then we have
11	n 1
(-	£П(У)) e A and therefore v. ( £ - f i (y) • а± (y)) >
Jo i=1
by (2') . We now have
GJ | NJ
0	(T(fir...,f ))	> v. (T(f ,...,f )(y) )
Jo	Jo
n
= v . ( I f, (y)-a.(y))
Jo i=1
2
> m.j.S
and this inequality yields
2	2 n
|-5- || (f1 ,...,fn) ||	= |-6-J HfJ
208
<	j.S.n.max { ||f 1 ||	||fn|| }
<	j.5.n.(max { |fl (y) [....|fn(y)|	} +	)
2
=	-j.fi.m-n + e
<	n-0- (T(f1	)) + e
Jo
2 , r
As e > 0 was arbitrary, we conclude that	IHf•] ' ‘ ‘ ‘ 'fn) II -
< 0. (T(f.,...,f )). Hence, T is open onto its image. □
J
18.5 Theorem. Let p : E + X be a bundle with a locally compact
base space X. Assume that all stalks of p have dimension n, where
n e IN is fixed. Then the bundle p : E + X is locally trivial if and
only if the bundle space E is Hausdorff.
Proof. By definition, every locally trivial bundle over a Hausdorff
base space has a Hausdorff bundle space.
Conversely, assume that E is Hausdorff and assume that all stalks
have dimension n. Given a point xq e X, we
hood U of x such that the bundle
have to find a neighbor-
p 1 (U) -+ U and the
P/P 1(U)
trivial bundle pr^ : U xJKn + и are isomorphic.
Let a.],...,an be a base of Ex . As the bundle p : E + X is full
о
by (2.12), there is a neighborhood W of xq and sections a^,...,an e
e rw(p) such that a^(xQ) = for all 1 s i < n. Applying (17.2)
we can find a neighborhood V c W such that {a^(y),...,an(y)} is
linearly independent and thus a base of E^ for every у e V. Moreover
by ( 18.4) we can find a compact neighborhood и с V of xQ such that
the C(U)-submodule of Гу(р) generated by a1|u,...,an|u and
C(U/Kn) are isomorphic. As the set {a^(y),...,an(y)} is a base of
Ey, the C(U)-submodule generated by |u,...,an|u of Гу(р) is stalk-
wise dense. Hence the Stone-WeierstraB theorem (4.3) applied to the
bundle P/p_1(u) : P 1 (U) + U shows that the C (U)-submodule generated
by a1 |U, . . . ,an | U is equal, to Гу(р), i.e. Гу(р) and С(ОДКП) are iso-
morphic. Now apply (14.10) to complete the proof. □

1 9. The space Mod(Г(p),C(X)).
In this section we shall discuss the existence of C(X)-module homo-
morphisms between the space r(p)
of sections in a bundle p : E + X
and C^tX). For bundles of Banach spaces it will turn out that this
question is closely related with the structure of the closure of the
"unit ball" {a e E : 11 a11 <	1}.
Let us start with a full bundle p : E + X over a completely
regular base space X. If S is any family of precompact subsets of
Г(р) whose union generates, then we know from the remarks proceeding
(11.21) that there is a bundle q : F + X such that Mod(г(p),С^(X)),
equipped with the topology of uniform convergence on elements S e S,
is topologically and algebraically isomorphic to a (X)-submodule
of F(q). The stalks of the bundle q : F + X are subspaces of
Ex ~ ^Ex' ' w^ere Ex denotes the stalk of p : E -> X over x e X.
Note that the choice of the stalks of q : F -> X does not depend on
S, although of course the topology on F does.
Let us try to describe the bundle q : F + X in greater detail. By the
remarks in (15.1), we may identify (the dual space of Ex) with
a subspace of r(p)’ and by (11.20), this embedding is topological, if
we equip E^ with the topology of uniform convergence on elements
of S(x) = {ex(S) : S e S} and Г(р)’ with the topology of uniform
convergence on S.
Now let T : Г(р) + Cj^CX) be a continuous (X)-module homomorphism.
Then T corresponds to a section XT e r(q), and T and XT are related
by the equation
210
Т ( a) (х)	=	(х) (a(x)) for all х е X, а е Г(р)
Moreover, may also be viewed as the unique bundle morphism
Лт : E + X x Ж from E into the trivial bundle pr^ : X><K + X which
represents T by (10.7).
Furthermore, as ^p(x) is an element of for every x e X and as
may be identified with a subspace of Г(р)', we also may view XT as a
mapping into Г(р)'.
The following result is a generalization of (10.23):
19.1 Proposition. Let p ; E * X be a full bundle over a completely
regular base space x. Then т : Г(р) -> С^(Х) is a continuous C^(X)-mo-
dule homomorphism if and only if there is a uniquely determined
° ( Г (p) ', Г (P) ) -continuous mapping Л : X Г (p) ' satisfying
(1)	^T(x) e for every x e X.
(2) Ат(X) is an equicontinuous subset of Г(p)’.
such that T(a)(x) = XT(x)(a(x)) for all x e X and all a c r(p)-
Moreover, if S is a total and directed family of precompact subsets
of Г(р) and if we equip Mod(Г(p),Cb(X)) and Г(p)' with the topology
of uniform convergence on S, then the mapping хт : X + Г (p)' is contin-
uous and
X_ : Mods(r(p) ,Cb(X) ) Cb(X,rs(p)')
T	- xT
is a continuous and injective C^iX)-module homomorphism which is open
onto -its image.
211
Proof. Let T : Г(р) + С, (X) be a continuous С, (X)-module homomor-
b	b
phism and let XT : X + Ftp)' be as explained in the above remarks.
By construction, we have XT(x) e E' and T(a)(x) = X (x)(a(x)) for
all a e F(p) and all x e X. This last equation also shows the
a(Г(p)Г(p))-continuity of XT, as for every a e r(p) the mapping
x + X (x) (a) = XT(x)(a(x)) = T(a)(x) belongs to C(X). Moreover, the
set XT(X) is equicontinuous, as we have
(*) {a e Ftp) : |XT(x)(a)| <1 for all x e X}
=	{a	e	Ftp)	:	| XT(x) (a(x) ) |	< 1	for	all	x e	X}
=	{a	e	F(p)	:	|T(a)(x)| <1	for	all	x e	X}
=	{a	e	Ftp)	:	||T (a) || s 1 }
and the last set is open by the continuity of T.
Conversely, let X : X + F(p)' be a a(Г(p)', Г(p))-continuous mapping
satisfying conditions (1) and (2). Define
TA : Ftp) - Cb(X),
where T (a) (x) = X(x) (a(x)) for all x e X. Then T (a) : X + Ж is
A	A
continuous for every a e F(p), because we have T (a)(x) = X(x)(a(x))
A
= X(x)(a) and because the mapping X : X + F(p)' is a(Г(p)',Г(p))-con-
tinuous .
Further, the mapping T (a) : X + Ж is bounded, since X(X) is equi-
A
continuous and hence weakly bounded.
Using (*) again, we see that the equicontinuity of X(X) implies the
continuity of T : Ftp) + С, (X). Obviously, T. is а С, (X)-module
Ad	Ad
homomorphism.
Now let S be a directed and total family of precompact subsets of
F(p) and let X : X -> Г(p)' be any a(Г(p)',F(p))-continuous mapping
212
satisfying (1) and (2). It is an easy consequence of (III.4.5) in
ESch 71] that under these conditions the a(r(p)',r(p))-topology and
the S-topology agree on X(X) . Thus, the mapping x : X r(p) ' is
continuous for the S-topology. It follows that
X_ : Mods(r(p) ,Cb(X) )	+ Cb(X,r(p^
T
lip
is an injective (X)-module homomorphism (note that XT is bounded
for every T e Mod(r(p) ,Cb(X)) by ESch 71,111.4.1 ]) .
It remains to show that x_ is continuous and open onto its image:
A typical neighborhood of 0 in Mod^(Г(P),Cb(X))
looks like
{T : sup ЦТ (a) ||	< 1 }
<7eS
for a certain S e $, and a typical neighborhood of 0 in Cb(X,r(p)^ )
is given by
{F e С. (X,г (p) '	: sup sup |F(x) (a) | < 1}.
D	d xeX aeS
An easy calculation shows that sup ||T(a) || £ 1 if and only if
aeS
sup sup |A™(x)(a)| £ 1 and the proof is complete. □
xeX aeS
19.2 Remarks. (i) Under the conditions of (19.1), we let
MP :=	{(x,XT(x)) : T e Mod(г(p),Cb(X)),x e X} с ХхГ(p)£ ,
equipped with the topology induced by the product topology and we
let
яР : MP + X
be the restriction of the first projection. It follows from ( 8.4(ii))
that Mp is a subbundle of the trivial bundle XxF(p)J, . Moreover, an
213
application of (1.6(viii)) yields that the bundle	+ X and the
bundle q : F -> X constructed in section 11 to represent Mod^(r(p) ,
C^tX)) are isomorphic. Let us point out that, in particular,
Mod^(г(p),Cb(X)) may always be represented as a space of sections
in a bundle over X with a Hausdorff bundle space.
(ii) If r(p) is barreled, especially if p : E -+ X is a bundle of
Banach spaces, and if the union of S generates r(p), we may
substitute the condition (2) in (19.1) by
(2') XT(X) is bounded in r(p)'
(see [Sch 71, iv.1.6]).
In this case, Mod^(Г(p),0^(X)) is isomorphic to the space of all
sections in the bundle	-► X.
We still know very little about the size of the stalks of the bundle
P P
л :	-> X. In fact, there are examples such that all stalks
consist of 0 only. Let us desribe some elements of the и E'
p	XeX X
which certainly do not belong to
We shall again use the notation introduced in section 16. Especially,
if v. : E -> 1R is a seminorm of the bundle p : E + X, then v  : E + Ж
denotes the largest lower semicontinuous function less than or equal
to vj . We define the "bad" part of the bundle p : E + X as follows:
Let F be the intersection of all closed subsets A с E such that
A n Ex is a non-empty linear subspace of Ex for every x e X. Clearly,
F contains the closure of {0 e Ex : x e X}. Using the same proof as
in ( 16.4, (ii) -+ (iii)) , one can show that F is the smallest closed
subset of E such that
214
(i)	Ex n F is a linear subspace of Ex for every x e X.
(ii)	Vj(a) = 0 for all j e J implies a e p.
If we define
Fx := Ex n F	for every x e x'
then we have:
19.3 Proposition. Let p : E + X be a full bundle over a completely
regular base space x.
(i) The stalk over x e X of the bundle л^ :	+ X ts contained
in {x}xF° c {x}xE^.
(ii) If the stalks of the bundle л^ :	-+ X are all equal to
{x}xEx, x e Xj i.e. if for every x e X and every ф e Ex
there is a continuous	-module homomorphism
T : Г(р) -> C|JX) 3U°b that
T(a)(x)	=	ф(а(х))	for all а e Г(р),
then E is a Hausdorff space.
Proof, (i) : Let x e X and let (х,ф) e (л^) (x). By construction
of the bundle л^: M? + X we can find a continuous С, (X)-module homo-
3	b
morphism T : r(p) + C^tX) such that XT(x) = ф. By the remarks pro-
ceeding (19.1), the mapping XT may be viewed as a bundle morphism
XT : E + Xx IK by defining
Ат(а) = (p (a) , A (p (a) ) (a) ) .
Let A = AT^({(y,O) : у e X}). As XT is continuous, the set A is
closed. From A n = XT(y) \o) we conclude that A n is a linear
215
subspace of	E^ for every у e X and	hence	F	c A- This implies
F о E = F	s A n E = X (x)“1 (0) =	ф“1 (0)	,	i.e. ф e F°.
(ii) : If	the stalks of the bundle	+ X are all equal	to
the E^, x e X, then F° and Ex coincide for all x £ X by (i). Using
polars, we conclude that Fx = {0} and hence F = {0 e Ex : x e X}
is closed in E. Now (16.4) yields that E is Hausdorff. ц
Of course, we would like to show that the stalks of the bundle
: Mc -»• X are identical with the family (F°)	. I do not know
Э	XX £A
an answer to this question at all. However, for a certain type of
bundles of Banach spaces, the situation is less hopeless:
19.4 Definition. A bundle p : E + X is called separable, if
there is a countable subset A с r(p) such that {a(x) : a £ A} is dense
in Ex for every x e X.	□
19.5 Examples. (i) The trivial bundle pr^ : Xx X + X is always
separable; more generally, if E is a separable topological vector
space, then pr^ : XxE + X is a separable bundle.
(ii) If E is separable and if X is locally compact and a-compact,
then every locally trivial bundle p : E + X is separable.
(iii) If p : E + X is a bundle of finite dimensional vector spaces,
if X is a compact metric space and if E is Hausdorff, then the
bundle p : E + X is separable.
(Indeed, let An := {x e X : dim Ex < n}. Then An is closed by (18.3)
and we have A c A	for all n £ B. As X is metric, we may find a
• n n+1
countable family (в ) of closed subsets of X such that
J n,m m rIN
216
и В
„ n,m
n eJN
A \ A . .
n n-1
From (18.5) we conclude that
the bundle p
Ip-1
(B J : P
n ,m
is locally trivial. Thus, for every n e JN
and
every m e
' 1 (B ) -
n,m
JN we can
В
n,m
find finitely many closed subsets C
the bundles p : p (C
r r n,m,j
n,m
n,m, j
countable subsets A^ . c F„ (p)
,m,j n,m,j
a' e m j} is dense in
Stone-WeierstraB theorem
{o'(x) :
From the
tion map
a + ol^njmjj : Г(р) + !(-,
subspace
Cb(Cn,m,j
can find a
n,m, j
are trivial. By (i)
such that the set
for every x e C
(4.2) we conclude
such that
we may find
n,m,j‘
that
onto
the restric-
a dense
(p) maps r(p)
n,m, j
of Гс (p) . As Гс (p) is (topologically)
n n,m,j	n,m,j
, Ж ), we conclude that this space is metrizable. Hence we
isomorphic to
{a iC
n, m, j
the set {a(x) :
countable subset A . с Г(р)
n,m,j
.} contains A'
3	n,m,j
.} is dense in E
n,m, j
such that the closure of
Finally, we set
An,m,j
a e A,
In particular, we have that
for every x e C
-1	n,m,j
и и
n eJN m eJN
An,m,j
n,m
For more examples and results concerning
separable
bundles, we refer
E
x
a e
A
x
и
to the papers of M. Dupre (see for example [Du 73]). The paper just
mentioned contains also the idea of the proof of (ii) as it was given
above. Note
however that M.Dupre uses
a more special type of bundles.
From now on we shall equip the spaces Ftp)' and Mod(Г(p),С^(X))
always with the topology of pointwise convergence and we shall denote
these spaces by rg(p)' and Modg(Г(p),С^(X)), resp.. Moreover, we
shall again use the notations of section 15, which we shall recall
for convenience:
B° : unit ball of F(p)1 ,
217
Вх : unit ball of с Г(р)' t
В
A
и Bx -
xeA
19.6 Proposition. Let p : E + X be a bundle of Banaoh spaces over
a oompaot base spaoe X. Then the bundle p : f + X is separable if and
only if 0 has a countable neighborhood base in Bv •
X
Proof. If p : E + X is separable, then choose a countable subset
A c Г(р) such that {a(x) : a e A} is dense in Ex for every x e X.
For every a e A we let
Aa := {ф e вх : IФ(а)I ' 1>•
Then Ад is a closed neighborhood of 0 in B^. Moreover, if ф e Ex is
given, then
ф e n a if and only if |ф(a)| £ 1 for all a e A
a£A °
if and only if |ф(a(x))| <1 for all a e A
if and only if |ф(a)| < 1 for all a e Ex
if and only if ф = 0.
We conclude that
n A =	{0}.
СИА
Since (A )	, is countable and since B„ is compact, we conclude that
a aeA	X
0 has a countable neighborhood base in В .
Л
Conversely, assume that 0 has a countable base (un)nej]* T^en f°r
each n e IN we may pick elements an -j , • • • , an m e Г(р) such that
'	' n
{ф e Bx : |ф(ап,j)| * 1 for 1 < j < mR} c ur.
Let A c Г(p) be the linear subspace over the rational numbers gene-
rated by {an j : n e IN and 1 s j s mn}. Clearly, the set A is
218
countable. We wish to show that {a(x) : a e a} =: A is dense in
x
Ex for every x e X. As the closure of Ax coincides with A°°, we have
to show that Ф = 0 whenever ф belongs to E^, has norm less than or
equal to 1 and satisfies ф(Ах) = {0}.
If Ф (Ax) = {0 }, then | ф ( an j ) (x)' | = | ф ( on j ) |	=0 for all n e И and
all 1 < j < mn- We conclude that ф e Un for every n e JN and therefore
ф e n U = {O}.	□
n.£JN
We are now ready for the construction of C(X)-module homomorphisms
T e Mod(г(p),C(X)). This construction will use to a large extent the
ideas of Douady and dal Soglic-Hferault as they were presented in
section 3.
19.7 Proposition. Let p : E + X be a separable bundle of Banaoh
spaces over a compact base space X and let К с В be a subset such
that
(i) О e К
(ii) Bx n К is closed, convex and symmetric for all x e X
(iii) If и с Г (p)' is open, then the set
{x e X : Bx n К n U	0} is open in X.
Then for every x e X and every ф e К n В there is a continuous
о
function ri : X -> Г (p) ' such that ri(x ) = ф and П(х) e В n К c E'
S	О	О	XX
for every x e X.
Proof. From (19.6) we know that 0 has a countable neighborhood base
(U )„ », in B„. We may assume that the U , n e IN,have the following
nneJNX	n	'
property:
There exists a sequence (vn)nelj of open subsets of Г (p)' such
that:
219
(1)	U = Bv n V for each n eJN.
И X И
(2)	V is a convex, symmetric neighborhood of 0.
(3)	Vn+1 + Vn+1 c Vn
A selection : X + is a mapping such that n(x) e g* for every
x ex. Let V be an open neighborhood of 0 in Г (p) ' . We say that
a selection rj : X + is V-continuous, if every x e X has an open
neighborhood W such that n(y) - ri(x) e V for all у e W.
In the following we abbreviate: К* = К n g^.
We shall divide the proof of (19.7) into a series of lemmas:
19.8 Lemma. Under the assumptions of (19.7) we have
(i)	If a selection П :x + By й V -continuous for every n e JN,
x n
then n is a continuous mapping.
(ii)	If (n ) is a sequence of V „-continuous selections
m me JN	n+2
such that
(CF) For every к e JN there is an N eK such that for all
pairs m^,m2 e JN with m^,m2 > N and all x e X we have
(X) " ^n2(x) e V
holds, then lim defined by (lim rim) (x) = lim rj^tx)
m->-“	m->-“	т->-°°
exists and is V -continuous.
n
(iii) Let (n )	„ be a sequence of selections such that
n n eJN
(1) Пп is Vn~continuous
(2)	nn+1(x) ” Vx) e Vn-1 for a11 n e ]N'
then lim r, exists and is continuous.
n-> n
Proof, (i):	If n(x) = 0 and if V is any neighborhood of 0, then
there is an n e JN so that Bx n vn c V. As n is Vn~continuous, we can
220
find a neighborhood W of x such that п(у) ~ r,(x) = nW _ 0 = r,(y)
e Vn for all у e W. Hence the selection q is continuous at x.
Now assume that r|(x) / 0. Let W be an open neighborhood of x. Then
BX\W is closed and does not contain n(x). Therefore the set
Bv \ B„... =	\ {0} is an open neighborhood of r,(x). We conclude
л X \W W
that Bjj is a closed neighborhood of r,(x) and that
n {B^ : x e W°} = Bx-
Note that this intersection is directed by inclusion.
Moreover, for an element ф e rs(p)' we have
ф e n (n<x) + V ) n Bx iff ф e Bx and ф - n<x) e for all n
n
iff ф = n(x)
Hence the family of sets
{Вд n (n(x) + Vn) : x e W°, n e JN}
is a filtered system of closed neighborhoods in Bv of n(x) having
Л
intersection {r|(x)}, i.e. it is a neighborhood base of n(x) in Bx-
Now let V be any open neighborhood of n(x). Then there is an open
neighborhood of x. and a natural number n e JN such that
Bw n (r) (x) + Vn) с V.
As n is Vn~continuous, we can find a neighborhood W c	of x such
that ri(y) - r|(x) e Vn for all у e W. We conclude that
n(W) c (n(x) + vn) n bw
с (n(X) + Vn) n Bw
С V,
i.e. r, is continuous at x.
221
(ii) : The family (Vn n ^x^neJN is a neighborhood base at 0 in Bx-
Hence, by assumption, the sequence (rim(x))mejj is a Cauchy sequence
in Bx> As Bx is complete (being compact) lim ^(x) exists.
m-»-oo
Define
П (x)	:= lim (x) .
in->-00
Then n is a selection, which is V -continuous:
1	n
Indeed, let x e X. Then there is a natural number N e JN such that
Пщ (x) - rjjn (x) e vn+3 for a11 mi ,m2 * N and a11 x e x- For every
x e X, this implies
nN(x) “ n(x) = nN(x) “ lim r|m(x)
_	m->oo
e Vn+3
c Vn+2•
Now let x e x. As n„ is V .-continuous, there is an open neigh-
o	'N n+2	r
borhood W of x such that пы(у) - г,ы(х ) e V .7 for all у e W. For
О	W	W О	Пт
a given у e W this implies:
n(y) - n(xo) = n(y) - nN(y) + nN(y) - nN(xo) + nN(xo) - n(x)
Vn+2
Vn+2
Vn+2
(iii)
V
n
Firstly, note that for m n we have
nm(x) - Vx)	=	Vx)	-	nm-1(x)	+ nm-1(x) - •••	+	nn+1<x) - Vx)
£	(Vm-2	+	Vm-3> +	Vm-4 + Vm-5 +	+ Vn + Vn-1
c	(V .	+	V .) +	V _+...+ V	+	V .
m-4	m-4	m-5	n n-1
С (V C+V c) + ...+V	+ V .
m-5	m-5	n n-1
V o.
n-2
222
Непсе (ii) shows that lim n = n exists and that n is V -continuous
n	n
n->-°°
for every n e JN. Thus, the selection n is continuous by (i). Q
19.9 Lemma. If f. : X + [0,1], 1
n
such that £ f. = 1 and if n- ’• x
i=1	n 1
uous, then the selection У f. -n.
i=1	1 1
s i < n, are continuous functions
+	1 s i s n, are V^^-contin-
is V-continuous.
ГЛ
Proof. Firstly, note that
n r.B =	{0}.
r>0 X
Hence there is a real number r > 0 such that r-В., c V , Fix
X n+m+1
xq e x. Then there is an open neighborhood W of x such that
(1)	rijjy) ~ hjjx ) e vm+i for 1 - i - n and у e W
(2)	|f^(y) - fi(xQ)| < r for all 1 s i < n and all у e W.
Let у e W be arbitrary. Then we have
( ?	(у) - ( ? f-rii) (x ) =
i=1	i=1 i i °
n
= I (f±(y)*П±(у) - f±(xo)-n±(x ))
n
= Z (f±(y) Т|±(у) - fi(xQ)-ni(y) + f±Cxo) .п±(У) - £±(хо) •гЧ(хо) >
n	n
= I (f±(y) - fi(X ))«T].(y)	+	£ f.(x ).(п±(у) - n.(x )).
i 1 X	X	X	x и x	xv-»
As the f^(xQ) sum up to 1, as n^(y) - ri£(xo) belongs always to Vm+^
and as V . is convex, we obtain
У f.(x ) - (rj. (y) - n . (x ))	eV.,
i о '1 " i о	m+1
As |f±(y) - fi(xo)I < r, we have (f±(y) - f±(xQ))•n±(y) e vm+n+1'
i.e.
(fi(y> - fi(xO)) -^(У)	£ Vm+1
223
Together, this yields
(.^ fi-niMy) - (.^ fi-ni)(xo) e vm+1 + vm+1
C Vm ,
n	m
i.e. У f.-n. isV -continuous. П
. i	m	LJ
1=1
19.10	Let ф e К . Then for every n e JN there is a V -continuous
о
selection n such that
(a)	n(x) e for all x £ X.
(b)	n(xo) = Ф-
Proof. Let
U = {x e X : Kx n (ф + Vn+3) / 0}.
Then U is an open neighborhood of xQ. Moreover, we define Фх = °
if x ф U and фх := ф. If x / x £ U is given, then let фх be
о
an arbitrary element of Kx n (ф + Vn+3)•
Now fix a continuous function f : X + [O', 1] such that f vanishes
on X \ U and takes the value 1 at x . Define
о
П : X	и К
XrX
by	n (x) = f (x) -nx.
Then we have n(x ) = ф. Furthermore, the selection n is V -continuous
О T	n
Indeed, let yQ £ U. As in the proof of (19.9) we choose a real number
r > о such that r«Bv с V Let V с U be any neighborhood of у such
that |f(y) - f(yQ)| < r for all у e V. Then for a given у E V we
have
n(y) - n(yo) = f(y)-ny - f(yo)*ny
= (f(y) - f(yo))-ny + f(yo)-(ny - ny°)
e Vn+3 + Vn+3 c Vn+2 c vn
224
If yQ | U, then f(yQ) = 0 < r. Hence we may find an open neighbor-
hood U' of yQ such that |f(y)| < r for all у e U'. Thus, for
every у e U' we have
n(y) - n(yo) = f(y)-n - f(yo)-nv
= f(У)-ny
e vn+3 c vn- □
19.11	Lemma. Let n e JN be a natural number, let x e X and let
n : X + К be a V -continuous selection. Then there is a V ,,-con-
n	n+1
tinuous selection n' : X + К such that
(i)	n(xo) = n' (xQ)
(ii)	n(x) - n'(x) e Vn-1 for all x e X.
Proof. Let x e X be fixed for a moment. By (19.10) there is a
V --continuous selection n such that n (x) = r,(x). As n is V -con-
П+ to	5C	2C	П
tinuous and as n is V --continuous, there is an open set U around
'x n+2	'	r	x
x such that for every у e Ux we have ri(y) - ri(x) e Vn and
Ч„(У) ~ ri(x) e V	We may assume that x does not belong to U
provided that x x . Moreover, for every у e Ux we have
П(У) “ ПХ(У) = П(у) - n(x) + n(x) - ПХ(У)
= n(y) - n(x) + Пх(х) - ПХ(У)
e Vn + Vn+2"
Now the Ux cover the compact space X; hence there are finitely many
elements x^,...,xn e X such that U и ••• и ux = X‘ si-nce xQOCCursin
1	n
exactly one of the Ux, this element belongs to {x^,...,x };w.l.o.g.
we may assume that x = x.. Let (f.)?_. be a partition of unity
о 1	11-1
subordinate to the covering (U )”=•]• Then f^ (xQ) = 1 and f^(xQ) = 0
for 2 < i < n. We define
n
225
Then n'(x )
о
n(x ). Moreover, the selection n
о	1
is V ,„-continuous
n+1
by (19.9). Finally, for a given x e X let
Jx := {i e {1,...,n} : x e Ux }.
Since x | U implies f.(x) = 0, we have
xi
I fjx) = 1.
ieJx
Moreover, as x e и implies ri(x) - n (x) eV + V n and since
x^	1	n n+£
V + V ,n is convex, we conclude
n n+z
n
n(x) - n' (x) = n(x) - У Г±(х).пх (x)
i=1	i
= У f. (x) • (n(x) - n (x))
i=1	1	xi
=	У	f±(x)	(n(x) -	nx	(x) )
ie J	i
x
vn-i a
We now finish the proof of (19.7):
Firstly, by induction using (19.10) and (19.11) , we find a sequence
of selections (n ) such that
'n' n e JN
(1)	rin is Vn~continuous .
(2)	nn+1 (x) ~ Vx) e Vn-1 for a11 x e x-
(3)	r,n(xo) = Фо for a11 n e 'IN‘
(4)	hn(x) e Kx for all n e JN and all x e X.
Now let n := Um П • Then n exists and is continuous by (19.8) . More'
n->-°°
over, r|(xQ) = фо and n(x) e Kx for all x e X, as Kx is always
closed. □
226
The following proposition states the converse of (19.7):
19.12 Proposition. Let p : E + X be a bundle of Banaoh spaces with
a compact base space X. Define
Kx := {ф e Bx : ф = ij (x) for some
Then К is convex and circled. If
x	J
{x e X : Rx n 0b Moreover, if
then every Kx is closed.
continuous selection r, : X + В }
Л
и C rs(p)' is open, then so is
p : E -> X is a separable bundle,
Proof. Obviously, the set Kx is convex and circled. To show the
closedness of К for separable bundles p : E -> X, let и c Bv
x	n X
and vn с Гд(р) ' , n e JN, be as in the proof of (19.7) . Moreover,
let ф belong to the closure of К . As Bx is metric, there is a
sequence of elements ф e К such that lim Ф = Ф. Picking an
n x	n^“ n
appropriate subsequence, we may assume that Фп+^ ~ Фп e Vn+1 ^or
all n eJN. We define recursively a sequence of continuous selections
П : X В such that
n	X
(1)	Пп(х) = Фп
(2)	nn+1(y) “ Vy) e Vn+1 for а11 П e У e X.
Choose any satisfying (1).
If hn is already	defined, choose any continuous	selection	:	x
+ В such that £	. (x) = ф	. Then we conclude	that г . (x) -	n	(x)
л	Пт I	Пт]	Пт I	П
e vn+1• As 5n+i	an^ hn are	continuous, there is	an open neighborhood
W of x such that	?n+1(y) -	hn(y) e Vn+1 for а11	У e w- Pick a
continuous function f : X + [0,1] such that f(x) = 1 and f(X \ W) =
= {0}. We now define
nn+1 f’^n+1 + (1 f) ,rln‘
227
Then we compute that rin+^ (x) =	(x) = фп+^ • Moreover, the
continuous mapping rin+-j : X + Bx is a selection, as Bx is always
convex.
The fact that V ,, is convex and contains 0 implies that
n+1
(rln+1 “ V (y) = f(y) ,(?n+1 (y) “ %(y)) e Vn+1 for a11 y e x' as
f vanishes on X \ W.
From (19.8(iii)) we conclude that the function q : X + Bx defined
by r|(y) = П (y) is a continuous selection. Obviously, we have
П->-со
r|(x) = lim F|n<x) = lim ф = ф, i.e. К* is closed.
П->°°	П-юо
Finally, let U c rs(p)' be open and assume that Rx n U 0. Then
° ~
there is a continuous selection n : X + Bv such that n(x ) e К n U.
X	О	X _
-1	°
As n is continuous, n (U) is an open neighborhood of xq and by
definition of the Kx, x e X, we have r|(x) e Kx n U whenever
x e n 1(U). This establishes the fact that {x e X : Kx n U / 0} is
open. □
19.13 Theorem. Let p : E + X be a separable bundle of Banach spaces
over a compact base space X. Moreover, let C be the smallest
closed subset of E containing the "unit ball" {a e E :	||a|| s 1 }
and having the property that C n Ex is convex for every x e X .
Finally, let К = (С n E )° с В .
Then for every x e X and every ф e Kx there is a continuous CCX.)-mo-
dule homomorphism T e Mod (Г (p) ,C (X) ) with ||T||	< 1 and
AT(x) = ф.
Converse ly, if T e Mod (Г (p) ,C (X) ) and if ||T|| s 1л then XT(x) e Kx
for every x e X.
Proof. By (16.9) and (16.10), the set К := и К satisfies the
x£X x
properties (i)-(iii) of (19.7). Hence, given a point x e X and an
element Ф e Kx, there is a continuous selection X : X + К such that
228
Х(х) = ф. By (19.1) and (19.2(ii) ) the mapping T : Г(р) + С (X) is a
Л
continuous C(X)-module homomorphism and we have XT (x) = X(x) = ф
Viewing X as a bundle morphism from from E into XxlK, we obtain from
(10.13) the equation
||T ||	= sup < 11 A (y) 11 : у e X}.
Л
As X(y) belongs to By for every у e X, we have always || X(y) || < 1
and therefore ||T|| <	1.
Conversely, again by (16.9) and (16.10) the family (K ) is the
X XfX
largest family such that {x e X : Kx n U 0} is open whenever
U с Г (p)' is open. Thus, by (19.12) the set < ф e Bx : ф = n(x) for
some continuous selection n : X + B„} is contained in К .
Now let T e Mod (Г (p) ,C (X) ) and suppose that ||T|| < 1. From (IQ.13)
we conclude that under these conditions we have for every x e X the
inequality || AT(x) ||	<1. Hence the mapping XT : X + Г(р)' maps
X into В and therefore is a continuous selection by (19.1), i.e.
X
XT(x) e Kx for every x e X. □
As we always have the relation K° = (C n Ex)°° = C n Ex, we obtain
as a corollary:
19.14 Corollary. Let p : E -+ X be a separable bundle of Banaoh
spaces over a compact base spaoe X. Then the set
{a e E: |X(a) | <1 for all bundle morphisms X : E + Xx Ж with || X ||<1 .
= {a e E: |T(a)(p(a))| < 1 for all a e Ftp) with a(p(a)) = a and all
T e Mod(F(p) ,C(X) ) with ||t|| < 1}
is the smallest closed subset of E which is stalkwise convex and con-
tains {a e E : 11a11 < 1}.	□
229
For every x e X let Gx be the largest closed vector subspace of C n Ex
(this space exists as C n Ex is convex and closed). From duality
between Ex and Ex we conclude that the vector space generated by
Kx is dense in G°. Hence we have:
19.15	Corollary. Let p : E + X be a separable bundle of Banaoh
spaces over a compact base space X. Then the stalks of the bundle
-* X defined in (19.2) are dense subspaces of G°.	□
In our next corollary, we characterize bundles with continuous norms
via some properties of Mod(Г(p),C(X)):
19.16	Corollary. Let p : E -+ X be a separable bundle of Banaoh
spaces and assume that the base space X is compact. Then the follow-
ing oonditions are equivalent:
(i)	p : E + X has continuous norm.
(ii)	If x e X and if ф e Ex, then there is a continuous C(X)-mo-
dule homomorphism T e Mod(Г(p),C(X)) such that Лт(х) = ф
and ||T|| = ||ф|| .
(iii)	If a e Г(р), then
norm(o) = sup { |T(a) | :	||T||	< 1, T e Mod(Г(p) ,C(X)) },
where the mapping norm(a) : X +1R is defined by norm(a)(x) =
= || a(x) || , where |T(a) | is defined by |t (a) | (x) = | T (a) (x) |
in IK .
(iv)	|fa|| = sup { ||T(a) II : T e Mod (Г (p) ,C (X) ) and ||T|| < 1}
and the set
К := {ф e Bv : ф = X_(x) for some x e X and some
T e Mod(r(p) ,C(X) ) with ||Т|| < 1}
is closed.
Moreover, under these conditions, the stalks of the bundle
230
ц-P : И? -> X defined in (19.2) are equal to E', x e X, and the
3	x
space Mod(г(p) ,C (X) ) separates the points of r(p)-
Proof. If p : E + X has continuous norm, then {a e E : ||a|| < 1}
is closed and stalkwise convex; thus (i) implies (ii) by (19.13) .
Obviously, (ii) implies (iii).
(iii) + (i): By (iii), the mapping x + ||a(x) || is a pointwise
supremum of continuous functions for every a e r(p) and therefore
lower semicontinuous. As it is always upper semicontinuous, we have
shown (i).
As (iii) implies || a|| = sip { ||T(a) || : ||T|| < 1 and T e Mod(r(p)
C(X))} and as (ii) shows that К = Bv, it remains to establish the
implication (iv) + (ii), i.e. we have to show that under the pre-
sence of (iv) the equation К = Bv holds.
X
Applying (iv), we obtain:
conv К =	K°°
=	{a e Г(р)	:	|ф(а)| <	1 for all ф e K}°
=	{а e Г(р)	:	|T(a)(x)|	< 1 for all x e X	and all
T e Mod(r(p) ,C(X) ) with )|T||	< 1}°
= {аеГ(р): | ] T(a) 11 < 1 for all T e Mod(Г(p) ,C(X)) with
||T||	< 1}°
= {а e Г (p) :	|| a ||	< 1 }°
As К n Bx is closed and convex, we conclude from (15.15(1)) that
К n В = conv(К n В ) = conv(K)n В = В ° n В = В and thus
X	X	X I X X
К = B„. □
A
The bundle constructed in example (16.3) shows that К is not closed
in general, even if we postulate in addition that || a|| =
= sup { ||T(a) ||	: T e Mod(r(p) ,C(X) ) ,	||t|| < 1}.
231
Let us conclude this section with a couple of open problems:
1	.) Suppose that E is Hausdorff and assume that all stalks of the
bundle p : E -> X are finite dimensional. Given a point x e X arid
ф e E^, is there a continuous С(X)-module homomorphism
T e Mod ( Г (p) ,C (X) ) such that XT (x) = ф?
2	.) Let us assume that the set К as it was defined in (19.16(iv)) is
closed and suppose that for every ф e	there is a T e Mod(r(p),
C(X)) such that Лт(х) = ф. Define a norm ||| • ||| on Г (p) by
HI a HI = sup { ||T (a) ||	: T e Mod(r(p),C(X)), ||T || < 1 }.
It is possible to show that Г(р) is a locally C(X)-convex C(X)-
module in this new norm. Moreover, the bundle p' : E' * X re-
presenting Г(р) in this new norm has up to isomorphy (not isometry)
the same stalks as the bundle p : E + X and, in addition, contin-
uous norm.
Question: Is the norm ||| • ||| equivalent to the orginial norm on
Г(р)? Is that true in the case where all stalks are finite dimen-
sional?

20. Internal duality of С(X)-modules
Let E be a Banach space and let be the dual of E equipped with
the topology of compact convergence. From the Mackey-Arens theorem
we know that E is (topologically) isomorphic to (E^)^. In this section
we shall study to what extend these results remain true for the
space Г(р) of all sections in a bundle and its "internal dual"
Modc(Г(p),C(X)). It will turn out that locally trivial bundles are
"internal Mackey spaces" in this sense and that for certain bundles
with continuous norm, the space Г(р) is at least algebraically
isomorphic to its bi-dual Mod(Modc(Г(p),C(X)),C(X)).
20.1 Proposition. Let p : E + X be a bundle and let S be any
directed family of bounded subsets of Г(p) whose union generates
Г(p). If a e Г(p) is a section, then the mapping
5 : Mods(T(p),Cb(X))	- Cb(X)
T	+ T(a)
is a continuous Cb(X)-module homomorphism. □
20.2	Proposition. Let p : E X be a full bundle over a completely
regular base space X and let S be a directed family of compact,
convex and circled subsets of Г(p) whose union generates Г(p).
If Ф : Mod^ (Г (p) ,cb (X) ) Cb(X)is a continuous Cb(X) -module
homomorphism, then there exists a (not necessarily continuous)'
selection s : X * E such that Ф(Т)(х) = AT(x)(s(x)) for all x e X
and all T e Mod(Г(p),С(X)).
Proof. Fix any point x e X and let S(x) := tex(S) : S e S}, where
233
ех : г(р) * Ех is the evaluation map. As ex is continuous, the family
S(x) consists of compact, convex and circled sets and as p : E -> X
is a full bundle, this family covers E^. Therefore the S(x)-topology
on Ex is finer than the weak-*-topology and coarser than the Mackey-
-topology t(E^,Ex). Thus, by the Mackey-Arens theorem, every con-
tinuous linear form on E^, equipped with the S(x)-topology, is of the
form ф + ф(а) , where a e Ex>
As in section 11, we let Nx°d = {Те Mod^(Г(p),С^(X)) :Т(а)(х) = О
for all а e Г(р)}. Applying (11.6) we conclude that the set
{f*T : T e Mod(Г(p),C, (X)), f e С, (X), f(x) = 0} is dense in NMod.
b	b	x
Moreover, if Ф is a (X)-module homomorphism on Mod^(Г(p),С^(X)),
we have 0(f«T)(x) = (f • Ф (T) ) (x) = f (x) • Ф (T) =0 whenever f (x) =0.
Mod
Thus, the continuity of Ф implies Ф(Т)(x) = 0 for all T e Nx
Let я : ModQ ( Г (o) ,C, (X) ) + ModQ(Г(p),C, (X))/NMod be the canonical
О	D	о	D	X
quotient map. Then Ф induces a continuous map
Ф : Mod(F(p),C, (X))/N™°d	- Ж
A	U	X
such that Ф °k(T) = Ф(Т)(х) for all T e Mod^(Г(p),Cb(X)). By the
remarks following (11.20), Mod^(Г(p),Cb(X))/Nx°d may be identified
with a subspace of L„ (e , Ж), i.e. with a subspace of E' equipped
О ( X) X	X
with the S(x)-topology. Under this identification, we have
k(T) = Am(x) and Ф becomes a continuous linear functional on a
T	x
subspace of Ex. Using the Hahn-Banach theorem, we may extend Фх to
a continuous linear functional on E', where E' carries the S(x)-to-
X	X
pology and by the above remarks, there is an element s(x) e Ex such
that Фх(ф) = Ф(э(х)) for all ф e E^.
Hence we have Ф(Т)(х) = Фх»л(т) = Фх(Ат(х)) = AT(x)(s(x)) for all
T e Mod^(Г(p),0^(X)) and x + s(x) : X + E is the selection we were
looking for. □
234
20.3	Remark. It is obvious from the proof of (20.2) that the
section s in unique if and only if {xT(x) : T e Mod(p(p),0^(X)) } is
dense in E' for every x e X.
Thus, the theorem of Mackey-Arens holds "internally"in the category of
С, (X)-modules, if we can show that the selection s : X + E con-
fa
structed in (20.2) turns out to be continuous and bounded. For
separable bundles of Banach spaces with continuous norm this is true.
To obtain a more general result, we need some remarks concerning
equivalent norms:
20.4	Definition. Let p : E + X be a fibered vector space. Two
norms || • ||	: E -»• JR and ||| • ||| : E -»• JR are said to be equivalent, if
there are constants m,M > 0 such that m. 11 a 11 < |||a||| < M. ||a|| for
all a e E. Q
20.5	Proposition. Let p : E + X be a bundle of Banaoh spaces over
a compact base space and with norm || • || . Moreover, let ||| • ||| be a
second norm on E. Then the following statements are equivalent:
(i)	The norms || • || and ||| • ))| are equivalent and the mapping
HI «HI : E + JR is upper semicontinuous.
(ii)	p : E + X is a bundle of Banach spaces with norm ||| • ||| .
(iii) The set
G = и {a e E : |||a||| < 1 }° с и E' c Ftp)'
xeX	x	xeX
is compact with respect to the weak-*-topology and the
closed convex hull G°° of G is a barrel in r(p)'-
Proof. (i) + (ii): Firstly, we show that the set {a e E :
HI a HI < e} is open in E:
235
Let m,M > 0 such that m- 11 a 11 < |||a||| < M. 11 a 11 and let aQ e E such
that H|ao||| < e. Choose any a e r(p) with a(p(aQ)) = aQ and let
U := {x e X : |||a(x) ||| < - -+ IП-^е-Ш-}.
Then U is an open neighborhood of P(aQ) and thus
0	:= {a e E : p(a) e U and 11 a “ a(p (a) ) 11 < -------
is an open neighborhood of a . Moreover, if a belongs to 0, then we
have
11Ы11 s III a(p(a) ) - a III + ||| a(p(a) ) |||
< J. .~_.lll Др-Щ. + e ..+
2	2
= e.
It is now easy to check that p : E + X satisfies the hypothesis of
(3.2) and thus is a bundle.
(ii) + (iii) : The compactness of G follows from (15.3) and by
(15.7(i)) the closed convex hull of G is equal to the dual ball
{a e F (p) : |||a||| = sup {|||a(x) ||| : x e X} s 1 }° c r(p)' and
therefore a barrel.
(iii) + (i) : As G is compact, we can find a constant M > 0 such
that G c {ф e Ftp) ' : ||ф|| s M} and as G°° is a barrel, there is a
constant m > о such that {ф e Г(р) ' :	||ф|| < m} c G°°. From
(15.15) applied to the bundle p : E + X equipped with the norm
we conclude that G°° n = {ф e : |||ф||| < 1}. Hence for every
x e X we have the inclusions
{ф e :	||ф||	< m} с {ф e E^ : |||ф||| < 1}
с{феЕ^: ||ф||	< M}.
236
If we take polars, we obtain
{a e E :	||a|| s c {a e E : |||a||| < 1 }
c {a e E :	||a|| s
or M. 11 a 11 s |||a||| - m * l|a|| f°r aH a e E. It remains to show
that the mapping |||*||| : E + ]R is upper semicontinuous, or, equiva-
lently, that the mapping x -+ ||| u(x) ||| : X->- ]R is upper semicontinuous
for every a e Г(р).
Thus, let a e Г(р). It is enough to prove that the set {x e X :
s HIa(x) HI > M} is closed in X, where M > 0 is defined as above.
Firstly, note that ||| a(x) ||| г M if and only if |ф(а(х) ) | > 1 for an
appropriate element ф e with |||ф||| s j-j- If we let
Аа = {а e Г(р)' : |ф(а)| > 1},
then we have
{x e X : HI a(x) HI > M} = {x e X : 1-G n n / 0}.
By the definition of M, the set 1»G is contained in В and therefore
M	X
j^«G n is a compact subset of B^ not containing 0. Since the mapping
у : В \ {0} X
A
ф + x iff ф e
is continuous by (15.4), the image of j^-G n A under Y is compact in
м а
X and therefore closed. Since this image is exactly the set {x e X:
III a(x) HI M}, our proof is complete. □
20.6 Proposition. Let p : E + X be
a compact base space X and with norm
then the mapping
a bundle of Banach spaces over
|| • || . If T c Ftp) ' is a barrel,
237
III-III : E
sup { | ф(а) | : ф e T n	}
is an upper semicontinuous norm on E which is equivalent to || • || .
Moreover, if for each weak-*-open subset U c r(p)' the set
{x e X : U n Ex n T 0} is open in X, then ||| • ||| : E + 3R is contin-
uous .
Proof. As T n Ex is a barrel in E^, the mapping ||| • ||| induces a
norm on Ex, i.e. the mapping |||«||| : E +1R is indeed a norm. Moreover
for every x e X we have
<а e Ex : |||a||| < 1 }° = T n E^,
i.e. the set G defined in (20.5(iii)) is equal to и T n E'.
x£X x
Let r > 0 be a constant such that r-T с {ф e Г(р)' : ||ф|| < 1}- Then
r«T is a weak-*-compact subset of Г(р)' and hence r-T n Bv is com-
pact, too. As G = ~(r*T n Bv) , the set G is also compact. Further let
x	X
s > 0 be a real number such that {ф e Г(р)' :	||ф||	< 1} c s-T. Then
we may conclude that В c s-G and therefore
X
{ф e Г(р) ' :	||ф||	< 1 }
c s -T.
This shows that G°° is a barrel.
The last statement follows from (15.11).	□
The following corollary may be viewed as a complement to (19.16):
20.6 Corollary. Let p : E + X be a bundle of Banach spaces and
assume that the base space X is compact. If the set
238
К = {ф е Вх : Лт(х) = ф for some х е X and some Т е Mod(r(p),С(X))
with ||Т || <	1 }
is dosed, and if conv К is a barrel, then there is a continuous
norm HI-HI : E -> 1R which is equivalent to || • || .
Proof. From (15.15) we conclude that К = и (conv К) n E'. Thus
xeX
(20.6) follows from (20.5), as (19.12) yields that for every open set
U с Г(p)' the set {x E x : К n n U / 0} is open. □
We now come to the following theorem of the Mackey-Arens type:
20.7 Theorem. Let p : E + X be a bundle of Banach spaces over a
compact base space X and assume that there is a closed subset
К of Г(р)' such that
0 e К с {феВ:ф= X (x) for some x e X and some
X	1
T e Mod ( Г (p) ,c (X) ) with ||T|| < 1}
Assume moreover that E^ n К is convex and circled for every x e X and
that conv К is a barrel in Г(p) ' •
Then the mapping
~ : Г(р)
Mod(Mods(r(p) ,C(X)) ,C(X))
5 ;	5(T) := T(a)
is an isomorphism of C(X)-modules, where S denotes any family of
compact subsets of Ftp) such that the union of S generates Г(p) •
Proof. From Q.5 .15) we know that К n E^ = (conv K) n E^, i.e.
К n E^ is a barrel in E^. The injectivity of ~ now follows easily
from (20.3).
239
Let ф : Mod^(Г(р),C(X)) + С(X) be a continuous C(X)-module homo-
morphism. Since in the Banach space Г(р) the closed convex hull of a
compact subset is compact, Ф is also continuous for the finer topolo-
gy of compact convex convergence on Mod(Г(p),C(X)). From (20.2) we
may now conclude that there is a selection s : X > E such that
Ф (T) (x) = Л (x) (s (x) ) for all x e X and all T £ Mod (Г (p) ,C (X) ) . It
remains to show that s is continuous.
Firstly, by (20.6) we may assume without loss of generality that
11 a 11 = sup {|ф(a) | : ф e conv К n Ep (a)
= sup {|ф(а) | : ф e К n Ep(a) }
and that under these conditions we have K = B . Further, by the choice of
X
K, for every ф e we can find a continuous C(X)-module homomorphism
T e Mod(Г(p),C(X)) with XT (x) = ф, although we are no longer allowed
to assume that ||t|| < 1.
Define a mapping
IK
s ; u Ex
xtX
s
-» s ;	§(ф) = ф(э(х)) if ф e E^.
Step 1 The restriction of s to В is continuous.
X
(Let MP := {(х,ф) : ф = XT(x) for some T e Mod(Г(p),C(X))} c Xxr^(p)'
and let туР :	+ X be the restriction of the first projection. As
we just remarked, {x}xe^ с mP. By (19.2), jjP :	+ X is a bundle and
the mapping
Л : Mod (Г (p) ,C (X) ) Г(яР)
T	A(T) ,- A(T) (x) = (x,XT(x))
is a topological isomorphism of С (X)-modules. As Ф°Л : Г(ттР)
C(X)
240
is a continuous C(X)-module homomorphism, we can find an unique
bundle morphism ц : MP + Xx Ж such that
( Ф°Л 1 ) ( X) (x)	= (pr2 oyoj) (x) for all x e X, X e r( irP) •
As ц preserves stalks and as the stalks of are just the {xJxE^,
x e X, we can find a linear map цх :	+ Ж such that ц(х,ф) =
= (х,цх(ф)) for every ф e E^.
Now suppose that T e Mod(г(р),C(X)). Then we may compute:
XT (x) (s (x) )	= ф(Т) (x)
=	Ф°Л"1(Л(Т))(x)
= pr2 °ц(Л(Т) (x))
= pr2»u(x,XT(x))
= Pr2(x'Ux<AT(x)))
= ux(AT(x)).
As the elements of the form X (x), T e Mod ( г (p) ,C (X) ) cover E', we
X	x
have
рг2°ц(х,ф) = цх(ф)
= ф(s (x) )
= Sopr (х,ф)	for all ф e E'.
Restricting ц to the set {(х,ф) : ф e Bx} c XxBx we obtain the
following commutative diagram
u
{(х,ф) : ф e В , x e X} + Xx Ж
pr2 +	+ Pr2
BY	+	Ж
X	§
It follows easily from (15.4) that the set {(х,ф) : ф e Bx, x e X}
241
is compact, if we equip Bv with the weak-*-topology. Since the subset
X
Bv с Г(р)' is equicontinuous, the weak-*-topology and the S-topology
on Bx agree. Thus the set {(х,ф) : ф e Bx, x e X} is compact in the
relative topology of M? and therefore the projection pr? : {(х,ф) :
: ф e В , x e X} Bv is a quotient map as it is a continuous sur-
X	X
jection between compact spaces. As ц and p^ : Xx Ж Ж are contin-
uous, the continuity of § follows.)
Step 2. If s(xq) = 0, then the set {x :	||s(x) ||	< e} is a neigh-
borhood of x .
о
(Let
U := {ф e Bx :	| S(ф) | < e /2}.
Then U is open by step 1. Moreover, s(xq) = 0 implies ё(ф) -
= ф(з(х )) = 0 for all ф e В , i.e. Bx c U. Hence the set {x e X :
о	о
: Bx c U} is an open neighborhood of xQ by (15-6). As for every x e X
with Bx c U we have || s (x) || = sup { | ф (s (x) ) | : ф £ Bx} =
= sup { | §(ф) | : ф e Bx} < e/2 < e, the larger set {x e X : || s(x) || <
< e} is a neighborhood of xq, too.)
Step 3. The mapping s : X + E is continuous.
(Let xQ e X and let 0 be an open neighborhood of s(xq). Pick any
section a e Г(р) such that a(xQ) = s(xq). Then there are an open
neighborhood U of xq and a real number e > 0 such that
{a e E : p(a) e U and ||a
a (p (a) ) 11 < e} c 0 •
To show the continuity of s, it is enough to check-that the set
{x e X :	11 a(x) - s (x) |[ < e}
is a neighborhood of xQ.
242
Let Ч1 = 5 - Ф : Mod(Г(р),С(X)) + С (X) . Then there is an unique
selection r : X + E so that ч'(Т) (x) = xT(x) (r (x) ) for all x e X and
all T e Mod(Г(p),C(X)). An easy computation shows that r = 0 - s
and therefore step 2 applied to r instead of s completes the proof. □
20.8	Corollary. Let p : E + X be a separable bundle of Banaoh
spaces with continuous norm and a compact base space. Then the
mapping
Г(р) -
Mod(Mods(Г(p),C(X)),C(X))
a
a
is a bijection. □
Our next corollary deals with locally trivial bundles. Firstly, how-
ever, we need a lemma:
20.9	Lemma. Let p : E + X be a bundle of Banaoh spaces over a
compact base space X and suppose that the bundle p : E -> X, viewed
as a bundle of topological spaces, is locally trivial. Then there is
a compact subset A c Modg(Г(p),C(X)) suoh that
(i)	К = {XT(x) : x e X and T e A} is compact
(ii)	О e К and К n is circled and convex for every x e X.
(iii)	conv К is a barrel in r(p)'.
Proof. Let A.,..., A be closed subsets of X such that the interiors
1 n
of the A. cover X and such that Гд (p) is isomorphic to C(A,E) as a
Ai
topological vector space for a certain Banach space E . Since the restric-
tion map
eA. : r(p) " rA.(p)
a -> a /
'A.
i
243
is a quotient map by (4.5)
we may embed C(A^,E)' into r(p)' via the
mapping
e± :	С(А±,Е)'
(ф)(a)
+ Г(Р)'
= Ф°31°ед (°)
where S. : гд (p) + C(A.,E) is a suitable continuous and open C(X)-
1 Ai	i
-module isomorphism. For every i e {1,...,n} and every ф e E' we de-
fine a mapping
: A.
Ф 1
* С(А±,Е)
where
Пф(х)(т) = Ф(т(х)) for all т e C(A.,E).
It is straightforward
to check that n, is
Ф
е1°^ф
maps x into E'
where
as usual, Ex
continuous and that
p 1 (x) .
For every i e {1,...,n} let f^ : X + [0,1] be a continuous function
such that f^ vanishes on X \ A? and such that
max f.(x)	=	1	for all x e X.
1 ^i<n 1
If we define
rs(P)'
by
r f . (x) • (e. °nJ (x) if x e A.
A,	.(x)	= /	1	1 *
<0	if x i A±
then X , . is continuous and X , .(x) e E' for every x e X.
ф, i	ф, i	x	-1
For every 1 s i < n we define a mapping
244
m± : Е' л Mods(r(p),C(X))
ф ->	T •
V
An easy calculation shows that rtb is linear. Moreover, we have
||T	(a) ||	<1 iff | X . (x) (a(x) ) | <1	for all x e X
лф,1	Ф'
iff )f±(x) (e. "nф(х)) (a(x)) | <1 for all x e A±
iff | (e. °n ) ( (f i -a) (x) ) | <1 for all x e A.
'	1 ф	X	'	J-
iff Iф((S±оЕд (f±•ст)) (X)) I <1 for all x e A±
As the set {[(SpeA ) (f^a) ](x) : x e A^} is compact in E, we con-
clude that itb is continuous if E' carries the topology of compact
convergence. As the set {ф e E' : ||ф|| < l}is compact in this
topology, the image
В,	:=	{T.	: || ф !| < 1 }
1	Л X
Ф,1
of the unit ball of E' under iru is compact, too.
Now let В = B^ и ... и Bn and let A be the closed, convex, circled
hull of B. As Г(р) and C(X) are Banach spaces, the space L (Г(р),С(Х))
of all linear operators from Г(р) into С(X) is quasicomplete and so’
is its closed subspace Mod (Г(p),C(X)). Thus, A is a compact convex
and circled subset of Mod (Г(p),C(X)).
Obviously, 0 = T, e A.
0, i
Let
К = {Лт(х) : T e A, x e X}.
Then, by definition, for every i e {l,...,n} and every ф e E' we
have A, . e К, О e К and К n E is convex and circled for every x e X.
Ф, i	x	-1
Moreover, the set К is compact: Since A is compact, it is enough to
245
show that the mapping (x,T) + XT(x) : ХхД + К is continuous.
Let (x^,T^). be a converging net in ХхД and let x = lim x^ and let
£	id
T = lim T.. We show:
iel 1
\,(x)	= lim X (x ) .
iel i
Indeed, let a e Г(р). As lim T^ = T, there is a j & I such that
iel
||Т±(ст) - T(a) || < e/2 for all i % j Moreover, as T(a) e C(X),
there is a j > j such that |T(a)(x) - T(a)(x^)| < e/2 for all
i > j2- For all i > j2 > j we have
I AT (x±) ( a) - XT (x) ( a) |	= lTi<o)(x±) - T(a)(x)|
< |Ti(a)(xi) -Т(а)(х£)| + |Т(а)(х±) -
- Т(а) (x) |
<	||Т± (а) - Т(а) || + е/2
<	е/2 + е/2 = е.
It remains to show that conv К = K°° is a barrel, i.e. that K° is
bounded in Г(р).
For 1 s i < n let
B± = {x e X s f±(x) = 1}.
Then c A? and the cover X as we have chosen the f^ so that
max {f^tx) : 1 < i < n} = 1 for all x e X.
Let 6. : Г. (p) + Г_ (p) and 6! : C(A.,E) + C(B.,E) be the restric-
1 A £	D£	11	1
tion maps. Then 6^ and 6^ are quotient maps by (4.5). Moreover, there
is a (topological) isomorphism R. : r„ (p) + C(B.,E) such that the
1 В £	1
diagram
Г (p)
ГА.(Р>
rB,(p)
С(А±,Е)
+ 6^
С(В±,Е)
246
commutes for every i e {1,...,n}.
Given x e and a e K°, we compute
|| ( (R. °eR ) (a) ) (x) || = || ( (6'oS. oE ) (a) ) (x) ||
11 1 X5	11	11 11 H.	1
= I, ( (S£°ea ) (a) ) (x) || by the definition of 6^
= sup {|ф(Ц31»Ед ) (a)) (x)) |: ф £ E', ||ф|| < 1}
=	sup { | n , (x) ( (S oE ) (ф) ) | : ф e E' , || ф || < 1 }
<p	i
by the definition of n,
Ф
=	sup { I (е± (Пф (x) ) ) (ст) I : ф e E', ||ф|| <	1}
by the definition of e^
=	sup <|Аф^±(х) (a)	: ф e E', ||ф|| < 1}
since f^(x) =1 on
s	1	since X, . e К and as a e K°
ф,1
Thus, we conclude that llR^ogg (a) || < 1 for every a e K°. Since R^ is
a topological isomorphism, there is a constant > 0 such that
|| eB (a) || < Mi for every a e K°. Let
M = max {M.,.. . ,M }.
1	nJ
As the B^ cover X, we conclude that ||a|| < M for every a e K° and
our proof is complete. □
20.10 Corollary. Let p : E + X be a bundle of Banaoh spaces over
a compact base spaoe and assume that p : E -> X, viewed as a bundle of
topologioal vector spaces, is locally trivial. Then the mapping
~	: Г(р) Mod (Mod$ (Г (p) ,C (X) ) ,C (X) )
is a bijection, where S denotes any family of oompaot subsets of
Г(p) suoh that the union of S generates Г(р).	□
For bundles with finite dimensional stalks we have the following
result:
247
20.11 Corollary. Let p : E + X be a Banaoh bundle with a oompaot
base spaoe X and assume that all stalks are finite dimensional.
(i)	If the mapping ~ : r(p) + Mod (Mod^ ( Г (p) ,C (X) ) ,C (X) ) is bijec-
tive, then E is a Hausdorff space. Further, for a given x e X
and a given ф e there is a continuous С (X)-module homomor-
phism T e Mod(Г(p),C(X)) such that Лт(х) = ф.
(ii)	Conversely, assume that E is Hausdorff. If the base spaoe X is
metrizable or if there is an n e JN suoh that dim	= n for
all x e Xj then the mapping ~ is a bijection.
Proof. (i) From (20.3) and the fact that E^ is finite dimensional
for every x E X it follows that E^ = {XT(x) : T e Mod(Г(p),C(X))}.
Now (19.3(ii)) yields that E is Hausdorff.
(ii)	If X is metrizable, then p : E + X is separable by (19.5(iii))
and therefore is a bijection by (20.8).
Now suppose that dim E^ = n for all x e X, where n e JN is fixed. Then
p : E + X is locally trivial by (18.5). In this case, (ii) follows
from (20.10).	□
20.12	Definition. Let p : E + X be a bundle. Then p is called a
Maokey bundle, provided that the mapping ~ : Ftp) + Modcc(Modcc(Г(p),
C(X)),C(X)) is a homeomorphism, where the subscript "cc" refers to the
topology of uniform convergence on compact, convex circled subsets. □
20.13	Remarks. It is easy to see that ~ is continuous whenever
every compact convex and circled subset of Mod(г(p),C(X)) is equi-
continuous. This is especially the case if p : E + X is a bundle of
Banach spaces.
Before we give a very meager set of examples of Mackey bundles, we
248
shall establish:
20.14	Proposition. Let p ; E + X be a bundle of Banach spaces over
a compact base space X. Moreover, let \ c Modg(г(p) г С(X)) be a com-
pact subset.
(i)	The set Кд = {XT(x) : x e X, T e A} is compact.
(ii)	p : E + X is a Mackey bundle if and only if there is a compact
subset A c Modg(г(p),C(X)) such that the closed convex circled
hull of Кд is a barrel.
Proof. (i) was already shown in the proof of (20.9).
(ii): Suppose that p : E + X is a Mackey bundle. Then the mapping
: Г(р) + Modcc(Modcc(г(p),C(X)),C(X)) is open. Hence we can
find a compact, convex and circled subset A c Modcc(г(p),C (X) ) such
that j|T (a) || < 1 for all T e A implies 1| a 11 < 1. Clearly, the
set A is also compact in Modg(г(p),C(X)) and it remains to show that
K? is bounded.
A
Thus, let a e K°. We show that |la|| < 1- Indeed, as a e K°, we
A	A
know that |T (a) (x) | = | AT (x) (a (x) ) | < 1 for all x e X and all T e A,
i.e. ||T(a) || < 1 for all T e A and therefore || a || < 1 by the
choice of A.
Conversely, suppose that A c Mod^ (Г (p) ,C (X) ) is given such that the
closed convex circled hull of Кд is a barrel in r(p)'• We may
suppose that A is circled. Hence the set Кд is circled, too, and
the closed convex circled hull of K. is equal to K°°. Since K?° is a
A	A	A
barrel, there is a constant M > 0 such that ||ф|| < M implies
ф e K°° for all ф e Г (p) ' . Now let us assume that ||T(a) || s 1 for
all T e A. Then we may conclude that |XT(x)(a(x))| < 1 for all x e X
and all T e A, i.e. a e K°. This implies |ф(а)| < 1 whenever
249
||ф|| < М, i.e. || а || < 1. Thus, we have shown that ||T(a) || < M for
all T e A implies ||a|| <1.
As every compact subset A c Modg(Г(p),C(X)) is compact in the strong-
er topology Modcc(Г(р),c(X)) and as the closed convex circled hull
of A is also compact, we just verified the openess of ~. Since the
map ~ is continuous by (20.13) and since it follows from (20.7) that
is bijective, the proof of (20.14) is complete. □
20.15	Examples. (i) If X is compact and if p : E + X is a locally
trivial bundle of Banach spaces, then p is a Mackey bundle (see
(20.9))and (20.10)).
(ii) If p : E * X is a bundle of Banach spaces, if X is compact,
if E is Hausdorff and if all stalks have dimension n for a fixed
n e JN, then p : E * X is a Mackey bundle (see (18.5), (20.9), (20.10)).
20.16	Proposition. Let p : E + X be a Mackey bundle of Banach
spaces, where X is compact. Then for every x e X and every ф e
there is a eontinuous C(X)-module homomorphism T e Mod(Г(p),C(X))
such that lT(x) = фл i.e. the stalks of the bundle	X
representing Mod^(Г(p),c(X)) are isomorphic to E^, x e X, where S
denotes again a family of compact subsets of Г(p) whose union gene-
rates Г (p) .
Proof. By (20.14) there is a compact subset A c Modg(Г(p),C(X))
such that the closed convex circled hull of К = {Am(x) : x e X and
A T
T e Aj is a barrel in Г(p)'. As Г(p) and C(X) are Banach spaces, the
closed convex circled hull of A is compact, too. Hence we may assume
w.l.o.g. that A is convex and circled. Moreover, by [Sch 71, III.4.2]
the set A is equicontinuous. Therefore, by multiplying A with a
suitable constant M > 0, we may assume that ||t|| s 1 for all T e A.
Under these conditions we have
250
(1)	К. с В„ and К. is closed
Ал	А
(2)	Кд п Вх is closed, convex and circled for every x e X.
(3)	K°° = conv K, is a barrel in r(p)'•
A	A
where the closedness of Кд nBx follows from (2O.14(i)).
Applying (15.15), we are allowed to conclude that кд n Bx =
= (conv Кд) n Bx is a barrel in for every x e X.
Now let x e X and let ф e	be given. Then there is a real number
r > 0 such that г.ф e Кд n Bx- By the definition of Кд we can
find a continuous C(X)-module homomorphism T' e A such that
г.ф = XT,(x). Let T := l.T'. Then we finally have ф = XT(x). □
20.17	Theorem. If p : E -+ X is a Maokey bundle of Banaoh spaces
over a oompaot base spaoe with norm || • || j then there is an equival
ent norm ||| • ||| on E suoh that for every x e X and every ф e E'
о
there is a T e Mod (г (p) ,C (X) ) with At(xq) = ф and |||ф||| = ||[T||| .
Furthermore, the mapping ||| • ||| : E +1R Й continuous.
Proof. As in the proof of (20.16), let A c Modg(г(p),C(X)) be a
compact, convex and circled subset such that
(1)	К. c B„ and K. is closed.
AX A
(2)	Кд n Bx is closed, convex and cirlced for every x e X.
(3)	K°° = conv K. is a barrel in r(p)'•
A	A
Again, proposition (15.15) implies
(4)	Кд n Bx = (conv Кд) n Bx for each x e X.
We now define a norm on E by
НЫН = sup { | ф (a) I : ф e Кд П Bp(a) }.
251
As it was shown in (20.6), the mapping |||.||| : E +JR is an equiva-
convex, circled and closed, we
longs to {a e Ex : |||a||| s 1}°
= К, n В for a certain x e X.
A x
have HIфHI < 1 if and only if ф be
if and only if ф e (Кд n Bx)°° =
Next, we show that |||T||| < 1 for every T e A- Indeed, if a e r(p) is
given such that |||a||| s 1, then we compute
|| |T (a) HI = sup {|T(0)(x) | : x e X}
= sup { I Хт (X) ( a (X)) I : x e X }
< 1
as Хт(х) e Кд n Bx and as a(x) e (Кд n Bx) °.
Further, let us start with x e X and let ф e E' \ {0}. Then the
o	Y x
“1	о
element |||ф||| «ф belongs to К. n В and therefore is of the form
A xo
Xc(x ) for a certain S e A. Since |||S||| < 1 by the above argument
О О
and since
|||S|||	= sup { HI As (x)|||: x e X}
* il|Xs(xo)|||
= 1НФ|||
= 1 ,
by (10.13) (recall that Xs may be considered as a bundle morphism
between E and the trivial bundle XxJKi),we conclude that |||S||| = 1.
Now define T e Mod (Г (p) ,C (X) ) by T := |||ф||| *S. Then XT (xq) = ф and
IIItIII = 111Ф|Ц.
Finally, the continuity of |||*||| : E + ]R follows exactly as in the
proof of (19.16, (ii) + (iii) +
□
(i) ) .
21. The dual space Г(р)' of a space of sections
In the proceeding sections we always used the dual space r(p) ' of
a space of sections in a bundle to construct subbundles, C(X)-module
homomorphisms etc. In this last section, we would like to
reverse these questions: Suppose that we already know the "intern
dual" Mod(г(p),С(X)), what can be said about r(p)' itself? Of course
we can expect reasonable answers only if Mod(г(p),C (X) ) is large
enough to separate the points of r(p) and it will turn out that we
need more than this.
Our first observation is the following: Given a C(X)-module E, then
the dual space E' is also a C(X)-module, if we define a multi-
plication on E' by
(f-ф) (a) = ф(Е«а)	for all f e Cb(X),a e Е,ф e E'
but there is no reason to expect that E' is locally C(X)-convex even
when E has this property. An example for this phenomenon is E = C(X)
itself.
On the other hand, given a C(X)-module homomorphism T : E + C^tX) ,
we may map the dual space M(X) of C^tX) into E' via the function
T0- : M(X)
E '
ц	тдц , Т0ц(а) = ц(Т(а))
and in certain cases the images of this mapping will generate E'.
In these cases, we shall obtain something close to a "integral
representation" of linear functionals on E. If p : E + X is a bundle
253
a typical linear functional looks like
T0U : Г(р) Ж
a
f lT(x)(a(x)) du
X
where T e Mod(Г(p),C(X))
and where ц £ M(X).
21.1 Definition. (i) Let E,F,G be topological vector spaces and
let b : ExF + G be a bilinear mapping. If b is separately contin-
uous on ExF and if for every bounded subset В c F the family of
linear maps b(-,u) : E + G, u e B, is equicontinuous, then b is
called hypoeontinuous.
(ii) If E and F are C^tX) -modules and if b : EXF -> G is bilinear
and satisfies
b(u,f«v) = b(f«u,v)	for all u e E, v e F, f e C^tX)
then we say that b is compatible with the C^tX)-module structure.
(iii) If in addition G is an C(X)-module, too, and if
b(u,f«v) = b(f«u,v) = f«b(u,v) for all u e E, v e F and
f e Cb(X),
then we call b a bilinear mapping between Cb(X)-modules.	Q
In the following we denote the dual space of C^tX), where X is
a topological space, by M(X).
Again, if E is a topological vector space and if S is any directed
family of bounded subsets of E, we denote the topology of uniform
convergence on elements of S defined on a space of mappings with
domain E by adding the subscript S.
21.2 Proposition. Let E be a topological Cb(X)-module, let S be
a directed and total family of bounded subsets of E and let S' be
254
be a directed family of bounded subset of Cb(X) whose union generates
C^tX) . Assume that {T(S) : S £ S , T e Mod(E,C(X)) } is contained in
S' .
(i) The mapping
S : Mods(E,Cb(X))xMs,(X)	-
(Т,ц)	ТЙц; (Тйц) (a) = ц(Т(а))
is a hypocontinuous bilinear mapping between C^tX)-modules
(ii) If S covers Ej and if a e E is given, then the mapping
: Mods(E,Cb(X) ) M (X) Ж
(т,ц)	ц(т(а))
is hypocontinuous, bilinear and compatible with the Cb(X)~
-module structure.
(iii) If Mod(E,Cb(X))separates the points of Ej then the linear
span of the image of S is a(E',E)-dense in E'.
Proof, (i) For all f e Cb(X), all
T e Mod(E,C, (X)) we have
b
((f.T)Su)(a) = y((f«T)(a))
= y(f«(T(a)))
=	(f-y)(T(a))
= (T S (f • у)) (a)
= y(T(f-a))
= (Тйц)(f•a)
= (f•(Тйц))(a)
a e E, all ц e M(X) and all
by the definition of the mult
plication on Mod(E,Cb(X))
by the definition of the mult
plication on M(x)
as T e Mod(E,Cb(X))
by the definition of the mult
plication on E'
establishing the fact that S is a bilinear mapping between C(X)-mo-
255
dules.
If we fix T e Mod(E,(X)), then
TS- : Ms, (X)
is continuous. Indeed, if S e S , then the set S' := T(S) belongs to
S'. Moreover, if p e M(X) belongs to the open neighborhood
{v e M(X) : |v(f) | < 1 for all f e S' }
of 0 in M (X) , then we have the inequality | (T pi) (a) | = |pi(T(a)) | <
< 1 for all a e S. Since the set
{ф e E' : |ф(a)| <1 for
is a basic neighborhood of 0 in EJ,
T0- is continuous.
all о £ S}
, we have shown that the mapping
Now let А с M$,(X) be bounded. As S' covers C^tX), the corollary to
[ Sch 71, III.3.4] yields an M > 0 such that ||ц||	< M for all ц e A.
Hence, if S e S is given and if T e Mod(E,С. (X)) satisfies
D
IIT (a) || <1 for all a e E, then for every a e S and every ц e A
we have |Tpi(o)| = |pi(T(cr))| < || ц || • || T (a) || < M.l = 1 showin9
the equicontinuity of the set {-gp : ц e A}.
The proofs of (ii) and (iii) are now straightforward. □
The following results state a converse of (21.2(ii)):
21.3 Proposition. Let E be a topological C^tX)-module and let
S and S’ resp. be directed and covering families of compact subsets
of E and C^fX-Jj resp. Furthermore, suppose that the mapping
256
Е ->
Mod(Mod (Е,С, (X))
S b
a
a ; a(T) = T(a)
is bijective. If
b : Mod (E,C, (X) ) xM , (X) Ж
S b s
is bilinear, hypoeontinuous and compatible with the C^tX)-module
structure, then there is a unique e E such that
Ь(Т,ц) = y(T(ab)) for all (Т,ц) e Mod(E,Cb(X))xM(X)
i.e. we have b = b.
%
Proof. Fix a T e Mod (E ,Cb (X) ) . Then the mapping ц -»• Ь(Т,ц) :
М^, (X) + Ж is continuous and linear. Hence there is a unique
Ф(Т) e Cb(X) such that b(T,p) = ц(Ф(Т)). As b is bilinear, the
mapping Ф : Mod(E,Cb(X)) + Cb(X) will be linear. Moreover, this
mapping is a Cb(X)-module homomorphism, as the following calculation
shows:
For all ц t M(X) we have
U(Ф(f-T))
b(f-Т,ц)
b(T,f.Ц)
(f-ц)(Ф(T))
u(f-Ф(Т) ) ,
i.e. ®(f»T) = f-Ф(Т).
Further, if we equip Mod(E,Cb(X)) with the S-topology, then Ф is
continuous. Indeed, we have
||Ф(Т) II <	: 1	iff	|Ц(Ф(Т))| < 1	for	all ц with	Hull * 1
		iff	| b (T, pi) I s 1	for	all ц with	Hull * 1
		iff	T e {T' : Ib(T,	u) I s	1 whenever	Hull -<1}
257
But this last set is open by the hypocontinuity of b. As the mapping
is surjective, there is a ab e E such that Ф(Т) = Т(д^) for all
T e Mod(E,Cb(X)) and for this ab we have Ь(Т,ц) = ц(Ф(Т)) = ц(Т(аь))-
The uniqueness of ab follows easily from the injectivity of .	□
21.4	Proposition. Let p : E + X be a Mackey bundle and let G be
a topological C^tX)-module. If
b : Modcc(r(p) ,Cb(X) ) xMc(X) + G
is a hypocontinuous bilinear mapping between -modules, then
there is a unique continuous C^tX)-module homomorphism
Sb  Gc - Г(Р>
such that ХоЬ(Т,ц) = n(ToSb(X)) for all T e Mod(Г(p),Cb(X)), all
ц e M(X) and all X e G' .
Proof. Let X e G'. Then, applying
Sb (X) e Ftp) such that (Х»Ь)(Т,ц) =
Sb : G' + r(p) is linear. Moreover,
f e Cj^CX) and X e G’, we compute
(21.3), we can find a unique
ji(T<>Sb(X)). Obviously, the mapping
given ц e M(X) , T e Mod ( Г (р),<^ (X) )
(T(Sb(f-X))	=	(f-X)ob(T,y)
=	X(f-b(T,y))
=	X(b(f-T,y))
= u(f-T(sb(X))
= u(T(f-Sb(X)).
As this holds for all ц and all T and as Mod(Г(p),Cb(X)) separates
the points of Ftp), we conclude that
S. (f-X)
D
f•Sb(X)
for all f e С, (X), X e G'.
b
258
i.e. is a (X)-module homomorphism.
It remains to show that Sb is continuous. Let U c r(p) be an open
neighborhood of 0. As p : E + X is a Mackey bundle, we may assume
that U is of the form
U = {a e r(p) : !|T < o) || < 1 for all T e A},
where A c Modcc(p(p),Cb(X)) is a compact convex and circled subset.
Define В := {ц e M(X) : ||ц|| < 1}. Then the restriction of b to A*B
is continuous. Thus, the image of A*B under b is compact in G and an
easy calculation shows that S^tx) e U if and only if x e {y e G' :
|у°Ь(Т,ц)| < 1 for all (Т,ц) e A*BJ. Since the latter set is open in
G'c, this shows the continuity of S^. Q
We are now in the position to identify the dual space r(p)' of r(p)
as a certain "tensor product" in the category of (X)-modules:
21.5	Theorem. Let p : E + X be a Maokey bundle and let G be a
quasicomplete topological c^tX)-module. If
b : Modcc(r(p) ,Cb(X) ) xMc(X) G
is a hypocontinuous bilinear map between c^tX)-modules, then there
is a unique continuous C^tX)-module homomorphism
b : Г (p) '	+ G
cc
such that Ь(ТВц) = Ь(Т,ц) for all T e Mod(Г(p),Cb(X)) and all
ц e M(X).
Moreover, if p : E + X is in addition a bundle of Banaoh spaces
(more generally: if Ftp) is barreled or bornological), then Г(р)дС
is uniquely determined by this property in the following sense:
259
Given a quasioomplete topological C^tX)-module т and a hypooontin-
uous bilinear mapping between C^tX)-modules
T : Modcc(r(p) ,Cb(X) ) xMc(X) -» T
such that every hypooontinuous and bilinear mapping between Cb(X)-
modules b : Modcc ( г (p) ,Cb (X) ) xMc (X) -+ G into a quasioomplete topological
C^tX)-module, is of the form b = Бот for a unique continuous C^tX)-
module homomorphism Б : T + G, then T is (topologically) isomorphic to
r(p)ic-
Proof.	Let Sb : G’ + Г (p) be the mapping constructed in (21.4) and
let b : Г(p)' + (G^)' be its adjoint. As G is quasicomplete, the
topology of compact convergence on G' is the Mackey topology. Hence
we obtain (G^)' s G and b will be continuous for the Mackey topolo-
gies on Г(р)' and G resp. As the original topology on G is coarser
than the Mackey topology t(G,G'), we conclude that b : r(P)gC + G is
continuous.
Clearly, Б will be a (X)-module homomorphism. Moreover, for all
A e G' we have
Mb(Teu)) =	(T»u)(Sb(A))
=	u(T(Sb(X)))
= Л°Ь(T,ц) ,
i.e. Б (T ц) = b(T,ц).
From (21.2(iii)) we conclude that Б is uniquely determined.
The second half of (21.5) follows from general category theory,
if on recalls that r(p)^.c will be quasioomplete whenever Г(p) is
barreled or bornological. □
In the special case where the bundle p : E + X is trivial, the dual
of Ftp) (and hence the tensor product over C^tX) between Mc(X) and
Modcc(г(p),Сь(X)) ) may be represented more explicitly (see [Gr 55],
[Si 59], [We 59], [C& 66], [Su 69] and especially [Pr 77]). We shall
use a different approach in this paper which makes use of the
M-structure of Г(р).
Before we start the final pages of these notes, I would like to
remark that all the following ideas are based on a joint work to-
gether with Klaus Keimel done in 1976, which however never was
published. To indicate this fact, we shall assign a new head line
to these final pages:
Appendix
Integral Representation of Linear Functionals on
Spaces of Functions
by
Gerhard Gierz and Klaus Keimel
From now on, let p : E + X be a fixed bundle of real Banach spaces
over a compact base space X. We shall concern ourselves with the
following problems:
Problem A. Given a continuous linear functional ф e Г(p)', find
a regular Borel measure ц on X and a function n : X -> Ftp) ' such
that the mapping x + n(x)(a) is ц-integrable for every a e F(p) and
such that
Ф (a)	=	/ n (x) (a) djj
X

In the special case where the bundle p : E > X is trivial, the dual
of Ftp) (and hence the tensor product over C^tX) between MC(X) and
Modcc(г(p)(Сь(X)) ) may be represented more explicitly (see [Gr 55],
[Si 59], [We 59], [C& 66], [Su 69] and especially [Pr 77]). We shall
use a different approach in this paper which makes use of the
M-structure of Ftp).
Before we start the final pages of these notes, I would like to
remark that all the following ideas are based on a joint work to-
gether with Klaus Keimel done in 1976, which however never was
published. To indicate this fact, we shall assign a new head line
to these final pages:
Appendix
Integral Representation of Linear Functionals on
Spaces of Functions
by
Gerhard Gierz and Klaus Keimel
From now on, let p : E + X be a fixed bundle of real Banach spaces
over a compact base space X. We shall concern ourselves with the
following problems:
Problem A. Given a continuous linear functional ф e Г(p)', find
a regular Borel measure ц on X and a function n : X ->- Ftp) ' such
that the mapping x + n(x)(a) is ц-integrable for every a e F(p) and
such that
Ф (a)	=	/ n (x) (a) djj
X
261
Problem В. Choose r| such that r|(x) e	and || r|(x)|| < 1 for all
x e X. In this case we would have ^(x) (a) = ^(x) (a(x)) for all x e X,
i.e.
ф(а) = J n(x> ( ст (x) )
X
and
T : Г(р) + Г(Х,ц)
ri
а + T (а); Т (а)(х) = п(х)(а(х))
is a continuous (X)-module homomorphism, where 1°°(Х,ц) denotes the
space of all bounded ц-integrable functions, equipped with the
supremum norm.
Problem C. Choose n as in problem B, but try to obtain in addition
that T (a) is Borel measurable for every а e Г(р).
It turns out that different technics can be used to solve these
problems. We could apply a Strassen desintegration theorem in the
form stated by M.Neumann in [Ne 77], we could work with vector valued
martingales as it was done by M.Metivier in [Me 67] or we could use
a vector-valued Radon-Nikodym theorem. As we think that the last
method is the most instructive one, we shall develop a Radon-Nikodym
theorem which is taylored to our problem. Here, of course, most of
the work was already done by various other authors. We shall
follow the ideas of J.Kupka as they were carried out in [Ku 77].
Let us start with an element f e C(X). Recall from (13.18) and (13.19)
that the operator
f :	Г(р)'	Г(р)'
ф + f-ф ; (f-ф) (ст) = ф (f «а)
262
belongs to the Cunningham algebra Cu(r(p)') of r(p)' and that
: C(X)
f
+ Cu(r(p)')
+ f
is a norm preserving mapping between Banach algebras
(Banach lattices
resp.) .
If A с x is closed in X, then Ыд = {a e Г(р) : а/д = 0} is an M-ideal
of r(p) by (13.6) and thus there is an L-projection
PA :	r(p)'	-	na	* rA(P>'-
21.6	Proposition. рд =	inf	{f	:	f e	C(X), 0 <	f	<	1	and	f^ft	=	1}
=	inf	{f	:	f e	C(X) , 0 <	f	<	1	and	=	1
for	some open	neighborhood	U = A}.
Proof. Let
1Д = {f : f e C(X), 0 s f s 1 and f/A = 1}
and
1Д = {f i f e C(X), 0 s f s 1 and f/ft = 1 where U is open, A c U}.
As 1д and I are closed under multiplication, we may use (13.2) to
see that inf 1Д and inf 1д exist in Cu(r(p)') and are idempotent.
Moreover, I = I, implies inf I, s inf I,.
A A r	A	A
Let f e 1Д, ф e Г(р)' and a e Г(р). Then (1 - f)«a e Ыд and there-
fore ( (id - f) »рд) (ф) (a) = Рд(ф) ( (1 - f) «a) =0, as Рд(ф) e N°.
Because a and ф were arbitrary, we may conclude that £°РД = Рд-
This yields рд £ f in Cu(r(p)') as we have 0 s f s 1, i.e.
p, < inf I..
rA	A
It remains to show that inf 1д < рд. This statement is equivalent to
263
(inf I )(ф) e Ыд for every ф e r(p) ' , because this would imply
p.’inf I = inf I,, i.e. p > inf I as inf I is idempotent.
A	A	A	A	A	A
Let ф e r(p)'i let a e and let e >0. Define
V := {x e X :	|)a(x) ||	< e}
Choose an open set U с x such that А с и с и с v and let f : X -+ [0,1 ]
be a continuous function vanishing on the complement of V and taking
the value 1 on U. Then we may conclude that ||g.a|) < e whenever
0 s g < f, g e C(X). Therefore for all g e 1Д with g < f we obtain
the inequality
Ig(ф)(ст)I = Iф(д-а)|
s ||ф|| -e
By (13.2), the net {й(ф) : h e I } converges to (inf 1Д) (ф) in the
norm topology of r(p)'. Since norm convergence implies weak-*conver-
gence, we have
(inf 1Д) (ф) (а)	<	||ф|| .e
and as e > 0 was arbitrary, we conlcude that (inf I )(ф)(a) =0, i.e.
(inf 1д)(ф) e N° .	□
21.7 Proposition. (i) If А,В с x are closed subsets, then
PAuB ” PA V PB-
(it) If (A.). is a family of closed subsets of X and if A = nA.,
1 leI	id
then p = inf p, .
A . _ A.
id 1
In both cases, the lattice operations are taken in the Banach lattice
Cu(r(p)')
Proof, (i) It is a well known fact from [AE 72] that the sum of two
a(Г(p)',Г(p))-closed L-ideals of Г(р)' is again а(Г(p)',Г(p))-closed
264
(the proof of this fact uses (13.4)
and the Krein-Smulian theorem).
Hence we have
NA + NB
(NA n NB)°
NAuB
and therefore p, v p.o = p, „.
А В rA uB
(ii)-. We conclude from (15.7(ii)) that
i.e. рд = inf рд .	□
i el i
We now extend the mapping A + рд :
Cl (X) + Си ( Г(р) ') to all Borel
subsets of X:
21.7 Definition. (i) If и с X is an open subset of X, then we
define py := id - pxxu-
(ii) For every subset м с x we let
p*(M) = sup {рд : A = A с M}
p*(M) = inf {py : M c u, U open}.
(iii) Let Bp(X) := {M e X : p*(M) = p*(M)}. If M c B^, then we define
PM := P*(M) = P*(M)' D
* X
21.8 Proposition. (i) The mappings p*,p : 2	+ Cu(r(p)') are
monotone and p* £ p*.
(ii) P*(X \ M) + p*(M) = id.
(iii) A subset M <= X belongs to в (X) if and only if for every
P
ф e Г(р)' and every e > о there are a closed set А с м and an open
set и э M such that ||p (ф) - Рд(ф) || < e.
Proof.
(i)	Obviously, the mappings p and p are monotone. Moreover
265
if М с x is given and if A с M is closed and if U э M is open, then
A	n	(X	\	U)	= 0 and	hence	рд л	=	0	by (21.6).	This	implies
the	inequality рд	<	id	- Px^g = Рц- As	A	and U were	arbitrary,	this
yields P (M) s p*(M).
(ii)	follows from the computation
p*(X	\	M)	=	sup {рд :	A	=	А с X \ MJ
=	sup {PxXu	:	M	C U, U open}
= sup {id - p^ : M c U, U open}
= id - inf {py : M c U, U open}
= id - p*(U).
(iii)	follows from the fact that for every ф e p(P)' the net {рд(ф) :
: A = A e M} (resp. {Р0(ф)
M e U, U open}) converges to p*(M)(ф)
(resp. р*(М)(ф.)) in the norm topology (see (13.2)).
21.9 Proposition. Let м , n e JN, be a oountable family of subsets
of X. Then we have
(i)	p*( n M ) = inf p*(M )
ndN	ndN
(ii) p*( и M ) = sup p*(M )
nelN	ndN	n
Proof, (i) Every vector lattice satisfies the equation sup D л sup E
= sup {dAe:deD,eeE} whenever these suprema exist (see
[Sch 77]). For given subsets M,N с x, this implies the equation
p* (M) л p* (N)
sup	{PA =	A =	А с M} л sup {pB : В = В c N}
sup	{PA a	PB :	: A = А с м, В = В c N}
sup	^PAnB	: A	= A c M, В = В c N}
sup	{PA =	A =	А с м n N}
266
= p*(M n N) .
Therefore, we may assume that M „ с м for all n eJN.
-1	n+1	n
Now let ф e Г(р) We have to show that p ( n MR) (ф) = (inf p (Мп»(ф) .
Let e >0. By (13.2)the net {рд(ф) : A = A c n Mn} converges to
pj n Mn)(Ф) in the norm topology of Г(р)'. Fix a closed subset
А с м such that
||pB ( ф) - P*( n M ) (Ф) || < e	whenever A c В = В <= nMn-
Moreover, using (13.2) again, for every n eJN there is a closed sub-
set A c X such that A c A c M and such that
n	n n
HpR (ф> " P/MrJ (ф) " < (7)П,е-
15	” П	Z
П
П
Let В := n A.. We claim that for all n e JN we have
П i=1	1
n 1
IIP (Ф) - p (M ) (Ф) II < у (1) x-e.
n	i=1
This inequality is obvious for n = 1. Suppose that the inequality
holds for n £B. As В . = A , n В and as M „ с м , we conclude
ь	n+1	n+1 n	n+1 n'
that p л PR = p »p = p and p (M ) »p (M ) =
An+1 n An+1 Bn Bn+1	*	n+1	* n
= PjMn+ib Since the Cunningham algebra Cu( (p) ' ) is commutative,
this yields
llpB (Ф) " P*<Mn+1 > (Ф) II = IIPA °PB (Ф) " P*(Mn+1 ) (Ф) H
n+1	n+1 n
= llpB °РД (Ф) - Рв °Р*(Мп+1)(Ф) +
n n+1	n
+ P*(Mn+1> °рв (ф) - P*(Mn+1> (ф) 11
n
< IIpb II • ||рА (Ф) - p (Mn+1) (Ф) II +
n	n+1
+ ||p (M ) °(p	(Ф) - p (M )(Ф))||
* Пт |	15	* Il
267
S фп+1.е + ||р*(мп+1) || - ||рв (ф) -
n
- Р*(МП> (Ф> II
„	n
/1'П + 1	. V ,Л1
< (2>	-£ + Z (2> ,£
and our inequality holds for n+1, too.
In particular, we have
(*)	11Рв (ф) - P*(Mn) (ф) || < £ for all n e IN.
n
Now let В = n в . Then Л с В с n M and thus
ngIN	n
||Рв(ф) - P*( n Mn) (ф) || < e.
As p (ф) = Um p (ф) and as inf. p (M ) (ф) = lim p (M ) (ф) by (9.2) ,
П->°° П	ndN *	n+eo *
we conclude from (*) that
||p (ф) - inf p (M ) (ф) ||	< e
neJN *
and the triangle inequality yields
|| inf p (M ) (ф) - p ( n M ) (ф) 11 < 2. e
nelN *	*
As e > 0 was arbitrary, we obtain inf p (M ) = p ( n M ).
* n * n
n^U
(ii) follows from (i) and (21.8(ii)).	□
21.10 Proposition. If А с X is closed, then рд = p (A) = p*(A).
In particular, we have А с в (X).
P
Proof. It follows from the definition of p that we have p = p (A).
Next, let и с X be open, let В с X be closed and assume that и с B.
Then we have p < p • indeed, В и (X \ U) = X implies p_ v pv.rT = id.
HU	'	H л \U
268
As PB v PX\U = pB + px\u  рв л PX\U' this implies PB + PX\U * id'
i.e. рв г id- pXxU = ₽u.
As X is compact, we have n {u
A c U, U open} = A. Therefore we may
use (21.7(ii)) to calculate
p*(A) = inf {py : A c U, U open}
< inf {p— : A c U, U open}
= PA '
We always have рд = p*(A) < p*(A), hence рд = p*(A).	0
21.11 Proposition. If p : E + X is a bundle of Banaoh spaces over
a oompaot base spaoe, then в (X) is a д-complete Boolean algebra
P
containing all Borel subsets of X. Moreover, the mapping
P_ •' B <x)
P
M
+ Cu(r(p)’)
* PM
is a <j-homomorphism between В (X) and the (complete) Boolean algebra
P
of all L-projections of г(p)'.
Proof. From (21 .9) we know that M e В (X) if and only if X \ M e
P
e В (X) and from (21.10) we conclude that В (X) is а-complete and
P	P
that p_ is a a-homomorphism. Finally, all Borel subsets are contained
in В (X) by (21.10). □
P
Now let ф e r(p)' be a continuous linear functional on r(p). For
every M e В (X) we define
^ф(м) := Рм'Ф’
ЦфМ := Црм(ф) II •
269
21.12 Proposition. If
M
n-
n e M, is a pairwise disjoint family of
elements of В (X), then
P
V , ( и
1 п£И
У
n=1 Ф n
for every
the sum converges
in
the
norm topology of
Г(p)' and
we have
ф ,,
* nc В
1 INJM) ||
n=1 ф n
У
n=1
u (M ) .
цф n'
In particular, the mapping v is a <j~additive г (p) '-valued
u is a g-additive real-valued measure on в (X).
Ф	P
Finally, we have \> (м) = 0 if and only if u (M)
Ф	Ф
0.
Proof.
PM Л PN
e (Id -
Firstly, let M,N e В (X) be two disjoint
P
= 0, and whence pM < id - pN- This implies
PN)(Г(p)') and Ры(ф) e PN(r(p)')• As pN
1|Рм(ф) II + l|PN <Ф> II •
a pairwise disjoint family of elements of
sets.
measure and
Then
Рм(ф)
is an L-projectlon,
we conclude 11PM(ф) + PN (ф) ||
Thus, if M , n e JN, is
n
В (X) , we use (13.2) +-o
calculate
u Mn)
(sup p ) (ф)
ndN i<n X
(P и M (ф))
i<n
(.I рм..(ф))
lim
lim
n->-°°
and
У
1=1
Рм <Ф>
M.
J u
Ф n
M
n
ф e Г (p) ' • Here
M )
n
M
n
e
Ф
p U Mn ( ф ’
и Mn)
Ф
Ф
и Mn) II
У p (Ф) II
=1 M1
270
= lim II 1 Рм <Ф> II
П+ео i=1	“1
= liltl I l|PM (Ф> II
п-юо i=1 i
1 l|PM <Ф) II
i=1 i
00
= □
We are now in the position to solve problem A: Joseph Kupka has shown
in [Ku 72, 4.9] that there is a function Пф : X ->- r(p) ' such that
V. (M)
Ф
meaning that
Рм(ф)(a)
Since р„(ф) = ф,
Л
Ф (a)
as desired.
We should remark at this point that (4.9.2) of [Ku 72] provides us
with more information, namely
u, (M)	= IIV, (M) II
Ф	11 Ф 11
=	/ || n, (x) || -dy (x) for all M e В (X)
ф	Ф	f
and hence || г)ф (x) || = 1 ц-almost everywhere.
M
M
пф’ацф
e
P
П (x) (a) .du (x)
Ф	Ф
we obtain
f Пф (x) (a) ^Цф (x)
X
for
for
all
all
M
a
e
e
в (X) , a € Г(Р) .
Г (p)
In order to solve problems В and C, we need more information on the
natur of n . To obtain this extra information, we find it convenient
to repeat the steps of the proof of j. Kupka's result.
21.13 Definition.
(i) Let (X,S,ij) be a finite measure space. By
271
№°(ц) we denote the space of all bounded ц-measurable and real-valued
functions on X.
(11) A mapping p : M (ц) -»• M (ц) Is called a lifting, provided that
(a) p is linear, positive and preserves multiplication and the
constant function 1,
(B) p(f) = f ц-а.е.
(у) f = g ц-а.е. implies p(f) = p(g).
(Hi) If, in addition, X is a compact space and if ц is a Borel
measure on X, then a lifting p is called almost strong, provided
that there is a subset N с X with p(N) = 0 and
p(f)(x)
f (X)
for all x e X \ N and all f e C(X).
21.14	Remark. Let X be a compact space, let ц be a regular Borel
measure on X and let p be an almost strong lifting. Then there is
a lifting p' : №°(p) + №°(ц) such that
(*)	p'(f)(x)=f(x) for all x e supp(ц) and all f e С(X).
and every lifting satisfying (*) is almost strong.
(Indeed, if ц is a regular Borel measure, then p(X \ supp(^)) =0; hence
Conversely, let N с X be a p-zero set such that every continuous
function f agrees with p(f) on X \ N and let M = N n supp(p). Intro-
duce a seminorm || • || on M°°(p) by defining
|| f || = inf {M e 1R : ц{х e X : | f (x) | > M} = 0}
(i.e. ||f|| is the essential supremum of f) and let
№° = {f e м“(ц) : ||f|| = 0}.
Furthermore, let ~ : f + f : M°°(p) + М°о(р)/№° be the quotient map.
272
Then the mapping f + ||f|| is well-defined and a norm on М°°(р)/№° and
М°°(ц)/№° is a Banach algebra in this norm. Moreover, if f e C(X),
then f = О if and only if f ,	. . = О and we have
1 /supp(u)
j| f II = sup {|f(x) | : x e supp(p) }.
Especially, the image of C(X) under ~ is isometrically isomorphic to
C(supp(p)) and therefore is closed.
For every t e M = N n supp(p) we define
et : {f : f e C(X) } Ж
f	f(t).
Then e is well-defined and continuous. Thus, using [IT 69, VIII.1,
Prop. 1], we may find a continuous extension
00	,	00
Xt : M (p)/N Ж
of e such that
Xt(f*g) = Xt(f)’Xt(9)	for all f,g e М°°(ц).
We now may define a new lifting p' : M°°(p) + М°°(ц) by
rp (f) (x)	if x e X \ M ,
p' (f) (x) = ч
%<£)	if x £ M .	)
21.15	Definition. Let X be a compact space and let ц be a regular
Borel measure on X. If p is a lifting satisfying (*) of (21.14) , then
p is called a strong lifting.	□
00	00
Now let p : M (p) + M (p) be a lifting, where (X,S,p) is a measure
space. Let S = {A e. X : xA e M°° (u) J be the ц-completion of S, where
Хд denotes the characteristic function of A. If xA belongs to M°°(p),
273
and therefore р(Хд) is idempotent, too. Thus, p defines an element
В e S such that р(хд) = Xb' th^3 case we write p*(A) = B.
21.16	Proposition. (i) Let (X,S,p) be a measure space and let
00	00	-	,
p : M (p) + M (p) be a lifting. Then the mapping
p* : S + S
= Xp*(A)
is a homomorphism of Boolean algebras satisfying
В*)	p*(A) = A ц-а.е.
у*) If A = В p-a.e., then p*(A) = p*(B) .
(ii) If X is a compact space, if ц is a regular Borel measure on x
and if p : M (p) + M (p) is a strong lifting, then we have in addi-
tion
I) p*(A) n supp(p) c a whenever A is closed
II) If x e supp(p) and if U is a neighborhood of x, then
x e p*(U).
Proof. For a proof of (i) we refer to [IT 69, 111.1].
Although (ii) is certainly well-known, too, we indicate a proof: Let
A c X be closed and let U be an open neighborhood of A. Choose a
continuous function f : X + [0,1] such that f(A) = {1} and f(X \ U) =
= {0}. Then we have v, s f and therefore x i*\ = p(x») - p(f)- This
ЛА	p* (A) K A K
implies v ...	, . = v *,..-v , . s p(f)»y , .. As p(f) and
F лр*(A)nsupp(ц) лр (A) Asupp(p) p Asupp(p)
f agree on supp(p), we obtain xp*(A)nsUpp(u) s f and as the °Pen set
U can be made arbitrarily small, this proves (I).
Property (II) now follows immediatly from (I) and De Morgan's rule. □
It may be shown that conversely properties (I) and (II) characterize
274
strong liftings.
The following proposition ensures the existence of liftings:
21.17 Proposition. (i) Let X be a compact space and let p be a
finite regular Borel measure on X. Then there is a lifting
00	00
p : M (p) M (p) .
(ii) Moreover, if X is metrizable, then there is a strong lifting
00	00
p : M (p) M (p) .
For a proof see [IT 69, IV.2, theorem 3] and [IT 69, VIII.4, theorem
8].	□
It is well known that liftings may be used in the proof of the Radon-
-Nikodym theorem (see [Di 51a] or [IT 69]). Let us repeat some of the
arguments here:
Let (X,S,p) be a finite measure space, assume that p is positive and
let p be a lifting. Moreover, let X be a second measure on (X,S)
which is p-continuous and which has the property
A (TH
(B)	{ : ' : E € S, p(E) / 0} is bounded in ]R.
P \E)
From the Radon-Nikodym theorem we know that there is a p-integrable
function p such that
A (E)	= f rpdp
E
for all E e S.
By the mean value theorem and (B) we may find a constant M > 0 such
that |n(x)| < M p-a.e. Hence we may assume without loss of gene-
rality that |n(x)| < M for all x e X. In this case p(p) is defined
and we may assume that q = p(q).
275
We now consider the set n of all partitions я = {F^,...,F } of X
satisfying
Fi = P*(F1)	0	( i = 1 ,...,n ) .
The set Ц is directed under refinement. For every я e n we let
„	_	г A (F)
% "	p(F)-XF-
21.18 Proposition.
The net (n )	„ oonverqes
4 тгеП	и
to n
uniformly on X.
Proof. Let e > 0 and define
An = {x e X : n-e s n(x) < (n + 1)-e}, n еИ.
Then An belongs to S and it is easy to see that p(n) = n implies
p*(An) = An. Let TT = {An : An / 0}. Then it is a partition of X be-
longing to П. Now let us take any refinement тг’ = {F^,...,Fm} £ П of
7t and let x £ X be arbitrary. Then we can find an index i e {1,...,ra}
and an integer n e И such that x £ F^ c An-
We compute
|n^,(x) - n(x)| =	- n(x)|
=	'I/ (n(t) ” n(x)>'dlj(t) ।
1 F
£ 777'f ) f |n(t) - n(x)|-dp(t)
i' f.
e.
Since x e X was arbitrary, we obtain
sup |n , (x) - n(x) | < e for all refinements it' of it.
xeX л
□
276
Let us return to our bundle p : E -> X and our continuous linear
functional ф : Ftp) -+1R.
The p-algebra В(X) contains the p-algebra of all Borel parts B(X)
and we have a regular Borel measure
p. : B(X) JR
Ф
M * Црм(ф) II
and a Ftp)'-valued measure
^Ф
B(X) Ftp) '
M +
^(Ф)
on B(X). Moreover, if p e F(p) is a section, then we may define a
Borel measure
on B(X)
by
A
p
A
p
B(X)
JR
M
p (ф) (p) = v (M) (p).
M	ф
It is obvious
that X
p
p-continuous for every p e F(p) and that
v, , p ± and the X
ф ф	p
Further, the set
may
be
extended to the p -completion B(X)~ of B(X).
Ф
Xp(E)
^ф(Е) ’
e B(X)~,p (E) / 0}
is bounded by ||o||
By (21.17)
we may choose a
lifting p : M (u)
M (F). As above
let П be the
directed set of
all partitions
7Г = {F1 , . . . ,FnJ of X
such that p
(F.) = F. / 0
E
1
. . ,n.
We define
П : X Ftp)'
Уф(А)
= aL XaX<a)
and for every p e r(p) we define
277
aL ^(а)‘Ха
Obviously, we have n (x) = n (x)(a) for all x e X, p e Г(р). More-
7Г r a	я
over, (21.18) shows that the net (n )	„ converges uniformly to a
ir,a kJI
function e M°°(p) with ri^ - p(n^) such that
A (E)	= f n "dp .
a	£ 'a рф
Hence the net (n )	„ is a a(Г(p)',Г(p))-Cauchy net. As we have
ir теП
|| D (x) || < 1 for all x e X and all
uniformly towards a function n : X
Ф
for all x e X and all a e Ftp).
e II, the net (n )	„ converges
я я? П
Ftp)' with n (x)(a) = n (x)
ф	a
Taking all these pieces of information together, we obtain
Рм(ф)(о)
Л (M) = [ n (х) -dp (х) = [ n (x)(a)dp (x).
° M ° ф M Ф	Ф
If we let M = X, the we obtain again the solution of problem A.
21.19 Proposition.
For a given a e Ftp)' and a given lifting
00
P : M (рф)
00
- м (рф)
we let
к := n {(M n supp(p ))	: x e M = p* (M) }
X	ф
for all x e X.
Then for all x e X we have n,(x)
Ф
о
N
e
A '
x
Proof.
Let x
X and let Me x be a subset such
that x e M = p (M).
Then я
{M, X
M} is a partition of X belonging
to Ц . Let tt ’
{F1,...,Fn} be
a refinement of tt. Then there is
n}
e
an i e {1
such that x e. F.
с M. Let E e B(X) be a Borel part of X such that
and
p (E) = p (F.)• Then it follows that v,(E) = v,(F.)
ф	ф i	ф	ф i
and hence
x e Ec f .
v(x)
%<E>
278
Moreover, as u± is a regular Borel measure, we know that
Ф
p.(X П supp(p )) = 0 and thus v (X n supp(p ))
Ф	Ф	ф	ф
0. This yields
V . (E n supptp ))
/ X _ ф________	Ф
Птт' kX	Цф(Е n supptp^) )
Now note that E n supp(pJ
Ф
(M n supp(p )) ,
Ф
which gives us
^ф(Е n supp(%)) = РЕп3ирр(Цф)(ф)
Р(МПЗирр(цф))—°PEnSUpp(Рф)(ф)
e P(Mnsupp(p ))-(Г(р) >
= № Ф -
(Mnsupp ( p ) )
Ф
and therefore
<X) e N(Mnsupp(p ))
Ф
As tt‘ was an arbitrary refinement of я and as N°	, x— is
*	(Mnsupp(p )
g(Г(p)',Г(p))-closed, we obtain
n , (x)	= lim n (x) e N?,,	,	.
ЧФ	леП %	(Mnsupp(Цф))
Finally, the mapping A + N° , A e C1(X), preserves arbitrary inter-
sections by (15.7(11)) yielding that
Пф(х) e N° .	□
Y	V
We now come to a solution of problem B:
21.20	Theorem. Let p : E + X be a bundle of Banaoh spaces over a
oompaot base spaoe X and assume that every finite, regular Borel
measure on X admits a strong lifting. (This is in particular the case
if X is metrizable) . If ф : Г(р) -+ ]R is a continuous linear functio-
nal on Ftp), then we can find
regular Borel measure p, on X
Ф
|| Пф,хН s 1 for all x
a finite
a family Пф x e Ex> x e X
such that
e X.
279
The
mapping
x * П , v (o(x) )
Ф /X
X+Bis v -integrable for
every
iii)
For every Borel
set M с X we have
Рм(ф)(о)
%,x(a(x))
a e Г (p) .
M
in partioular
Ф(а)
пф,х(а(х))
Proof.
Let а e Г(p)'
and as before
by Рф(М)
lifting.
If
let u, be the measure defined
Ф
Further, let p : M (ц.) ->-M (ii) be a strong
Ф	Ф
constructed
we conclude
||рм(ф) II •
the mapping n : x + Г(р) ' is
Ф
(21.19) that Пф(x) e
from
for every x
{x}
for every
зирр(рф).
Indeed, let
A be
a closed neighborhood of
x e supp(p^)
as above, then
e X. We show:
. Then
A
x
x

x e
implies that
x e
supp(p ) n P (A) c A.
Ф
If we let M
p (A)
then we know that p*(M) = M and
x e
(зирр(Цф)
n M) c A.
As A was arbitrary, this yields Ax = {x}
and whence n±(x) e № = E'
ф	X X
for every x e supp(^).
Now define a family Пф x e E^, x e X, by
Г'ф ,x
(X)
x e зирр(Цф)
x e X \зирр(Цф).
Then || Пф x||	<	||Пф(х) ||	< 1 for all x e X. Moreover, since Цф is a
regular Borel measure, the set X \ зирр(Цф) has measure 0 and there-
280
fore n, = П±(х) pi.-a.e. Hence for every a e Ftp) we have
Ф 	(p	(p
Рм(ф)(а) = J n (x) (a) -dp (x)	= J n (a(x) ) .dp, (x) .	□
The following theorem is a partial solution of problem C:
21.21	Theorem. Let p ; E + X be a separable bundle of Banach spaces
over a compact base space x and assume that every finite regular
Borel measure on x admits a strong lifting. If ф : Г(р) -+ 1R is a
continuous linear functional on F(p), then we can find a family
E, e E' x e X. and a finite req
ф,Х X1	1	J	V
that
i) И ?ф,х II & 1 ^or al1 x e x-
ii) the mapping x •* Еф x(a(x))
bounded for every о e Г(p).
iii) for every Borel part M с X
lar Borel measure ц on x such
Ф
: X + ]R is Borel measurable and
have
РМ(Ф)(о)
/ E (a(x))-dn (x),
м	T
in particular
Ф(а) =	/ v (a (x)) «dpi (x) .
v Ф ,x	Ф
Proof. Let <an)ne]N be acountable family of sections of Г (p) such
that {an(x) : n e JN} is dense in Ex for every x e X. Further, let
(n,	)	„ be a family of elements of E', x e X, such that the con-
ф,x xeX	x
ditions i), ii) and iii) of (21.20) are satisfied. Then the mapping
x + n, (a (x) ) is pi.-integrable for every n e JN. Hence
Ш f Il	Ш
we can find
a Borel set A
n
с X with pi (Ar) = 0 such that the mapping
x "* XX\A ^х),Г|ф x(an(x)) is Borel measurable. Let A
we still have р1ф (A) = 0. Now define
и A . Then
u_, n
n cN
281
5. х
ф f X
Ча v
Ф f X
О
if х е X \ А
if х е А
With this definition the properties (i) and (iii) are satisfied.
Define an operator
T : Г(р)
T(a)(x)
= 5. x(a(x))•
(p / X
It remains to show that T(a) is Borel measurable for every a e Г(р)-
By the choice
of the 5,
ъф ,x
this is clear for the a , n e TJ.
x e X
Moreover, the operator T is a C(X)-module homomorphism and we have
IlTtoJlL s INI r Where Ц-l^ denotes
(recall that the elements of M°°(ii ) are
Ф
fore T(a) is Borel measurable for every
00
the supremum norm on M (ц,)
Ф
bounded by definition). There-
element a belonging to the
closed С(X)-submodule
generated by the set {an
n e TJ} and thus
for every a e Г(р) by the Stone-WeierstraB theorem (4.3).
□
We conclude this book with a description of the dual space of
C(X,E), where E is a Banach space and where X is compact (see
[Gr 55], [Si 59], [Ca 66], [We 69], [Su 69] and [Pr 77]).
21.22	Definition.	Let E be a Banach space and let X be a compact
topological space. A linear operator u : С(X) + E' is called
dominated, if there is a positive finite Borel measure ц on X
such that 11 u (f) 11 < J | f | - dpi for all f e C(X) .	□
X
Now let ф : C(X,E) + ]R be a continuous linear functional. We define
an operator Пф : С (X) + E' by
(1)	u .(f) (a)	=	*(f-c ) for all a e E
ф	a
where с : X -> E denotes the constant mapping with value a. As on
cl
the previous pages, let ц be the finite Borel measure on X defined
282
Ьу Рф<м) = |1₽м(ф) II • We claim that
II Уф (f) II s J |f I .dy	for all f e C(X) .
X
Indeed, let e > 0, let f e С(X) and define
An := {x e X : n*e s f(x) < (n + 1) -e}	, n e И.
Then for each compact subset К c An we have
Ip (Ф) (f’C ) | S ||р„(Ф) II *sup { II f (x) -all : x e K}
r\	d	Jt\
= Цф(К) • 11 а11 •sup { I f(x) I : x e K}
s Рф (An) • ||a || • sup { | f (x) | : x e Ar} .
As the р„(Ф), К e A , converge to p, (Ф) in the norm topology of
Jx	П	A
П
C(X,E)', we conclude that
|p (Ф) (f-с ) I S p. (A ) • Ha|| • sup { |f (x) I : x e A }.
d	y Г1	Г1
Note that the sets An, пей are pairwise disjoint. Hence (21.12)
yields
1иф(£)(а)|	=	^(f«ca)|
= I I P, (Ф) (f-c ) I
ne 2Z n	a
s Z 1рд (Ф) (f‘c=) I
n e И n
< II a II • ( 1 Pa (a ) ’sup { | f (x) | : x e A })
пей *
s Hall -(I	( f |f(x) l-dp. + e-p (A )))
n eE An	*	T
= Hall •( f |f(x) I dЦф + е-Рф(Х))
As a e e and e > 0 were arbitrary, we conclude that
l|u (f) II <	/ |f (X) I .du
r	r
and therefore u, is dominated.
Ф
283
Conversely, assume that u : С(X) + E' is dominated, i.e.
||u(f) ||	< J |f(x) | .dp
X
for a certain finite Borel measure p on X. In this case, we define a
linear functional ф^ on the С(X)-submodule M e C(X,E) spanned by the
constant functions via the formula
(2) Фи( Z fi‘ca >	= Z u<fi> <ai>'
i=1 i i=1
where the f^ belong to C(X) and where the a^ belong to E^. It is not
too difficult to check that ф is well defined and linear.
Yu
n
Moreover, if all the f^ are positive, if Д f^ = 1 and if ||a^|| < 1
for all i e {1,...,n}, then
IV Z fi ca ) I s 1 lu (f ±) (V I
i=1 i i=1
* Z IMVH
i=1
s У / f.(x)-dp
i=1 X 1
= J 1 f±(x)
X i=1
= Hull-
Repeating some arguments from the proof of the Stone-WeierstraB
theorem, we find that ф is bounded on M, as the elements of the
above form are dense in the unit ball of M. Thus, as M is dense in
C(X,E), the mapping фц may be uniquely extended to C(X,E). Ob-
viously, the mappings u + фц and ф + are mutually inverse to
each other. Thus, we have shown:
21.23 Theorem (Wells 1965). Let X be a compact space and let E be
a Banach space. Then the mappings defined in (1) and (2) are mutually
inverse isomorphisms between the dual space C(X,E)' of C(X,E) and the
space of all dominated operators u : С (X) -* E'.	□
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[Au	75]	Auspitz, N.E.: Dissertation, University of Waterloo (1975)
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Burden, C.W.: The Hahn-Banach theorem in a category of
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Dauns, j. and K.H.Hofmann:	Spectral theory of algebras
and adjunctions of identity. Math. Ann. 179, 175 - 202
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Topology 3, 1 - 18 (1965)
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Ergebnisse der Mathematik und ihrer Grenzgebiete 48,
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INDEX
adjoint of a multiplier: 152
approximation property: 131
automatically: 8
Banach algebra:	8
-controid of a: 77
-topological center of a: 158
Banach bundle: see bundle of Banach spaces
Banach lattice, center of a: 78
-topological center of a: 158
base space: 12,17,18
-of a function module:	45
-Borel subsets of the: 268
bilinear map: 253
-between С(X)-modules:	253 ff
-compatible with the С(X)-module structure: 253 ff
bounded convergence: 144
bounded selection: see selection, bounded
bundle, full: 22,54,97 , lOOf, 210,232
-, locally full: 22,27
-locally trivial: 56,200,202,208,242,246
- of Banach spaces: 21
- of compact operators: 127 ff
- of C(S)-module homomorphisms: see bundle representation
- of finite dimensional vector spaces: 188,208,215,231,247
- of fi-spaces: 12
- of operators: 112 ff
- reduced: 76
★
- representation of a C -algebra: 158
- -compact operators: 127 ff
- -C(X)-module homomorphisms: 106 ff, 133 ff, 209 ff, 249
- -operators: 122 ff
-, separable: 215, 217 ff, 242,280,234
- space; 12 (see also: bundles with Hausdorff bundle space)
- stalk of a: 9
-, trivial: 12,215
- with complete stalks: 19,42,100,186
292
-with continuous norm: 166 ff, 202,229,242,250
-with Hausdorff bundle space: 185,188,204,208,214,215,247
★
C -algebra, bundle representation: 158
Center of a Banach lattice: see also Banach lattice, center of a
-,topological: 152 ff
centroid of a Banach algebra: 77
closure of the unit ball of a bundle: 190 ff
compact convergence: 114
compact, convex circled convergence: 114
compact convex sets as a lattice: 137 ff
compact operators: 112, see also bundle representation of compact
operators
continuous lattice: 136 ff
continuous norm of a bundle: see bundles with continuous norm,
countable family of seminorms: 42,186
Cunningham algebra: 145 ff, 262 ff
C(X)-convex: 16,65,69ff
С(X)-module:	13, 62ff, 115,150 ff, 253 ff
-, reduced: 76
C(X)-module homomorphisms: 95 ff,196,210,232
-, bundle representations: see also bundle representation of С(X)-module:
existence of: 218,225 ff
С(X)-morphism:	95
С(X)-fi-module:	62 ff
С(X)-П-morphisms:	95
С(X)-submodules:	80 ff,180,182
Dauns-Hofmann-Kaplansky multiplier theorem: 150 ff,155
dimension is lower semicontinuous:	204
directed family of seminorms: 11
directed set in a lattice: 136
dominated linear operator: 281 ff
dual, internal: 252,232,196
duality between subsets of the unit ball of a bundle and Bx:159 ff,193 ff
dual space of Г(р): 252 ff
293
e-n-continuous:	30 ff
e-n-thin: 29 ff
e-tubes: see tubes
equivalent norms of a bundle: 234,250
evaluation map: 19,24 ff, 42 f, 97
extreme points of an L-ideal: 146
fibred fi-space:	10
fibred vector space: 10
full bundle: see bundle, full
fully additive: 39,82 ff
function module: 45 ff
-uniform fi-function module 51
hull-kernel-topology: 142
hypocontinuous: 253 ff
ideal: M-ided: 146 ff, 164,182
M-ideal, primitive: 149 ff
M-ideal, stalkwise: 182
M-fi-ideal: 154
-, primitive: 154
L-ideal: 146 ff
fi-ideal: 60 f, 63
-, stalkwise: 86
integral representation of linear functionals: 159,252,260 ff,278
280.
internal dual: see dual, internal
internal Mackey space: 232
isometry of bundles: 103
isometrical isomorphy of bundles: 103
isomorphy of bundles: 100
-, locally: 200
Klein's bottle: 56
lattice of closed sets: 137 f
of closed linear subspaces: 140 ff
of compact convex sets: 137 ff
of open sets: 137 f
Lawson-continuous:	138 ff
Lawson-topology: 138 ff
L-ideal: see ideal
lifting: 271 ff
-, strong: 272 ff
limit of a net in a continuous lattice: 140
-of an ultrafilter in a continuous lattice: 145
local linear independence: 201 ff
locally countable family of seminorms: 25,29,100
locally C(X)-convex: 69,75,115
locally finite: 39
locally full bundle: see bundle, locally full
locally isomorphic: see isomorphy of bundles
locally paracompact space: 28
locally trivial bundle: see bundle, locally trivial
294
local section: see section, local
L-projection:	145 ff, 262 ff
Mackey-Arens-theorem: 232,238
Mackey-bundle: 247 ff, 257 ff
M-bounded: 152 ff
Moebius strip: 55
M-ideal: see ideal
M-fi-ideal: see ideal
morphism between bundles: 95 ff, 102
- between trivial bundles: 106 ff
multiplier: 152 ff
norm of a bundle morphism: 102
fi-B-space: 8
П -center: 79
-	, topological: 157
fi-function module: see function module
fi-ideal: see ideal
fi-morphism: 60
fi-space: 8,60
-	, fibred: 10,17
-	, topological: 8,14,60
fi-subbundle:	80
fi-subspace: 13
operators of finite rank:
pointwise convergence: 114
precompact convergence: 114
prime element: 140
prime lemma: 140
primitive M-ideal: see ideal
projection: 145
quotients of bundles:	86 ff
quotients of С(X)-modules:	86 ff
Radon-Nikodym theorem: 261 f,274 f
restriction of a bundle: 18
Scott-continuous: 136, 162 f
Scott-open: 137
Scott-topology:	137
section: 11
-, local: 12,42 f
selection: 10,42 f
-, bounded: 11
Seminorm of a boundle:	11
separable bundle: see bundle, separable
spectrum of a lattice: 142
stalk of a bundle: 9
- of a function module: 45
-of a locally trivial bundle: 200
stalkwise closed: 80 ff
stalkwise dense: 39
stalkwise fi-ideal: see ideal
stalkwise product: 9
295
standard construction of bundles: 45 ff
Stone-Weierstrass theorem for bundles: 39,41
Stone-Cech-compactification:	63
subbundle: 80 ff, 179 ff
-, stalkwise closed: 86 ff
submodule: see С(X)-submodule
tensor product of С(X)-modules:	258 ff
3-ball-property:	147 ff
topological center: see center, topological
topological fi-center: see fi-center, topological
topological fi-space: see fi-space, topological
total family: 113
trivial bundle: see bundle, trivial
tube: 12 ff
type:	8
uniform fi-function module: see function-module
Varela's lemma: 66
vector space, fibred: 10
way below: 136
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