SAKS SPACES AND APPLICATIONS TO FUNCTIONAL ANALYSIS
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NORTHHOLLAND MATHEMATICS STUD I ES
28
Notas de Matemhtica (64) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester
Saks Spaces and Applications to Functional Analysis JAMES BELL COOPER JohannesKepler Universitat. Linz, Austria
1978
NORTHHOLLAND PUBLISHING COMPANY  AMSTERDAM NEW YORK OXFORD
@ NorthHollandPublishing Company  1978 All rights reserved. No part of this publication may be reproduced,stored in a retrievalsystem. or transmitted, in any form or by any means, electronic, mechanica1,photocopying. recording or otherwise, without the prior permission of the copyright owner.
NorthHolland ISBN: 0 444 8.51 00 3
PUBLISHERS:
NORTHHOLLAND PUBLISHING COMPANY AMSTERDAM*NEW YORK*OXFORD SOLE DISTRIBUTORS FOR THE U.S.A.ANDCANADA
ELSEVIER / NORTH HOLLAND, INC. 52 VANDERBILT AVENUE, NEW YORK, N.Y. 10017
Library of Congress Cataloging in Publication Data
Cooper, James Bell. Saks spaces and applications to functional analysis. (Notas de matemhtica ; 64) (NorthHolland mathematics studies) Bibliography: p. Includes index. 1. Saks spaces. 2. Functional analysis. I. Title. 11. Series. @l.N86 no. 64 [@I3221 510'.8s [515'.73] 781921 ISBN o  w A  ~ ~ ~ o o  ~
PRINTED I N THE NETHERLANDS
PREFACE
This monograph is concerned with two streams in functional analysis

the theories of mixed topologies and of strict topo
logies. The first deals with mathematical objects consisting of a vector space together with a norm and a locally convex topology which are in some sense compatible. Their theory can be regarded as a generalisation of that of Banach spaces, complementary to the theory of locally convex spaces. Although closely related to locally convex spaces, the theory of mixed spaces (or Saks spaces as we propose to call them) has its own flavour and requires its own special techniques. The second theory is concerned with a number of special topologies on spaces of functions and operators which possess the common property that they are substitutes for natural norm topologies which are not suitable for certain applications. The best known example of the latter is the strict topology on the space of bounded, continuous, complexvalued functions on a locally compact space, which was introduced by Buck. The connection between these two topics is provided by the fact that the important strict topologies can be regarded in a natural way as Saks spaces and this fact allows a simple and unified approach to their theory.
The author feels that the theory of Saks spaces is sufficiently welldeveloped and useful to justify an attempt at a first synthesis of the theory and its applications. The present monograph is the consequence of this conviction.
V
vi
PREFACE
The book is divided into five chapters, devoted successively to the general theory of Saks spaces and to important spaces of functions or operators with strict topologies (spaces of bounded cont nuous functions, bounded measurable functions, operators on Hilbert spaces and bounded holomorphic functions respectively
. An
appendix contains a categorytheoretical
approach to Saks spaces. Each chapter is divided into sections and Propositions, Definitions etc. are numbered accordingly. Hence a reference to 1.1.1 is to the first definition (for example) in
9 1 of Chapter
abbreviated to 1.1.
I. Within Chapter I this would be
At the end of the book, indexes of notations
and terms have been added.
An attempt has been made to make the first five chapters as selfcontained as possible so that they will be useful and accessible to nonspecialists. However, some results have been given without proof, either because they are standard or because they seemed to the author to provide useful insight on the subject but the proofs were unsuitable for inclusion. Some of the latter are relegated to Remarks. In the first Chapter, only familiarity with the basic concepts of Banach spaces and locally convex spaces is assumed. In Chapter I1 GILLMAN and JERISON has been applied


V the principle of
namely to make them com
prehensible to a reader who understands the words in their respective titles. The monograph should therefore be suitable say for a graduate course or seminar. However, by providing each Chapter with notes (brief historical remarks and references to
vii
PREFACE
related research articles) and a list of references, an attempt has been made to produce a work which could also be useful to research workers in this and related fields. In addition to those references which we have used directly in the preparation of this book, we have tried to include a complete bibliography of papers which relate to Saks spaces and their applications, including some where the connection may seem rather remote.
A s a warning to the reader, we mention that all topological
spaces (and hence also locally convex spaces, uniform spaces etc.) are tacitly assumed to be Hausdorff.
It goes without saying that the author gratefully acknowledges the contributions to this book of all those mathematicians whose published works, preprints, correspondence or conversation with the author have been used in its preparation. Special thanks are due to Frxulein G. Jahn who made a beautiful job of typing a manuscript in a foreign language under rather trying circumstances.
J.B. Cooper Edramsberg, October, 1977
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LIST OF CONTENTS
Preface
V
I Mixed topologies :
1 4
1
Basic theory;
2
Examples;
20
3
Saks spaces;
25
4
Special results;
37
5
Notes;
63
References for Chapter I.
I1
111
68
Spaces of bounded, continuous functions:
75
1
The strict topologies;
77
2
Algebras o € bounded, continuous functions:
89
3
Duality theory;
101
4
Vectorvalued functions;
112
5
Generalised strict topologies:
119
6
Notes;
136
References for Chapter 11.
14 1
Spaces of bounded, measurable functions:
155
1
The mixed topologies;
156
2
Linear operators and vector measures;
172
3
Vectorvalued measurable functions;
176
4
Notes;
190
References for Chapter 111.
ix
192
X
IV
LIST OF CONTENTS 195
Von Neumann a l g e b r a s :
1
The a l g e b r a of o p e r a t o r s i n H i l b e r t s p a c e s ;
197
2
Von Neumann a l q e b r a s :
203
3
Notes:
22 1
References f o r Chapter I V .
V
S p a c e s o f bounded h o l o m o r p h i c f u n c t i o n s :
e:
223
225 226
1
Mixed t o p o l o g i e s on
2
The mixed t o p o l o g i e s on H m ( U ) :
230
3
The a l g e b r a Hm:
24 0
4
The H m  f u n c t i o n a l c a l c u l u s f o r c o m p l e t e l y non
5
unitary contractions:
24 9
Notes:
2 60
R e f e r e n c e s f o r C h a p t e r V.
267
Appendix :
.
263
C a t e g o r i e s of S a k s s p a c e s :
268
A.2.
D u a l i t y f o r c o m p a c t o l o g i c a l and uniform s p a c e s :
272
A.3.
Some f u n c t o r s :
282
A.4,
D u a l i t y for s e m i q r o u p s and g r o u p s :
2 84
A.5.
E x t e n s i o n s of c a t e g o r i e s :
295
A.6.
Notes:
305
A. 1
R e f e r e n c e s f o r Appendix.
308
Conclusion
31 3
Index of n o t a t i o n
319
Index of t e r m s .
323
CHAPTER I

MIXED TOPOLOGIES
Introduction: The objects of study in this chapter are Banach spaces with a supplementary structure in the form of an additional locally convex topology. The motivation lies in the interplay between certain mathematical objects (topological spaces, measure spaces etc.) and suitable spaces of (complexvalued) functions on them. These often have a natural Banach space structure. However, by passing over from the original spaces I
to the associated Banach spaces, one frequently loses crucial information on the underlying space. A good example (which will be the subject of our most important application of mixed topologies) is the Banach space Cm(S) of bounded, continuous, compl xvalued functions on a locally compact space it is impossible to recover
of cm
S)
S
S
where
from the Banach space structure
(in contrast to the case of compact spaces
S).
As we shall see in Chapter 11, this situation can be saved by enriching the structure of Cm(S) with the topology convergence on the compact subsets of
S.
' I ~ of
uniform
The class of spaces
that we consider can be regarded as a generalisation of the class of Banach spaces (we can "enrich" a Banach space in a trivial way, namely by adding its own topology). In fact these spaces can be regarded as projective limits of certain spectra of Banach spaces with contractive linking mappings (just as one can regard (complete) locally convex spaces as projective limits of arbitrary spectra of Banach spaces) and we shall lay particular emphasis on this fact for two reasons: for
1
2
I. MIXED TOPOLOGIES
purely technical reasons and secondly because, in applications to function spaces, we shall constantly use the fact that our function spaces are constructed out of simpler blocks which correspond exactly to the members of a representing spectrum of Banach spaces. As an example, dual to the fact that one can consider a locally compact space as being built up from its compact subspaces, we find that one can construct the space Cm (S) from the spectrum defined by the spaces {C ( K )1 as K runs through these subsets.
One of our main tools in the study of our enriched Banach spaces will be a natural locally convex topology

the mixed
topology of the title of this chapter. It turns out that this can be regarded as a generalised type of inductive limit. The latter were systematically studied by GARLING in his dissertation. For this reason, we begin with a treatment of this theory in the generality suitable to our purposes.
For the convenience of the reader, we now give a brief summary of Chapter I. In the first section, we give a basic treatment
of generalised inductive limits. Essentially, we consider a vector space with two locally convex topologies which satisfy suitable compatibility conditions. We then introduce in a natural way a "mixed topology" and this section is devoted to relating its properties to those of the original topologies However, a closer examination of the definitions and results shows that, for one of the topologies, only the bounded sets
INTRODUCTION
3
are relevant. We have taken the consequences of this observation by replacing this topology by a "bornology", that is a suitable collection of sets which satisfy properties which one would expect of a family of bounded sets. We really only use the language (and not the theory) of bornologies and introduce explicitly all the terms that we use. In section 2, we give a list of examples of spaces with mixed topologies. Some of these will be studied in detail (and in more generality) in the following chapters. Others are introduced to supply counterexamples. All are used to illustrate the ideas of the first section. In section three, we define the class of enriched Banach spaces mentioned, restate the results of sections 1 in the form that we shall require them for applications and describe the usual methods for constructing new spaces (subspaces, products, tensor products etc.). It is perhaps not inappropriate to mention here that one of the main reasons for our emphasis on spaces with two structures (a norm and a locally convex topology) rather than on locally convex spaces of a rather curious type is the fact that it is important that these constructions be so carried out that this double structure is preserved and not in the sense of locally convex spaces. The fourth section is devoted to attempts to extend the classical results on Banach spaces to enriched Banach spaces (e.g. BanachSteinhaus theorem, closed graph theorem). The results obtained are perhaps rather unsatisfactory since they involve special hypotheses but, as we shall see later, they can often be applied to important function
I. MIXED TOPOLOGIES
4
spaces. In any case, there are simple counterexamples bhich show that such results cannot hbld without rather special restrictions.
I.1. BASIC THEORY
As announced in the Introduction to this chapter, it is convenient for us to use the language of bornologies. We begin with their definition:
1.1.
Definition: Let E be a vector space. A
ball in E is an
absolutely convex subset of E which does not contain a nontrivial subspace. If B is a ball in E l we write EB for the 03
linear span U nB n=l
11
of B in E. Then
/IB
:
x
is a normon E. If (EB,II
+ inf
llB)
(X > 0 : x
E
XB)
is a Banach space, B is a
Banach ball.
Note that any absolutely convex, bounded subset of a locally convex space is a ball. The following Lemma qives a sufficient (but not necessary) condition for it to be a Banach ball.
1.2.
Lemma: Let B be a bounded ball in a locally convex space
(E,T). Then if B is sequentially.complete for lar if it is Tcomplete), B is a Banach ball.
T
(and in particu
Proof: Let (x,)
be a Cauchy sequence in ( E B l
since B is bounded, (x,) so
that xn
If
E
>
+
5
BASIC THEORY
I. 1
x for
Then,
is TCauchy. Hence there is an x
(xm 
We show that IIxn
T.
]IB).
11
0, there is an N E N
so that
B
E
0.
xllB
xn) belongs to EB
for m,n 2 N. Since B (and so also E B ) is sequentially complete and so sequentially closed, we can take the limit over n to deduce that xm

x
belongs to EB for m 2 N.
Definition: If E is a vector space, a (convex) bornology
1.3.
on E is a family B of balls in E so that (a) E =
U
(b) if B
E
B;
> 0, then
8, X
XB E 8 ;
(c) €3 is directed on the right by inclusion (i.e. if B,C (d) if B A
B then there exists D E B with B
E E
u C
2 D);
B and C is a ball contained in B , then C
E
B
.
subset B of E is Bbounded if it is contained in some ball
in 8 . A
basis for B is a subfamily B 1 of B so that each B
a subset of some B1
E
E
B is
B1.
is complete if B has a basis consisting of Banach balls.
(E,B)
B is of countable type if B has a countable basis.
If
(E,T)
is a locally convex space, then B,,
the family of
all Tbounded, absolutely convex subsets of E is a bornology on E

the von Neumann bornology. In many of our applications B
will be the
von Neumann bornology of a normed space ( E l l [
This is of countable type (the family {nB is the unit ball of E is a basis).
II
I b E
where B
11 ) .
II II
I. MIXED TOPOLOGIES
6
Now we consider a vector space E with a locally convex topology r and a bornology B of CounKable type which are compatible in the followinq sense: and 8 has a basis of rclosed sets. Then we can
8 18,
choose a basis (B ) for B with the following properties: n for each n; (a) Bn + B n L Bn+l (b) each Bn is rclosed; (if (C,)

is a countable basis for 8 , we can define (Bn)
inductively as follows: take B1
=
C1. Once B1,...,Bn have
been chosen, we can find a Tclosed ball in B which contains Bn + Bn + Cn+l. This is our Bn+l). In future, we shall tacitly assume that a given basis (B,)
has the above properties.
Definition: We define the mixed locally convex structure
1.4.
y = Y[E,TI
Let U
=
as follows: m
(Un)n=l be a sequence of absolutely convex Tneighbour
hoods of zero and write m
.....
U (U, n B1 + + Un fl Bn). n= 1 Then the set of all such y ( U ) forms a base of neighbourhoods
y(u)
:=
of zero for a locally convex structure on E and we denote it by
y [ 8 , ~ ] (or simply by y if no confusion is possible).
In the case that 8 is the bornology defined by a norm on E, we write y [ l l
11 , T I
for the structure y [ E , r ] .
The following Proposition gives a natural characterisation of y .
I. 1
1.5. Proposition: (i) y is finer than (ii) y and
7
BASIC THEORY T:
coincide on the sets of B ;
T
(iii) y is the finest linear topology on E which coincides with
T
on the sets of 8.
Proof: (i) if U is a Tneighbourhood of zero, then U 3 y (Un) where Un := 2"U. (ii) if B B

B
C_
8 , we can choose a positive integer r so that
E
Br. A typical neighbourhood of the point xo
E
B
for the topology induced by y on B has the form B fl (xo + y (Un)). Then
Ur n (BB) z Ur
n B r L y((Un)) and
so
(xo + Ur 1 fl B i (xo + y((Un)) n B. (iii) let T~ be a linear topoloqy on E which coincides with
T
on the sets of B. We show that y is finer than 'r1. Let W be a neighbourhood of zero for
T,
and choose neighbourhoods Wn
.
of zero so that Wo = W and Wn + Wn C_ Wnl (n 1 1 ) There are Tneighbourhoods (Un) of zero so that Un n B n G Wn. Then, for any n
(u, n
B ~ ) +
..... + (unnB
~ C_ )
w
and so y((Un)) C_ W.
1.6. Corollary: (i) y is independent of the choice of basis (Bn); (ii) if
T
(i.e. if
and T
' I ~are
suitable locally convex topologies on E
and T~ are compatible with Blthen y [ B , ~ l = y [ B , T l l
if and only if
T
and
T~
coincide on the sets of B.
I. MIXED TOPOLOGIES
8
The localisation property of y expressed in 1.1.5 implies that the continuity of linear mappings is determined by their behaviour on the bounded sets of E.
1.7. Corollary: Let H be a family of linear mappings from E into a topological vector space F. Then H is yequicontinuous if and only if HIB is Tequicontinuous for each B
E
B. In
particular, a linear mapping T from E into F is continuous if and only if TiB is Tcontinuous for each B
E
8.
Proof: We must show that if W is a neighbourhood of zero in F then
n
T' (W) is a yneighbourhood of zero in E. TE H But ( T 1 ~ 1 (W) l = T' (w) f7 B and so we can apply T EH TEH 1.5. (iii).
n
n
In view of the above property, the following Lemma of GROTHENDIECK will be useful:
1.8.
Lemma: Let ( E , T ) be a locally convex space, B an absolute
ly convex subset of E and T a linear mapping from B into a locally convex space F. Then TiB is rBcontinuous if and only if it is TBcontinuous at zero.
Proof: Let V be an absolutely convex neiqhbourhood of zero in
F, x a point of B. We must find a neighbourhood E so that T((x
+
U)
(l B ) C Tx
+
U
of zero in
V. We choose U (absolutely
1.1
BASIC THEORY
9
convex) so that T(B n (U/2)) C_ V/2. Then if y xy
E
B

B
=
E
((x+U) n B)
2B and the result follows from the inclusion
T(2B n U) C V.
As we shall see later, the topology y is, in the interesting cases, never metrisable (or even bornological)
. However,
it does, sometimes, have one useful property in common with such spaces.
1.9.
Proposition: Suppose that 8 has a basis of Tmetrisable
sets. Then a linear mapping from E into a topological vector space F is continuous if and only if it is sequentially continuous.
Proof: For any B
E
B , TIB is sequentially continuous and so
continuous. The result then follows from 1.7.
In the following Propositions, we characterise certain properties (boundedness, compactness, convergence) with respect to y directly in terms of 8 and
1.10.
Proposition: A sequence (x,)
T.
in E converges to x in
(E,y) if and only if Exn) is 8bounded and xn
Proof: We can suppose, that x = 0. By show that if xn
+
0 in (E,T), then
+
x in
(ErT).
1.5 it suffices to
{xn) is 8bounded. If this
were false, we could find a subsequence (x ) so that x "k "k
4
Bk.
I. MIXED TOPOLOGIES
10
Since Bk is rclosed,we can choose a Tneighbourhood of
4 Bk + 2Uk and we can suppose that zero Uk so that x "k (k > 1). Then for each k > 1 k' + k' 'k1 m
y((un)) =
u
n= 1
..... + un n B ~ )
( u l n B~ +
m
c u (B1 + p= 1
..... +
Bkl
+
Uk
.....
+
+
'k+p)
c Bk + 2Uk. Hence xnk
4
that (x,)
is a ynullsequence.
1.11.
y ( (Un)) for each k which contradicts the fact
Proposition: A subset B of E is ybounded if and only
if it is Bbounded.
Proof: Suppose that B is Bbounded. Then B E Br for some r. Let (Un) be a sequence of absolutely convex neighbourhood of zero. Then there is a K > 1 so that B L KU,.
Hence
B S K(Urn Br) E K y ((Un 1 and so B is ybounded.
Now suppose that B is ybounded. If B were not Bbounded, we could find a sequence (x,) positive integer n. Now n'x n Bbounded  contradiction.
in B so that xn +
4
nBn €or each
0 in ( E , y ) and so {n
1
xn) is
1.12. Proposition: A subset A of E is ycompact (precompact, relatively compact) if and only if it is Bbounded and rcompact (precompact, relatively compact).
1.1
BASIC THEORY
11
Proof: This follows immediately from 1.11 and 1.5.(ii).
We recall that a locally convex space is semiMonte1 if its bounded sets are relatively compact. It is Montel if, in addition, it is barrelled. In the next Proposition, we characterise semiMonte1 mixed spaces. As we shall see below, nontrivial mixed topologies are never barrelled

and so
never Montel.
1.13. Proposition: (E,y) is semiMonte1 if and only if B has a basis of .rcompact sets.
Proof: This is a direct consequence of 1.12 and the definition.
We now consider the completeness of (E,y). It is customary to use here a completeness theorem of RAIKOV which generalises KBTHE'S completeness theorem. However, this result is rather inaccessible so
we give a direct proof, using a method of DE WILDE and HOUET.
1.14. Proposition: (E,y) is complete if and only if B has a basis of .rcomplete sets.
Proof: The necessity of the given condition follows from 1.5. (ii) and 1.11. Now suppose that we have a basis (B,)
(i,;)be 5, choose xo 4
for B consisting of
.rcanplete sets. Let
the locally convex completion
of E. If E
\ E. Then for each n, xo
#
4
2Bn
I. MIXED TOPOLOGIES
12
and so by the HahnBanach theorem ([561, p.39, Folg. 2 1 , there is an fn
E
(i) I
only finitely many
with fn
E
0 Bn and fn(xo) = 2. Since
fn are nonzero on a given B ,
Ifn) is
A
equicontinuous on E (1.7) and hence also on E. Then by the theorem of ALAOGLUBOURBAKI ([311 , p.250, 5 10.10(4) 1 Efn) A
is u (El ,E) compact and so has a a(E' ,G)limit point f. Then f vanishes on each Bm and so on E. Thus f = 0 which contradicts the fact that f(xo)
=
2.
We now make some remarks on the case where 8 is the bornology associated with a norm on E. Then there are three natural locally convex topologies
T,Y
and
II III
the norm topology,
on E and we discuss their distinctness. We first note that the equality
'I
= y means essentially that T is already a mixed
topology i.e. we have gained nothing by mixing. On the other hand the condition y = T 11
11
means that we are in the trivial
situation (trivial from the point of view of mixed topologies)
of a normed space "enriched" by its own topology (since y[ll
II,'I]
=
y [ l l I I , T ~ ( 111).
The following result shows that if
y belongs to the traditional classes of wellbehaved locally
convex.spaces, then we have this trivial situation.
1.15. Proposition: If y is bornological (in particular, metrisable) or barrelled, then T =
T
II II.
Proof: If y is bornological, then the identity mapping from (E,y) into ( E , I I
11) , being
bounded (1.11)
is continuous
I. 1 and so y 1 'I
II II
. The
being a barrel in ( E , y )
11
13
inverse inequality is obvious.
If y is barrelled, then B
are assuming that B I I
BASIC THEORY
,
II I l l
the unit ball of (El11
11) ,
is a yneighbourhood of zero (we
is 'Iclosed  strictly speaking, this
need not to be the case. However, it follows easily from the compatibility conditions that we can find an equivalent norm
so that this condition is satisfied).
The essential property of y is that given in 1.5.(iii). The neighbourhood basis used in the definition was chosen so that this would hold. However, in applications, we shall frequently require a much less obvious description of yneighbourhoods of zero. Let U be as in 1.4 (except that it is now convenient to index from zero to infinity) and put
Y ( U )
m
:=
u0 n 0 (un + n= 1
B~).
Then the family of such sets forms a neighbourhood basis for a locally convex topology on E which we denote by y[B,'11.
Proof: Firstly, 7 is coarser than 'I on each set Bk. k ., ( B +~ un) n B~ and this is a For y(U) n B~ = (u0 n n=1 T neighbourhood of zero. Hence by 1.5. (iii), y is finer I Bk than y.
n
Now we show that
y is coarser than
T.
Let y((Un)) be a typical
yneighbourhood of zero. There exists a decreasing sequence
I. MIXED TOPOLOGIES
14 m
of cneighbourhoods of zero so that
(Vn)n=O
(n
1 + Vnl E Untl 'yn ( (Vn) 1 y ( (U,) 1 .
Choose x
C_
..) . We
0,l ,2,.
=
shall show that
y( (Vn)) .
E
Then x
for each n, x has a decomposition y, + zn zn
E
x,+. and
x1
Vn. Define
Znl
. .+ xn+ zn
:= y1 , xn := y,
y l + (y2 y 1 ) +.
=

E
Vo and,
where y,
E
Bn,
Yn1 (n > 1). Then
. .+ (ynynl1 +zn
 y, + zn
=
x
 xn + zn.
We have
xn
and
xn  yn

= z n1


zn
Y,1
Vn1
+
E
Bn
Bn1 E Bn+l*
Hence xn E Un+l Bn+l * If no is so chosen that x On the other hand, Zno
E
vn E
E
B
E
+
"0
Vn1
+
Vn1
then zno = xyno
Un+1
E
Bno+l.
V n o s Uno+2.
Hence we have x = x
1
+...+
xn+zn
E
U2 n B2+ ...+Uno+l
n Bno+l
un0+2
n Bno+2
1.17. Corollary: y has a basis consisting of rclosed s e t s .
Proof: If Un is an absolutely convex cneighbourhood of zero, then
(2lU )
n
+
Brit
(2'Un)
+ B n C Un + B,
03
and so y((2'Un))
E Uo n
n= 1 and this implies the result.
((2'Un)
+ Bn) c_y((Un))
1.1
15
B A S I C THEORY
We now consider duality for ( E , y ) . E has three dual spaces:

E; E'
Y
Ei
the dual of the locally convex space ( E , . r ) ; the dual of the locally convex space ( E , y ) ; the dual of the space ( E , B ) , that is, the space of linear forms on E which are bounded i.e. bounded on the sets of 8.
Then E; E E ' E Ei
Y
and we regard each of these spaces a s a
locally convex space with the topology of uniform convergence on the .rbounded sets (resp. the ybounded sets, resp. the sets of B). Since B is of countable type, E ' is metrisable and it is also clearly complete (since the uniform limit of bounded functions is bounded). Hence it is a Frgchet space. Our next results characterise E ' and its equicontinuous sub
Y
sets in terms of E;
and E i .
Proposition: (i) E'; is a locally convex subspace of Ei
1.18.
(ii) E7; is the closure of E:
;
in E i and so is a Frgchet space.
Proof: (i) follows directly from 1 . 1 1 . (ii) E' is closed in Ei since the limit of a sequence in E' Y Y is continuous on the sets of B and so is in E ' by 1 . 7 .
Y
We show that E;
B and
E
is dense in E'. Let B be a .rclosed ball in Y be a positive number. If f E E ' then there is an
Y
absolutely convex .rneighbourhood U of zero so that If(x)l if x
E
B n U i.e. f belongs to E ( B n UIo (polar in E",
IE
the
algebraic dual of E). Now the polar of B n U is the closure
I. MIXED TOPOLOGIES
16
of
1 / 2 (Bo
+ Uo) in u(E",E). But this set is closed since
Uo is u (E",E)compact by the theorem of ALAOGLUBOURBAKI and so
(B
n u)Oc
+
Hence f belongs to E(UO 0
to EU if x
E
C_
+ u0.
BO
Bo) and so there is a g belonging
ElT such that fg belongs to EB'
i.e. If (XI g(x) I <
B.
1 . 1 9 . Corollary: Let
T
~
,
Tbe~
locally convex topologies on E
which are compatible with €3 and suppose that 'rl and the same dual. Then
1.20.
E
y[B,'rl]
and
T~
have
~ [ B , T ~ ]have the same dual.
Proposition: A subset B of E is yweakly compact if
and only if it is €3bounded and a(E,E;)compact.
Proof: The condition is clearly necessary. It is sufficient since if B is Bbounded then, regarded as a subset of the dual of E'
Y'
it is equicontinuous and so the weak topologies
defined by El and its dense subspaces E{ coincide on it
Y
(see
[ 6 1 1 , p.83, 111.4.5).
1.21.
Corollary: (E,y) is semireflexive if and only if B
has a basis of u(E,E;)compact
Sets.
1 . 2 2 . Proposition: A subset H of Ei is yequicontinuous if
and only if it satisfies the following condition:
I. 1
17
BASIC THEORY
for every strong neighbourhood U of zero in Ei, there is a 'requicontinuous set HI in E'T so that H G U
+ H1.
Proof: Sufficiency: choose B
E
B,
E
>
0 . It is sufficient
to find a Tneighbourhood V of zero so that if x f E H, then
I f (x)I S E. We choose V so that H c (E/~)BO+ (E/~)v'G E(V n
E
B n V,
B)O
(for the last inclusion, cf. the proof of 1.18.(ii)). Necessity: suppose that H is yequicontinuous and U is a strong neighbourhood of zero in El
Y'
We can suppose that U
=
0 Bk
for some positive integer k . Then there is a yneighbourhood of zero y ( (Un)) so that
. Then
1.23. Corollary: Let (xn) be a nullsequence in E'
Y
Ixn) is yequicontinuous.
Proof: If U is a strong neighbourhood of zero in E;,
then all
but finitely many of the elements of the sequence lie in U. Hence we can apply 1.22.
1.24. Corollary: Let A be a compact subset of E'
Y'
yequicontinuous.
Then A is
I. MIXED TOPOLOGIES
18
Proof: Since E' is a Fri?chet space, A is contained in the
Y
closed, absolutely convex hull of a null sequence ( 1 5 6 1 , Ch.7,
9
2 , Lemma 2). By 1.23 the range of this sequence is equi
continuous and hence s o is its a(E',E)closed absolutely
Y
convex hull and this set contains A .
If ( F , T ~ )is a locally convex space and E is a subspace, there is a natural vector space isomorphism between F'/Eo and El induced by the restriction mapping from F' onto E'. Since the latter mapping is continuous for the strong topologies, the map from F'/Eo onto E' is continuous. In general, it is not a locally convex isomorphism. This is the case, however, when E has a mixed topology.
1.25. Proposition: Let (E,y) be a locally convex subspace of ( F , . r l ) . Then the strong topology on F'/Eo as the dual of (E,y) coincides with the quotient of the strong topology on F'.
Proof: We show that the natural mapping from E' onto F'/Eo is
Y
continuous. Since El is a Fr6chet space, it suffices to show
Y
that every sequence which converges to zero in E' is bounded Y in F'/Eo. But such a sequence is yequicontinuous (1.23) and
so is the restriction of an equicontinuous set in F' (HahnBanach theorem). The image of such a set in F'/Eo is bounded.
1.26.
Corollary: If, in addition, E is dense in F, then every
bounded set in F is contained in the closure of a bounded set in E.
I. 1 Proof: By 1 . 2 5 ,
BASIC THEORY
19
the natural vector space isomorphism between
the duals of E and F is an isomorphism for the strong topologies and this is equivalent to the statement of the Corollary (bipolar theorem).
1 . 2 6 can be used to give an alternative proof of 1.14.
1.27.
Remark: It follows from the results of this paragraph
that the spaces of type (E,y[B,'C]) can be internally characterised as the class of locally convex spaces (E,T) with the following properties: a) BT is of countable type: b) an absolutely convex subset V of E is a neighbourhood of zero if and only if V n B is a Tneighbourhood of zero for each B
E
8,.
Hence they form a generalisation of the class of (DF)spaces of GROTHENDIECK. They have been studied from this point of view by NOUREDDINE who calls then Dbspaces (see [411). NOUREDDINE has shown that they possess many of the properties of (DF)spaces. We mention without proof the following: I. if (Vn) is a sequence of neighbourhoods of zero in
E, there is a sequence (an) of positive scalars so that nanVn is a neighbourhood of zero. 11. every continuous linear mapping from E into a metrisable locally convex space F is bounded (i.e. it carries some neighbourhood of zero in E into a bounded set in F).
20
I. M I X E D TOPOLOGIES 111. t h e c l a s s of Dbspaces i s c l o s e d under t h e
f o r m a t i o n o f q u o t i e n t s , c o u n t a b l e l i m i t s , s u b s p a c e s of f i n i t e codimension and even s u b s p a c e s of c o u n t a b l e codimension when t h e s p a c e i s complete. IV.
a Dbspace E h a s t h e p r o p e r t y (B) of PIETSCH and
so e v e r y a b s o l u t e l y summable sequence i s t o t a l l y summable
( i . e . i s a b s o l u t e l y summable i n t h e normed s p a c e (EB,II f o r some B
E
B.,).
\IB)
As a consequence, t h e theorem o f DVORETSKY
ROGERS h o l d s i n a s i m p l e Dbspace
( i . e . a Dbspace whose von
Neumann bornology i s g e n e r a t e d by a norm): i f E i s i n f i n i t e d i m e n s i o n a l , t h e n E p o s s e s s e s a sequence which is summable b u t n o t a b s o l u t e l y summable.
1.2.
EXAMPLES
A. L e t F be a F r g c h e t s p a c e and d e n o t e i t s d u a l by E . On E
we c o n s i d e r t h e s t r u c t u r e s : 8 := t h e c o l l e c t i o n of a b s o l u t e l y convex, e q u i c o n t i n u o u s s u b s e t s of E ( e q u i v a l e n t l y , t h e bounded sets o f E , p r o v i d e d w i t h t h e s t r o n g t o p o l o g y as t h e d u a l o f F ) ; U(E,F)

t h e weak t o p o l o g y d e f i n e d on E by F.
Then t h e t r i p l e
( E , 8 , a ( E , F ) ) s a t i s f i e s t h e c o n d i t i o n s of
1 . 4 and so w e can form t h e a s s o c i a t e d mixed t o p o l o g y . W e
i d e n t i f y t h i s t o p o l o g y . F i r s t l y , t h e d u a l of
(E, a(E,F)) is,
of c o u r s e , F and on F t h e t o p o l o g y of uniform convergence on B i s t h e o r i g i n a l t o p o l o g y . Hence, by 1 . 1 7 , t h e d u a l of
EXAMPLES
1.2
21
(E,y) is F. Now the u(E,F)equicontinuous subsets of F are the bounded, finite dimensional subsets. Hence, by 1 . 2 2 and a standard characterisation of precompact subsets of a locally convex space (1561, p.58), the yequicontinuous subsets of F are precisely the precompact subsets i.e. y is the topology
T
C
(E,F) of uniform convergence on the compact
subsets of F.
B. If S is a set and p
E
Lp(S) denotes the space of
[l,m],
complex functions x on S so that llxll < P
llxllm (lp(S)I T~
11
[Ip)
:=
sup (Ix(t)l : t
E
m
where
s). consider the topology
is a Banach space and we
of pointwise convergence on S. Then (lp(S),I[ Ilp,rs) satis
fies the conditions of 1 . 4 and we can define the corresponding mixed topology. For p > 1 , lp(S) is the dual of l/p
+ l/q
= 1 (p
>
unit ball of lp(S) ness !
)
1 ) and T~
q = 1 (p = m)
eq (S) where
. Hence, since on the
coincides with u (lp(S),lq(S)) (compact
we are in the situation of A and so y = rC(lp,lq).
Similarly, for p co(S) := {x
= 1,
l1 (S) is the dual of
E lm(S)
: for each
E
> 0, there is a finite
subset J of S so that Ix(t)l < for t and so y is rC(l I ,c0).
4
J)
E
I . M I X E D TOPOLOGIES
22 C.
If p
<
p l , then
L p ( S ) 5 Lp' (S)
on l p ( S ) t h e mixed s t r u c t u r e
and s o w e can c o n s i d e r
11
Ilp,rP1 I where T PI d e n o t e s t h e topology induced on L p ( S ) by (1 ( 1 T h i s mixed topology PI i s d i s t i n c t from t h a t d e f i n e d i n B s i n c e t h e u n i t b a l l of y[

11 1 1PI compact.
LP(S) i s n o t
I t f o l l o w s from 1 . I 8 t h a t t h e
d u a l of Lp ( S ) under t h i s mixed t o p o l o g y i s t h e d u a l of L 1 ( S ) i s c o ( S ) .
D.
l q ( S ) ( p > 1 ) and
L e t T be a l o c a l l y compact s p a c e , S a c o l l e c t i o n of s u b s e t s
of T so t h a t U S i s d e n s e i n T. W e d e n o t e by C m ( T ) t h e s p a c e of bounded, c o n t i n u o u s f u n c t i o n s from T i n t o C. Then
11 [Irn
: x
sup I l x ( t ) l : t
i s a norm on C m ( T ) and (C" (T),I[
),1
E
TI
i s a Banach s p a c e .
I f A C T then
pA : x
+t
{sup I x ( t )1 : t
i s a seminorm on C"(T)
E
A)
and w e d e n o t e by T~ t h e l o c a l l y convex
s t r u c t u r e g e n e r a t e d by {pA : A
E
S ) . Then ( C m ( T ) , I I l L , r s )
s a t i s f i e s t h e c o n d i t i o n s of 1 . 4 and so w e can form t h e mixed topology
y[
11
II,,rS]
which we d e n o t e by
Bs.
E. L e t G be an open s u b s e t o f t h e complex p l a n e and d e n o t e
by H"(G)
t h e subspace of C m ( G ) c o n s i s t i n g of holomorphic
f u n c t i o n s . Then ( H m ( G ) , l l compact s u b s e t s of G and
l l m , ~ K ) where K i s t h e f a m i l y of 11 IL and r K are d e f i n e d as i n D )
s a t i s f i e s t h e c o n d i t i o n s of 1 . 4 . mixed t o p o l o g y by 8 .
We d e n o t e t h e c o r r e s p o n d i n g
23
EXAMPLES
1.2
F . L e t S be a l o c a l l y compact s p a c e . C o o ( S )
denotes t h e
s e t o f f u n c t i o n s i n C m ( S ) w i t h compact s u p p o r t . Thus Coo(S)
=
u
CK(S)
where K ( S ) d e n o t e s t h e f a m i l y o f
K&K(S)
compact s u b s e t s o f S and C K ( S ) d e n o t e s t h e f u n c t i o n s i n C m ( S ) which have s u p p o r t i n K. C K ( S ) , w i t h t h e norm i n d u c e d from C m ( S ) i s a Banach s p a c e . W e d e f i n e a b o r n o l o g y 8 on C o o ( S )
as f o l l o w s : B
E
8 i f and o n l y i f t h e r e i s a K
E
B i s a bounded b a l l i n C K ( S ) . I f S i s ocompact
K ( S ) so t h a t (i.e. the
u n i o n of c o u n t a b l y many compact s u b s e t s ) , t h e n 8 i s o f c o u n t a b l e ( C o o ( S ) , B , T ~ )s a t i s f i e s t h e c o n d i t i o n s o f 1 . 4 .
t y p e . Then
G.
L e t (F,)
be a s e q u e n c e o f Banach s p a c e s and l e t co
E :=
C
m
Fn;
F :=
n=l
n= 1
Fn
t h e v e c t o r s p a c e d i r e c t sum and p r o d u c t r e s p e c t i v e l y . On F
w e c o n s i d e r t h e p r o d u c t t o p o l o g y T and on E t h e von Neumann b o r n o l o g y of t h e d i r e c t sum l o c a l l y convex t o p o l o g y on E (so t h a t a s e t B C E i s bounded i f it i s bounded i n a subs p a c e o f t h e form
m
n;l o f 1 . 4 are s a t i s f i e d .
H.
Fn
f o r some m ) . Then t h e c o n d i t i o n s
L e t ' F b e a F r & c h e t s p a c e , G a Banach s p a c e and d e n o t e by
E t h e s p a c e L(F,G) o f c o n t i n u o u s l i n e a r mappings from F i n t o G .
I f 8 i s a b o r n o l o g y on F , c o n t a i n e d i n t h e von Neumann bornol o g y , we d e f i n e on E t h e f o l l o w i n g s t r u c t u r e s : Beq

t h e b o r n o l o g y g e n e r a t e d by t h e e q u i c o n t i n u o u s b a l l s i n E;
I. M I X E D TOPOLOGIES
24 T~

t h e t o p o l o g y of uniform convergence on t h e s e t s of B.
Then t h e c o n d i t i o n s of 1 . 4 a r e s a t i s f i e d .
I . L e t E be a @  a l g e b r a w i t h u n i t e l S ( E ) t h e s e t of s t a t e s
of E i . e . t h e p o s i t i v e l i n e a r forms f Then each
f
pf : x on E . I f x llxll
E
E
E ' with
f ( e ) = 1.
d e f i n e s a seminorm
S(E)
e
E
Cf
(X*X)I
1/ 2
E
= sup c p f ( x ) : f
E
S(E)}
and so w e can form t h e mixed topology
y[
11 I ~ , T ~ ]
where T~ i s
t h e l o c a l l y convex t o p o l o g y g e n e r a t e d by t h e f a m i l y cPf
: f E S(E)}.
J . L e t (X,d) be a m e t r i c s p a c e . For c o n v e n i e n c e , w e suppose
t h a t t h e a s s o c i a t e d t o p o l o g i c a l s p a c e i s compact. I f
x
E
C(X),
x i s Lipschitz i f
( A t h e d i a g o n a l s e t ) i s f i n i t e . W e d e n o t e by
s p a c e o f such f u n c t i o n s . On
Lip ( X I
L i p (X)
the
we consider t h e following
norms :
11 II II Then ( L i p ( X )

t h e supremum norm;

t h e norm
,11 ] I I
T~~
IL)
x
c$
max ilIxIIrn, IIxIIL)
s a t i s f i e s t h e c o n d i t i o n s of 1 . 4 .
1.3 K.
L e t (E,II
11)
SAKS SPACES
25
be a Banach s p a c e w i t h b a s i s (x,)
i.e.
i s a sequence i n E w i t h t h e p r o p e r t y t h a t for e v e r y x
t h e r e i s a unique sequence ( A n )
(x,) E
E
of scalars s o t h a t
m
x =
C
n= 1
X n x n . Then i t i s c l a s s i c a l t h a t n
lllxlll
:= s u p
C I I c x ~ x ~ I I:
n
E
IN
1
k= 1
i s a norm on E , e q u i v a l e n t t o
11 11.
On E t h e seminorms pn : x M11
n k= 1
'kXk
11
d e f i n e a l o c a l l y convex topology T. Then (E,111
111 ,T)
satisfies
t h e c o n d i t i o n s of 1 . 4 .
1.3. SAKS SPACES
I n t h i s s e c t i o n we c o n s i d e r s p e c i a l t y p e s of s p a c e s w i t h mixed t o p o l o g i e s

t h o s e whose bornology i s induced by a norm.
We propose t o c a l l them Saks s p a c e s s i n c e t h e y c o i n c i d e e s s e n t i a l l y w i t h t h e s p a c e s i n t r o d u c e d under t h i s name by
ORLICZ ( t h e p r e c i s e r e l a t i o n s h i p between t h e s e c o n c e p t s i s d i s c u s s e d i n t h e n o t e s ) . W e a r e concerned h e r e w i t h t h e b a s i c c o n s t r u c t i o n s on Saks s p a c e s . S i n c e t h e s e are based on t h e c o r r e s p o n d i n g c o n s t r u c t i o n s on Banach s p a c e s (which t h e y g e n e r a l i s e ) , w e r e c a l l t h e l a t t e r b r i e f l y (see SEMADENI [ 6 6 ] ) .
The c o n s t r u c t i o n of s u b s p a c e s and q u o t i e n t s p a c e s of normed s p a c e s i s wellknown.
If {(Ear11 l l a ) j a
A
i s a f a m i l y of normed
I. MIXED TOPOLOGIES
26
spaces, we define new normed spaces as follows:
II
denote by E the Cartesian product
Ea
and define extended
~ E A
norms
on E. Then if
(E,
El := {x
E
E
IIxII, <
m}
Em := {x
E
E : 11x11, <
m}
Ill)
rI1
and (EmrII ),1
:
are normed spaces. They are Banach
B 1 Ea aEA for E, and Em resp. They satisfy the universal
if (and only if) each Ea is a Banach space. We write and
B II Ea CIEA
property that one expects of a sum and a product if we restrict attention to linear contractions.
If A is a directed set and
is a projective spectrum (resp. an inductive spectrum) of normed spaces (i.e. each with then
= idE
a
7~
and each i
Ba (resp. iaa = id
Ea
aB
)
is a linear contraction
for each a and, if a
I @ Iyr
(resp iaY = iBY o iaB 1 1 , then we define the projective limit of the first spectrum as the (closed) subTI
ya =
T
Ba
7T
O
y@
space
and denote it by
BEm{Ea,TIBa}. Similarly, the inductive
limit of the second spectrum is the quotient of
B C Ea with ~ E A
1.3
SAKS SPACES
27
respect to the closed subspace generated by elements of the

form (x
Y
i
(x ) ) (we are regarding each space E as a B B B C E, in the obvious way). In fact we shall only
BY
subspace of
ClEA
require the following special representation of an inductive is a closed subspace of a given
limit: suppose that each E,
Banach space F and that A is so ordered that a
I
B if and only
if
E, 5 E (and then i is the natural injection): then the B a$ inductive limit is naturally identifiable with the closure
Of
in F (see SEMADENI [ 6 6 1 , 9 11.8.3, p.212).
E,
,!A
3.1. Lemma: Let ( E , T ) be a locally convex space,
with unit ball B
II II
11 [la norm
on E
Then the following are equivalent:
(a) B I 11~ is .cclosed; (b) (c)
11 11 11 11
is lower semicontinuous for T: =
sup{ p : p is a Tcontinuous seminorm with P
Proof:
(c) =>
5
II 11).
(b) and (b) =>
(a) follow immediately from
the elementary properties of semicontinuous functions. (a) =>
(c) : suppose x
E
E with llxll > 1
i.e. x
4
BII
We need only find a continuous seminorm p on E so that p and p(x) > 1. By the HahnBanach theorem, there is an f
so that Ilfll seminorm.
I
1 on B
II I1
E
and f ( x ) > 1. Then IlflI is such a
II. I 11 11 (E,T)'
28
I. MIXED TOPOLOGIES Definition: A Saks space is a triple (E,ll
3.2.
E is a vector space, and
11 11
T
11,~)
where
is a locally convex topology on E
is a norm on E so that B
II II’
the unit ball of (E,11
11) ,
is .rbounded and satisfies one of the conditions of 3 . 1 .
11,~)
If (E,II
and ( F , l l
111,~1)
are Saks spaces, a morphism from
E into F is a linear norm contraction from E into F
TiBll
II
if B I I
continuous.
is
II
A
Saks space ( E , I I
11)
is .rc:omplete. Then ( E , l l
11,~)
so that
is complete
is a Banach space (1.2).
In constructing Saks spaces, one occasionally produces triples (E,II
11,~)
of B I I
where all but the last condition (on the closure
1 1 ) is
satisfied. This forces us to take the following

precaution: we define B1
:= B I I
11 Ill of 1 1 1 , ~ ) is a
11
(closure in
T)
. Then the
(as defined in 1 . I ) is a norm
Minkowski functional
B1
on E so that ( E l l [
Saks space. The following Lemma
ensures that we do not lose any morphisms in this process :
3 . 3 . Lemma: Let T be a linear mapping from E into a locally
convex space F so that TI Then TIB
B I I II
is Tcontinuous.
is Tcontinuous.
1
Proof: By 1 . 8 , it suffices to show that T ~ is B continuous ~ at zero. Let U be an absolutely convex neighbourhood of zero in F and choose an open neighbourhood V of zero in E so that T(V n B 11
11) c
For if x
E
1 / 2 U. Then
T(V n B 1 ) C U.
V n B1 and we choose X E ]0,1[
then we can, by the continuity of T in B
so that Ax
II II‘
find y
E B E
II II’ II II
V n B
1.3
so that T(Ax) Tx
=

T(Ay)
A’(T(Ax)
SAKS SPACES
29
X/2 U. Then
E

T(Xy)) + Ty
E
U.
11,~) I\,T] 
3.4. The associated topoloqy: If ( E , \ l
is a Saks Space,
we can form the mixed topology
it is called the
y[ll
assuciated locally convex topology of E. Then a morphism between two Saks spaces is continuous for the associated topologies (1.5. (ii) and 1.7) and a Saks space is complete if and only if (E,y) is a complete locally convex space (1.14). We repeat that, despite these facts (and others to follow), the relevant structure is that of a Saks space and not a locally convex space (this is one of the reasons that we have been careful not to forget the norms in the definition of a morphism

thus a ycontinuous linear mapping need not be a morphism although it is, of course, a scalar multiple of one). Hence we shall stubbornly persist in defining notions like completeness, compactness etc. in terms of the structure as a Saks space even when these can be expressed in terms of y (using the theory of
5
3.5.
1.1).
Subspaces and quotient spaces: Let ( E l11
space, F a vector subspace of E . Then if
11,~)
11 I I F , ~ F
be a Saks denote the
norm (resp. the locally convex topology) induced on F , ( F , ( I is a Saks space. We shall see later that coincide with
y[
11
ll,T]
y[ll
IIFfTF)
I I F f ~ F 1 need not
IF.
If F is a yclosed subspace, then we denote by
F I I 11
and F~ the
structures induces on the quotient space E / F . The triple
I. MIXED TOPOLOGIES
30
(E/F,FII
ll,F~)
need not be a Saks space since it can happen
11)
that the unit ball of (E/F,FII
is not FTclosed. However,
by the process described before Lemma 3 . 3 , we can obtain a Saks space which we shall call the quotient Saks space.
I] 1 1 , ~ )
3 . 6 . Completions: Let (E,
ET
n
the completion of the locally convex space (E,T). We write B
for the closure of B 11 in
be a Saks space and denote by
ElT.
Then if
11 ]IA
11
in
ET
and $ for the linear span of
is the Minkowski functional of
locally convex structure induced on
6
from
sT, (;,]I
and
6 is the is a
complete Saks space. We call it the (Saks space) completion of E. It has the following universal property: for every morphism T from (Ell[1 1 , ~ ) into a complete Saks space (F is a unique morphism
from
(E,II /In,;)
into
extends T (for we can extend T firstly to
11 Ill
, T ~ ) ,there
F,II
lll,Tl)which
y uniform continui
ty and then to $ by linearity), As
an amusing example, consider the Saks space (E,ll II,IJ(E,E'))
where (E,ll
11)
is a normed space. The completion of (E,a(E,E'))
is (Em)+: the algebraic dual of E' (ROBERTSON and ROBERTSON [ 5 6 1 , Sat2 18, p.71). The closure of B in (El)?: is its bipolar, i.e. the unit ball of E", the bidual of (E,11
11).
Hence the completion
of (E,ll II,u(E,E')) is El', with the Saks space structure described in 2 . 1
(as the dual of El). Thus we can regard the bidual as
a completion.
1.3
3 . 7 . Products and projective limits: Let
be a family of Saks spaces. We can give normed space product of {E,},
and so is .c,closed.
T
~
IL,T,)
IT E,. C~EA If {
(E,,
,T,)
the
the , product of { T ~ ) .
TI B clEA 1111, For the same reason, (Em,II I l m , ~ m ) is
II ,tI
of Em is the product
complete if and only if each (E,,II (E,,II
11 1, (Ern, 1 1 Ic) , {
a Saks space structure by
considering on Em the topology Then the unit ball B
31
SAKS SPACES
IL,.cm) is.
the Saks space product of {E,}
We call
and denote it by
S
:~ EB~d E,,
T
c1 I f?
, a ,@
system of Saks spaces ( s o that the
E
A)
is a projective
' s are Saks space 6, morphisms), we can define its (Saks space) projective limit
(E,ll
11,~)
T
as follows: as in the first paragraph of this section,
we consider the space E of threads as a subspace of give it the induced structure in the sense of 3.5.
ll E, and ,€A It can easi
ly be checked that the unit ball of E is .c,closed
in
S
S
TI E,
and so E is complete if each E, is. We denote this projective limit by
Slim {E,,IT~,}.
Sums
and inductive limits of Saks
spaces can be defined without difficulty but we shall not require them. We recall that each Banach space can be regarded as a Saks space

namely the Saks space (E,II lI,.cll
11).
The following
result shows that the Banach spaces are in a certain sense dense in the Saks spaces and corresponds to the fact that complete locally convex spaces are projective limits of Banach spaces.
I. MIXED TOPOLOGIES
32
3 . 8 . Proposition: A Saks space (E,ll
11 ,T
is complete if
)
and only if it is the Saks space projective limit of a system of Banach spaces.
Proof: The sufficiency follows from the remarks above. Necessity: denote by S the family of Tcontinuous seminorms p on E which are majorised by
11 11.
Then S is a directed set
with the natural (pointwise) ordering and If p
E
S we denote by
E
P
P
p
:= (x E E : p(x) =
I q,
=
sup S ( 3 . 1
P
.
where
01, with the norm induced by p). If 6
A
let T~~ denote the natural contraction from E
then
)
the Banach space associated with p
(i.e. the completion of the space E / N N
11 1 1
into E P' q is a projective system of
A
p I q) PI Banach spaces and it is not difficult to show that (E,ll
11,~)
is its projective limit.
We remark that if (E,II
11,~)
is not complete, then the above
construction produces its completion in the sense of 3.6. As an example of a Saks space product, consider a family {Ta)acA of locally compact spaces and let S,
be a family of subsets
of T as in 9 I.2.D. Denote by T the topological sum of the
u S, (T, is regarded as a ~ E A subspace of T) Then the underlying vector space of S TI Cm (T,) CX€A can be naturally identified with C"(T) and this induces a Saks spaces (T,)
and by S the family
.
space isomorphism between (C" (TI,11
~ L , T and ~)
S II C" (Ta). This ,€A example displays the suitability of a Saks space product in a situation where any locally convex product is hopelessly inadequate.
1.3
33
SAKS SPACES
If T is a locally compact space, then I P K l ,K : C(K1)
+ C(K)
K,K1
E
K(T)
K C K1I
r
(where C ( K ) denotes the space of continuous, complexvalued functions on K and p
K1 , K
is the restriction operator) is a
projective spectrum of Banach spaces and its Saks space projective limit is. CC" ( T I I
II [I,, T ~ .)
3.9. Duality: The dual of ( E , l l
11,~)
is defined to be the
linear span of the set of morphisms from E into a: i.e. it is the dual of the locally convex space ( E , y

[
11 11 ,T 1) 
we denote
it by El. It is a Banach space. Suppose now that E is the Saks
Y
space projective limit of the spectrum (
1
:~ E ~
B
~
Eci, ci I B ,
ci
,B
E
A}
of Saks spaces. We assume, in addition, that rci(B
II I t )
T
ci
dense in B
II IIcl
for each
ci
is
(rciis the natural morphism from
E into E 1 . This condition is satisfied, for example, by the ci
canonical representation of E ( 3 . 8 ) . Then each ( E a ) regarded as a Banach subspace of ( E , injection i
aB
of
71
from (E
'
U Y
11 11)
into ( E B ) ;
I
can be
and the natural
( a 5 B ) is the transpose
Ba *
3.10. Proposition: (a) E;
is the Banach space inductive limit
of the spectrum { iciB
: (E,);
d (ED);,
(b) a subset H of (E,11
11)
I
c1
I
B 1 a,B
E
A).
is yequicontinuous if and only if
there is a sequence (an) with values in A and, for each n
E
N,
I. MIXED TOPOLOGIES
34
equicontinuous; n (ii) c Sup{IIfll : f E Hn} < m ; n (iii) H 5 C Hn (i.e. if f E H, f (i) Hn is
'ca
W
=
C
fn where fn
n= 1
E
Hn).
Proof: (a) By a standard result on the duals of locally convex
5 IV.4.41, the dual of
projective limits (see SCHAEFER [ 6 1 1 , (E,T) is the subspace
U (Ea,~cl) I of ( E l l ] C~EA
11)
I .
Hence by
1.18(ii) and the remarks at the beginning of this section, El is the inductive limit of the Banach spaces Y
(E ) '1. a Y
(b) We note firstly that if H' is a Tequicontinuous subset of E' then there is an
c1 E A
so that H' E Ed, and H' is
T
c1
equi
continuous. The sufficiency of the condition follows then from 1.22. On the other hand, if H is yequicontinuous then there are a 0 , ~ , in A ( a o < a,) and Ho Y
HI
aO
equicontinuous in E' (resp.
~,,equicontinuous in El) so that H
(E
T
G
Ho
+
E/2B, H C HI
> 0, B the unit ball of
Then if h ho
+
E
+
E/2B
El)
H, it has a representations €/2bo;
h, + ~ / 2 b , (ho
2 and Then h = ho + (hlho) + ~ / bo
E
Ho, h,
11 hl 
hoII
We define HI to be the set of all such (ho

E
H I , bo,bl
E
B)
S E.
h,) required in
the representations of the elements of H. Continuing inductively, we can construct a sequence (H,) ties.
with the required proper
1.3
35
SAKS S P A C E S
Spaces of linear mappings: Let ( E l l \
3.11.
IlrT)
and ( F r l l
lll,Tl)
be Saks spaces and denote by L(E,F) the space of ycontinuous linear mappings from E into F. Let C be a saturated family of normbounded sets in E. Then on L ( E , F ) we consider the following structures:
11 11

the uniform norm;
T~

the topology of uniform convergence (for T ~ ) on the sets of C .
Then
(L(ErF)
11
r
IlrTE)
is a Saks space. It is complete if
is and if C is the family of all bounded sets in E.
(FrII l l l , T l )
It is a Banach space if F is Banach.
3.12.
Tensor products: Let ( E J I
spaces. We denote by
1Ir~)
and ( F , "
be Saks
lllr~l)
the algebraic tensor product of
E 8 F
E and F. On E 0 F we consider the following structures:
/I He
SUP
: X
{I
(f 8
4 )( X I [
: f E
g
E
Eir
IIflI
F;!r
IlglI 5 1 )
S 1
(here we are using the norm of f (resp. g ) in E;' (resp. in F;')) T
8 8
'cl
: the projective tensor product
T~
:
of
T
and T,;
A
T
Then ( E 0 F,II
the inductive tensor product of T and IhrT
6
Tl)
and ( E
Q F,II
spaces (this follows from 3.1 since If T
G
(E
TI
i,
2
and Frll
8 TI). llrT
6
TI)
I~,T6 Q
T ~ .
' 1 ~ 1 are Saks
gl is continuous for
We denote their completions by and ( E
6,
F,ll
1 1 , ~ Qf
T ~ ) resp.
Using the
representation of the completion by projective limits, we can
.
I. MIXED TOPOLOGIES
36
4
give another construction of the tensor product E By F : Let
L.
A
be the canonical representations of E and I?.
Then
is a projective spectrum of Banach spaces and its Saks space
11,~
projective limit is naturally isomorphic to (E By F,ll
.2 @
T~).
We finish this section with some brief comments on spaces which generalise the class of Banach algebras (resp. C"algebras) exactly as the Saks spaces generalise the class of Banach spaces.
3.13. Definition: Let A be an algebra with unit e. A submultiplicative seminorm on A is a seminorm p with the
properties
p(xy)
I p(x)p(y)
If A has an involution x ++
(x,y
E
A)
and
p(e)
=
1;
xgr,p is a C"seminorm if, in
addition, it satisfies the condition p(xgrx)= {p(x)I2 A
Saks algebra is a triple ( A f l l I l f . r ) where
(Afll
11)
(x E
A).
is a Banach
algebra with unit and the locally convex topology can be defined by a family A
S
of submultiplicative seminorms so that
11 11
= sup S.
Saks C":algebra is defined in exactly the same way with the
additional requirements that A have an involution and the seminorms of S be C*':seminorms. It follows from this condition that (A,]]
11)
is then a @algebra.
1.4
31
SPECIAL RESULTS
If p is a submultiplicative seminorm, then
(as defined P in the proof of 3.8) has a natural Banach algebra structure. Hence a Saks algebra A has a canonical representation as a projective limit of a spectrum {IT
qP
:
where each
ii
iiq +iP'
P
p I q, p , q
E
S}
is a Banach algebra and the linking mappings
are unitpreserving homomorphisms. Similary, a Saks @algebra has a representation with @algebras
as components and C"
algebra homomorphisms as linking mappings. 3.14. The spectrum: If (A,]]1 1 , ~ )
by
My(A)
is a Saks algebra, we denote
the set of ycontinuous multiplicative functionals
f from A into a: with
f(e)
= 1.
My(A) is called the spectrum
of A. We regard M (A) as a topological space with the weak
Y
topology induced from
(A', a(A' ,A)1 . Then My ( A )
,
as a subspace
of a locally convex space, is completely regular. T ( A ) is a (topological) subspace of the spectrum of the Banach algebra (A,II
II)
1.4. SPECIAL RESULTS
In 1.5 we saw that the mixed topology could be characterised as the finest linear topology (and hence also as the finest locally convex topology) which agrees with T on the sets of €3.
I. MIXED TOPOLOGIES
38
In general, it is not the finest topology which satisfies this condition. In the Corollary to the following Proposition, we give sufficient conditions for this to be the case. Proposition: Let (E,g,r)be as in 1.4. Then the following 4.1. are equivalent: (a) B has a basis of rcompact sets; (b) there is a Frhchet space F so that E
=
F', with the
structure described in 2.A.
Proof: (b)
(a) follows from the BOURBAKIALAOGLU theorem.
(b): since (E,y) is semireflexive (1 . 1 3 ) , we can
(a)=7
identify E with the dual of the Fri!chet space E'. Then 8 is
Y
the equicontinuous bornology of E and
'I
= a(E,E') on the sets
of 8 by compactness.
4.2. Corollary: If (a) or (b) is satisfied then: (c) y[8,'11 with
T
is the finest topology on E which agrees
on the sets of 8;
(d) a subset A of E is yclosed (resp. open) if and only if
A rl B
is 'IIBclosed (resp. open) for each B
E 8.
Proof: In view of 4. 1,
this is a restatement of the Banach
Dieudonni! theorem " 3 1 1
, p.274).
1.4
39
SPECIAL RESULTS
W e remark t h a t a famous counterexample of GROTHENDIECK
( [ 2 6 ] , pp.9899)
which w e s h a l l n o t r e p r o d u c e h e r e shows
t h a t 4 . 2 ( c ) d o e s n o t hold i n g e n e r a l . See a l s o PERROT [471.
4.3. C o r o l l a r y : L e t ( E , l l
11,~)
be a Saks s p a c e i n which B
i s Tcompact and l e t A be a s u b s e t of E so t h a t
(A
>
II II
XA = A
0) ( i n p a r t i c u l a r , i f A i s a s u b s p a c e ) . Then A i s y  c l o s e d
i f and o n l y i f
A Cl B I I
11
i s Tclosed.
P r o o f : T o check t h a t A i s y  c l o s e d , A
n
nBll
A
n
nB
I~
II II
is
closed f o r e a c h n
= n(A
E
v e c t o r subspace of E . Then B and T~
N. But
BII 1 1 ) 
Now suppose t h a t w e have a s p a c e
a topology
w e need o n l y v e r i f y t h a t
(E,B,T) T
and t h a t F i s a
i n d u c e a bornology B F and
on F ( B F i s g e n e r a t e d by t h e b a l l s of B which
are c o n t a i n e d i n F ) . An i m p o r t a n t q u e s t i o n i s t h e f o l l o w i n g : do w e have t h e e q u a l i t y
W e c l e a r l y have
n
n
However, t h e r e i s a counterexample which shows t h a t e q u a l i t y d o e s n o t h o l d i n g e n e r a l (ALEXIEWICZ and SEMADENI [ 8 1 , p.133) b u t t h e following Proposition g i v e s s u f f i c i e n t c o n d i t i o n s f o r equality:
I. MIXED TOPOLOGIES
40
4.4. Proposition: Suppose that (E,8, T ) is such that (E,y)
is semireflexive and let F be a yclosed subspace. Then Y[B
IT]
IF
=
Y[BF"IF]

Proof: Consider the Frkchet space
G := E;.
Then
E
=
G'.
Since F is a(E,G)closed, F is the dual of the quotient space G/Fo (Fo is the polar of F in G). Hence it suffices to show that every yequicontinuous set H in G/Fo is the image, under IT,of
the quotient mapping
a yequicontinuous subset of G
(note that G/Fo is the dual of F under both of the topologies under consideration). Now we can choose a neighbourhood basis (Un) of 0 in G so that n(Un) is a neighbourhood basis of zero in G/Fo (since both are Frgchet spaces). Then by 1.22 there is, for each n, a Tequicontinuous subset Hn of G/Fo so that H C Hn We can
H H
:= _C
+
r(Un).
extend Hn to a Tequicontinuous subset
gn of
G. Then
n (%n + Un) is a yequicontinuous subset of F and IT@).For IT
and so
1
H
(HI E
IT'
=
n
IT
E
fl
(En+
1
= IT (IT
~
(
1
n ( H +~ n (un)) ( H +~ IT(U,)
(HI
Un
+
C _ IT( n
Fo)
(in+ un +
Fo))
( ( Hn +~ .rr(un))).
We have defined the mixed topology in 1.4 by displaying a neighbourhood basis of zero. However, it is often convenient
1.4
41
SPECIAL RESULTS
to characterise a locally convex topology by its continuous seminorms. It is not too difficult to show that the topology y can be defined by the following seminorms: choose for each
n
N a rcontinuous seminorm pn and define, for x
E
p(x) := inf
I
C
E
E
Pk(xk))
where the infimum is taken over all representations of x in the form
x,
... +
+
xn
where xk
E
Bk. Then the set of all
such seminorms determines y (cf. DE WILDE 1741 , where a corresponding representation for seminorms on inductive limits of locally convex spaces is given). This characterisation is not very useful for applications and we shall now show that in certain cases a much simpler representation for the y continuous seminorms on a Saks space can be given. Let (E,l[ 1 1 , ~ ) be a Saks space and let
S
be a defining family of seminorms
for r which is closed for finite suprema and is such that
11 I ]
= sup S.
(where pn
An
t
E S
Then for each pair (p,) and (1,)
and (An) of sequences
is a sequence of positive numbers with
m), Y
p : x
+sup
"1
pn(x)
is a seminorm on E. The family of all such seminorms defines

a locally convex topology
Y[ll
4.5. Proposition: Let (E,II
11,~)
11,~1 on E.
be a Saks space and suppose
that either (a) for every x x,y
E
E so that
x
E
E,
E
> 0, p
E S,
= y+z, p(z) = 0 and
there are elements [IyII I p(x) +
E:
I. MIXED TOPOLOGIES
42
Proof: We verify firstly the inclusion
Y [/I
IIrTl
7 [/I
2
IIrTl
which is valid without assumptions (a) or (b). We show that

7 [I/ ]],TI
A 7neighbourhood of zero in m
where (p,)
and (An)
Xn > 1
that
(this suffices by 1 . 7 ) .
is coarser than T on B
for
It I1 has B II I1
the form
are as above. But if N
E
is chosen so
N
n 1 N, then N
m
and the latter is a Tneighbourhood of zero in
€3
I1 11
We now consider the reverse inclusion under assumption (a). Every yneighbourhood of zero contains a set of the form W
U, n
where
Un
:=
P
(un +
n= 1 Ix : pn(x) I
nB
E ~ I
I1 It) (
E
We can assume, in addition, that Put with
A,
:=
p(x)
min ( ~ ~ , 1 / 2 ) X,, 5
1
p
where
and
p n ( z ) = 0. Then
z
E
Un
p,
1;’
(n
2 1).
y
E
S) (1.16).
for each n. Choose x
E
E
Then for any n there
p,.
of x where and
E
I pn+l
:= n / 2
:= s;p
is a decomposition x = y+z
>~ 0, pn
((~(1
nB
II II
I
pn(x)
+
so that x
(n/2) E
U
+
nB
1.4
Now we assume that B
II II
SPECIAL RESULTS
is .rcompact. Let U be a yopen
neighbourhood of zero. Then there is a
11
so that U n B 1 1
7 {x : p0(x)
Suppose that we can find
fi
ix
k= 1
where
Xo
:
= E,
of a
E
n+l
S
pk(x)
I
€1
pl,...,p,
I 1 ),
n
nB
po
n
B
E
S
II II c:
E
S and an
E
>
0
I1 II. so that
" nB
Xn = (n1) (n > 1). We shall prove the existence so that
{x : pk(x)
(71 k=
43
I
x,)n (n+l)B II
I I c u
n (n+l)B
Then we can construct, by induction, a sequence (p,)
II II. in S so that
for each n and so 03
iX : pk(x) I k= 1 which proves the result. we argue by contradiction.
To prove the existence of
If no such seminorm exists, then
has nonempty intersection with the .rcompact set for each
q E S. Hence, by the finite intersection property,
there is an
x
0
(n+l)BII11 \ U
E
xo
E
U n nB
(ntl)~
II I I
I1 I1
\
cq.
u n qEs
E U n (n+l)B
which is a contradiction.
I1 I1
Hence
q(xo) s n
I. M I X E D TOPOLOGIES
44
Using this result, we can obtain new sufficient conditions for the equality y[
11
llFrTF]
=
Y[
11
]ItT]
IF
to hold for a subspace F of a Saks space ( E l l ]
11,~)
(cf. 4.4).
4.6. Proposition: Let F be a subspace of a Saks space ( E l l \
and suppose that (F,ll
I]F,~F)
11,~)
satisfies (a) or (b) of 4.5.
Then y[
11
IIF"F1
= y[
11
i11'1
IF
Proof: It is sufficient to show that a y [ I I I]F,~F]continuous 1
seminorm of the form p := m i A n of a y [
11
pn
on F is the restriction
I~,T] continuous seminorm p on E .
But if
UPn
is an
extension of pn to a Tcontinuous seminorm on E then
5
:= sup A n '
nEN
zn has the required property. 11,~)
4.7. Corollary: Let ( E , l l
be a Saks space, F a subspace
so that the unit ball of (F,ll ),1
is TFcompact.
Then the restriction mapping induces an isometry from ( E , y ) '/Fo onto (F,y)'.
Proof: The restriction mapping from ( E , y )
'
into (F,y) I is
obviously a normcontraction and it is surjective by 4.6. We must show that if
f
E
then there is an extension with ]!TI/5 1 + ~ . Since B
II
(F,y)' with Ilfll
7 IIF
= 1
and if
E
of f to an element of (E,y) ' is TFcomp&ct, there is an
> 0,
45
SPECIAL RESULTS
1.4
a b s o l u t e l y convex yneighbourhood U of z e r o so t h a t
w
:=
1 (1+~ BII llF) +
U
c
{x
F :
E
I f(x)l
I
1)
W e can e x t e n d t h e Minkowski f u n c t i o n a l of W t o a y  c o n t i n u o u s seminorm
on E s o t h a t
1 BII
11 C
I X

: P(X)
1)
S

The r e s u l t t h e n f o l l o w s by a p p l y i n g t h e HahnBanach theorem t o f t o o b t a i n a l i n e a r form
?
on E which e x t e n d s f and i s
N
m a j o r i s e d by p.
Now w e d i s c u s s a theorem of BanachSteinhaus t y p e f o r mixed t o p o l o g i e s . S i n c e such s p a c e s are n o t , i n g e n e r a l , b a r r e l l e d t h e c l a s s i c a l approach c a n n o t be employed. I n d e e d , i f one c o n s i d e r s t h e i d e n t i t y mapping from ( L 1

which i s n o t y  c o n t i n u o u s ,
lll,~J
1
into ( R
, 11 I)
b u t i s t h e p o i n t w i s e l i m i t of t h e
p r o j e c t i o n mappings pn : (x,)
(~l,***r~~rOrOr)
which a r e c o n t i n u o u s , i t i s c l e a r t h a t a BanachSteinhaus theorem can o n l y hold under r a t h e r s p e c i a l r e s t r i c t i o n s .
4.8. D e f i n i t i o n : L e t E be a l o c a l l y convex s p a c e . Then E h a s t h e BanachSteinhaus p r o p e r t y i f f o r e v e r y sequence ( T n ) of c o n t i n u o u s l i n e a r mappings from E i n t o a l o c a l l y convex s p a c e F such t h a t t h e p o i n t w i s e l i m i t
T : x M l i m (Tnx)
(x
E
E)
e x i s t s , t h i s l i m i t i s c o n t i n u o u s . The f o l l o w i n g remarks are evident:
I. MIXED TOPOLOGIES
46
a)
in this definition, one can assume that F is a
Banach space: b)
inductive limits and quotients of spaces with the
BanachSteinhaus property also have this property: c)
the BanachSteinhaus property is possessed by abarrelled
spaces (and in particular by barrelled spaces) (recall that a locally convex space is abarrelled
if every barrel which
is the intersection of a countable family of neighbourhoods
of zero is itself a neighbourhood of zero): d)
if E is a Mackey space, it is sufficient to verify
the condition for linear forms (i.e.that El is a(E',E) sequentially complete).
4.9.
Definition: Let ( E , B , ' I )
be as in 1 . 4 . We say that E satis
fies condition C 1 if, for every n
E
N every xo
E
Bn, and every
Tneighbourhood of zero U there is a rneighbourhood V of zero so
that
v
n B~ c_ ((x0 + U I n B ~ ) ((x0 + u) n B ~ ) .
The importance of this property is that it allows us to deduce 'IIBcontinuity of an operator on some B
E
B from its
'I
I B'On
tinuity at a single point.
4.10. Lemma: Let (E,B,T) satisfy condition
C,
and let F be a
family of linear mappings from E into a locally convex space F. Suppose that the following condition is satisfied:
SPECIAL RESULTS
1.4
for each B
8 and each absolutely convex neighbourhood
E
W of 0 in F there is an xo so that
T( (xo
47
+
E
B and a Tneighbourhood U of 0
Txo
U ) fl B ) C
+ W. Then
F is yequicontinuous.
Proof: By Grothendieck's Lemna ( 1 . 8 ) we need only show that FIB
is
T I Bequicontinuous
at 0. Let W be an absolutely convex
neighbourhood of zero in F. Then there is a Tneighbourhood U of zero so that
T ( ( x ~+ for each T
E
T(V
u) n
B) EL T X + ~ w/2
F. Choose V as in 4.9..
n B) E. T ( ( x ~+ u ) n (Txo + W/2) =
4.11.
B

Then for each T

(x0 +
u) n
E
F,
B)
(Txo + W/2)
w.
Proposition: Suppose
a)
that (E,B,T) satisfies condition 1,;
b)
B has a basis B1 so that ( B , T I ~ )is a Baire space for
each B
E
B1.
Then (E,y) has the BanachSteinhaus property.
Proof: Let (T,)
be a pointwise convergent sequence of ycon
tinuous linear mappings from E into a Banach space F. Let B be a TIBBaire space. We shall find a point xo
E
E
8
B and a
Tneighbourhood U of zero so that Tn( (xo + U) n B)
C_
Tn(xo) + EBII1lF
for each n.
Then (Tn) will be yequicontinuous by 4 . 1 0 and this will Suffice
I . M I X E D TOPOLOGIES
48
t o prove t h e r e s u l t . L e t := i x
E
B :
IIT~x  T~xII
N
E
0
so t h a t An
0
B =
0
(xo + U) flB
a Tneighbourhood U of 0 s o t h a t
II TnoX 
TnoXo II 5 ~ / 3 if
x
+ u)
11 T
E
(x0 ~ X 
4.12.
T"0
XI]
f o r p,q 2 nl
U An. Hence t h e r e i s an nEN c o n t a i n s an i n t e r i o r p o i n t x NOW choose
d Then An i s ~ I ~  c l o s eand n
I ~ / 3
x E (xo
+
C_
.
A
"0
and
U ) n B. Then i f
n B I
IIT~x 
TnoXII +
C o r o l l a r y : Suppose
IIT~X ~
~
a) that (E,B,T)
b ) B h a s a b a s i s of TIBcompact o r
TI
B
+
II TnoXo ~ 1 T ~ x 1~ I I I
s a t i s f i e s C,
and
 m e t r i s a b l e and complete
s e t s . Then (E,y) h a s t h e BanachSteinhaus p r o p e r t y .
One problem which h a s proved t o be c e n t r a l i n t h e t h e o r y of f u n c t i o n s p a c e s w i t h s t r i c t t o p o l o g i e s i s t h a t of d e t e r m i n i n g whether t h e g i v e n s p a c e i s a Mackey s p a c e ( i . e . i f it h a s t h e f i n e s t l o c a l l y convex topology c o m p a t i b l e w i t h i t s d u a l ) . Most f a m i l i a r Mackey s p a c e s have t h i s p r o p e r t y by v i r t u e o f . some s t r o n g e r p r o p e r t y ( e . g . t h a t of b e i n g b o r n o l o g i c a l ) . O n c e a g a i n , w e must seek a n o t h e r approach f o r mixed t o p o l o g i e s . O u r f i r s t r e s u l t g i v e s a p o s i t i v e answer f o r Saks s p a c e pro
d u c t s of Banach s p a c e s . We r e q u i r e t h e f o l l o w i n g p r e p a r a t o r y result.
E
I. 4 4.13.
SPECIAL RESULTS
AEa),l
Proposition: Let { (Ea,II
49
be a family of Banach
spaces. Then ( S n Ea, B C Ed,)
1)

is a dual pair under the bilinear
mapping ((x,), (fa))
fa(xa).
C
~ E A
2)
Under this duality, B C Ed,, with its Banach space struc
ture is identifiable with the ydual of 3)
a subset C of
B C Ed, is yequicontinuous if and only
if it is normbounded and, for each subset J of A so that
Proof: We can regard
SII Ea;
C Ilf,lI ~ E AJ\ SnEa
E
IE
> 0, there is a finite
for each (fa)
E
C.
as the projective limit of the
spectrum defined by the spaces
{B
ll
E, : J
UEJ
J(A))
E
where
J(A) denoted the family of finite subsets of A. Now there is a natural isometry from
( B a!J
E,)'
onto
B
C
~ E EJ J
and the
result follows then from 3.10.
We shall require the following version of Schur's theorem:
4.14.
Proposition: A subset C of
1
(S) is relatively u (eel,em)
compact if and only if it is normbounded and for each there is a finite subset J of S so that each (x,)
E
C
a€S\J
IIxaII
IE
E
> 0 for
C.
In particular, there is a countable subset S1 of S so that
I. MIXED TOPOLOGIES
50
Proposition: Let {E,]
4.15.
be as in 4 . 1 3 with
Then (E,y) is a Mackey space (i.e. y
=
E := S
IT Ea.
T(E,E')). Y
Proof: It is sufficient to show that if C is a weakly compact
B C E;,
subset of
then C satisfies condition 4 . 1 3 . 3 ) .
We
first show that the support A1 of C is countable. (A1 is the
C exists with f B For each B E A1 we choose an x E E so that IIx 11 = 1 B B B B 0 for some (fa) E C. Then the mapping f (x ) B B set of those f3
B .c E;
from
E
A for which an (fa)
into L 1 ( A ) is
0
0).
E
(B c E;,
and
B IT E ~ ) u ( t l ( A ) , L ~ ( A ) )
continuous. For this is equivalent to the fact that for each
(Aa)
on
E
Lm(A)
is defined by an element of
B C EA
clearly is
the linear form

the element (1 x
cla
)
. The
B ll Ea

which it
image of C under this
mapping is thus weakly compact and so has countable support (4.14).
But, by construction, the support of this set is A1.
Thus we can consider the case where the indexing set is N. If C does not satisfy the condition of 4 . 1 3 . 3 ) ,
there is a
strictly increasing sequence of positive integers (nk ) and a sequence (f(k)) in positive and : f
E.
c
"k+ 1
c Ilfnkll 2 E for some n=nk+ I For each n, there is an xn E En so that IIxnII so that
11
/2
(n, < n
I
nk + l
'
= 1
SPECIAL RESULTS
1.4
and so
{(fn(xn)) : f
E
51
C ) is not weakly compact (4.14)

contradiction.
4.16. Proposition: Let
spaces and let (E, 11
11,~)
c ( E a r 11 ]la) IaEA
be a family of Banach
be the product
S
n
,€A
E,.
Then (E,y)
has the BanachSteinhaus property.
Proof: By 4.15 and 4.8.d), it is sufficient to show that the pointwise limit of a sequence (f,)
of continuous linear forms
is continuous. We use the isometry
E' y
B C Ela. Let ~ E A
=
a
fn = ( f n ) .
a
Then the sequence (fn)aEA is pointwise convergent to a continuous linear form f" on E, for each
c1 &
of uniform boundedness, sup{IIfnII
E
sup {
c
aEA
IIfnII
:
n
E:
N) <
:
n
A. By the principle
N 1 <
m
and so
m.
C 1 1 fall < m and so there is a ycontinuous linear ~ E A form f on E with flEa = fa for each c1 E A. A simple ~ / 3
Hence
argument shows that f is the pointwise limit of the sequence (f,).
In 4.15 and 4.16 we have shown that Saks spaces of a very special type have two important properties. Every Saks space is a closed subspace of such a space. However, these properties
are not inherited by closed subspaces. We shall now show that
I. MIXED TOPOLOGIES
52
under certain circumstances a Saks space is a complemented subspace of a Saks space product of Banach spaces case the properties
4.17.

in this
E
A, a 5 B)
are inherited.
Definition: Let
:
EB
Ea, a,B
L__3
be a projective system of Banach spaces, (Ell[1 1 , ~ ) its Saks space projective limit and
the natural mappinqs from E
into Ea. A partition of unity is a family ITa) of norm contractions where
Ta : Ea
E
a)
for each B
E
A,
b)
for each x
E
S II Ea, B
c)
for each x
E
E,
{a
E
C
A : E
so that
IT^
1)
A
Ta
Ta f 0) is finite;
o
C IT T (x 111 I I I x I I ; C~EJ a a
va(xa) = x (convergence in
CXEA
4.18.
Proposition: If a partition of unity exists then (E,y)
is a complemented subspace of

(S II Ea,y) ~ E A
.
Proof: We shall show that the natural injection x
bra (x)
which is ycontinuous from E into
has a left inverse. In fact the mapping
T
: (xu)
S X Ea
C Taxcl
is a left inverse. Firstly, this mapping is welldefined. For the right hand side converges (we need only show that for each B and this follows B from a)). Secondly, it is a norm contraction by b) and is TC
aEA
nB
o
Ta(xa)
converges in E
continuous on the unit ball of It is a right inverse by c).
S II E aEA

once more by b).
T)
.
1.4
53
SPECIAL RESULTS
4.19. Corollary: If a partition of unity exists then
1)
(E,y) is a Mackey space:
2)
(E,y) has the BanachSteinhaus property.
We recall that a locally convex space has the approximation property if the identity mapping
is uniformly approximable
on compact sets by continuous linear mappings of finite rank. A normed space has the metric approximation property if one can, in addition, demand that the approximating operators be contractions. A locally convex space which is the projective limit of a spectrum of normed spaces with the approximation property also has the approximation property. In the following, we give a corresponding result for Saks space projective limits. Unfortunately we require rather strong additional hypotheses but, as we shall see, these are often fulfilled in applications.
4.20. Proposition: Let
Inga
: EB d E,,
a,B
E
A, a
S
B)
be a projective system of Banach spaces, (E,ll I[,.r) its Saks space projective limit. Suppose that
that
a)
each E, has the metric approximation property;
b)
there is a norm contraction 1, : E
n,
i,  idE,
for each a
E
a
d E
so
A.
Then ( E , y ) has the approximation property.
Proof: Let K be an absolutely convex compact subset of E and
V a Tneighbourhood of zero (we can and do assume that 2K E B
I1 II)
*
I. MIXED TOPOLOGIES
54
There is an a E A so that V ( l B I I11 2
,(Bll ),1
E7TI
Then there is an operator T : , E with llTll I 1 and (id E, T,. Then := i, o T
T
If x
E
E,
'rr,(idE(x) Since

K then (id
(id,

Tx)
 T) ( K ) 11,~)
T) ( n , ( K ) )
T) (?r,(x))
II I, E B II 11 {
C_
EB
E
EB
II I,l
I
1.
. Hence
(id,
we have
(id,
)1,
0)*
. Let
II I,
and so
(Em, 11
>
of finite rank
is of finite rank and llyll
E EB
4.21. Corollary: Let
spaces, ( E l11

E,
(E
T) ( K ) C €IT' (BII ),1  T) ( K ) C V. 
be a family of Banach
its Saks space product. Then if each E, has
bhe metric approximation property, (E,y) has the approximation property.
Proof: We can regard E as the projective limit of the spectrum {B ll E, : J ~ E J
E
where F ( A ) denotes the family of finite
F(A)}
subsets of A . Now
ll E,
€3
ClEJ
has the metric approximation pro
perty and 4.20.b) is satisfied.
4.22. Corollary: Let
{.rrg,
: ~ ~E,,i a,B E A, a I
8)
be a projective system of Banach spaces, (Ell]1 1 , ~ ) its Saks space projective limit. Then if there exists a partition of unity, (E,y) has the approximation property.
Proof: This follows from 4.18, 4.21 and the fact that a complemented subspace of a locally convex space with the approximation property has itself the approximation property.
*
1.4

SPECIAL RESULTS
55
Now w e t u r n o u r a t t e n t i o n t o c l o s e d graph theorems f o r s p a c e s w i t h mixed t o p o l o g i e s . S i n c e t h e g r a p h o f t h e i d e n t i t y mapping:
(E,y)
11)
(E,ll
i s c l o s e d f o r any Saks s p a c e
i t i s once a g a i n clear t h a t any such c l o s e d g r a p h theorem must employ r a t h e r s p e c i a l r e s t r i c t i o n s . N e v e r t h e l e s s , w e can o b t a i n two u s e f u l c l o s e d g r a p h theorems, one based on a c l o s e d graph theorem f o r l o c a l l y convex s p a c e s due t o KALTON and one o b t a i n e d by combining a c l o s e d g r a p h theorem f o r born o l o g i c a l s p a c e s (due t o BUCHWALTER) and a c l o s e d g r a p h theorem f o r t o p o l o g i c a l spaces.
I f T i s a l i n e a r mapping from E i n t o F ( E , F l o c a l l y convex spaces), we put
i.e.
f ( T x ) ) E E')
D(T') = {f
E
F'
: (x
D ( T ' ) = (f
E
F'
: T"f
E
El)
(Tgc i s t h e a l g e b r a i c a d j o i n t
of T ) . W e n o t e t h a t T i s weakly c o n t i n u o u s i f and o n l y i f D ( T ' ) = F ' . A s i m p l e c a l c u l a t i o n shows t h a t
where
(r
T))O = {(T'f,f)
(r
TI ) O i s t h e p o l a r o f t h e graph
Now i f x
E
(D(T'))O
: f
E
D(T'))
r (T)
Hence i f T h a s a c l o s e d g r a p h ( s o t h a t
i n F'.
x = TO
= 0
E' x F'
.
( p o l a r i n F ) t h e n ( 0 , x ) v a n i s h e s on t h e
r i g h t hand s i d e of t h e above e q u a t i o n and so
have
of T i n
and so
D(T')O = (0)
(0,x)
( r ( T ) )O0
=
E
(r(T)Ioo.
r (T)
t h a t is, D(T')
we i s dense
I. MIXED TOPOLOGIES
56
4 . 2 3 . Proposition: Let E be a Mackey space for which E' is
u (E',E)sequentially complete. Then a linear mapping T from E into a separable Frechet space F is continuous if and only if it has a closed graph.
Proof: Since E is a Mackey space, it suffices to show that a linear mapping T from E into F is weakly continuous if it has a closed graph. By the above remarks we need only show that D(T') is u(F',F)closed. Since E' is weakly sequentially complete, D(T') is sequentially closed. For if (fn) is a sequence in D(T') which tends weakly to a linear form f then (T'(fn)) is u(E',E)Cauchy and so converges to a continuous linear form on E. But the limit is of course T"f and so f
E
D(T'). By the BanachDieudonng theorem, to show that D(T')
is closed it suffices to show that D(T') n Uo is weakly closed for each neighbourhood U of zero in F. But Uo is weakly metrisable (since F is separable) and so D(T') n Uo, being sequentially closed in Uo, is closed.
4.24.
Proposition: Let
(E,ll
11,~)
be a Saks space and suppose
that a)
or
b)
(E,ll
11,~)
satisfies the conditions of 4 . 1 2 ;
E is the Saks space projective limit of a spectrum of Banach spaces with partition of unity;
Then a linear mapping T from E into a separable Frgchet space is ycontinuous if and only if its graph is closed.
SPECIAL RESULTS
1.4
57
Proof: Under these conditions, E' is weakly sequentially complete ( 4 . 1 2 and 4 . 1 9 )
and so, by 4.24,
T is weakly con
tinuous. We show that it is continuous. By projecting down into the Banach spaces associated to the continuous seminorms of F we can assume, without loss of generality, that F is a separable Banach space. In addition we can assume that F has a Schauder basis (x,)
(since every separable Banach space is
a subspace of a Banach space with Schauder basis space C [ O , l ] Let (P,)


e.g. the
see BANACH C131, pp. 1 8 5 and p. 1 1 2 ) .
be the associated sequence of projection operators a n
pointwise on E and so by 4.12 and 4.19 it will suffice to show that Pn
o
T
is ycontinuous. But this is clear since Pn
o
T
is weakly continuous and takes values in a finite dimensional subspace of F .
4.25.
Corollary: Let (E,ll
11,~)
be a Saks space as in 4 . 1 2 and
suppose that (E,y) is separable. Then (E,y) is a Mackey space.
Proof: We must show that any weakly continuous mapping from E into a Banach space is continuous. But the range of such a mapping is weakly separable and so separable. Its continuity follows then from 4.24.
4.26.
Corollary: Let ( E , I I
I~,T)
be as in 4 . 2 4 and let ( F , B ) be
a separable locally convex space. Then if
T : E
__3
F
is
I. MIXED TOPOLOGIES
58
a linear mapping which is ya continuous where a is a locally convex topology on F so that B has a basis of aclosed sets, T is ya continuous.
Proof: The hypothesis on a and B implies that
is the topo
logy of uniform convergence on a family S of subsets of the adual of F (bipolar theorem!). Then if
M :=
uS , T
is
yo(F,M)continuous. The continuity of T follows from 4 . 2 4 since its graph is yB closed (at this stage, we can assume that F is a Banach space).
4.27.
Remark: Note that in the proof of 4 . 2 4 we have actually
proved that a locally convex space with the BanchSteinhaus property satisfies a closed graph theorem where the range space is a separable Frgchet space. On the other hand, a space with the latter property satisfies a BanachSteinhaus theorem for mappings into a separable Frgchet space. One may compare this with the fact that a locally convex space is barrelled if and only if it satisfies a closed graph theorem with Banach spaces as range spaces.
4.28.
( [61]
,
IV.8 . 6 )
.
Definition: Let ( E r g ) and (F,B,) be bornological spaces.
A linear mapping T from E into F has a Mackey closed graph if for each B
E
8, C
E
8,’
r ( ~ )n
( E x~ F ~ )
is closed in the normed space EB x FC.
1.4
Lemma: L e t ( E l l [
4.29.
11)
59
SPECIAL RESULTS be a Banach s p a c e ,
( F , B ) a complete
b o r n o l o g i c a l s p a c e of c o u n t a b l e t a p e , T a l i n e a r mapping from E i n t o F w i t h a Mackey c l o s e d graph. Then T i s bounded.
P r o o f : L e t CBn) be a b a s i s f o r B and f o r e a c h n p u t E~ :=
r (TI n
( E x F~
n
so t h a t En h a s a n a t u r a l Banach s p a c e s t r u c t u r e . Denote by pn t h e p r o j e c t i o n from
E
pn(En) = { x
E =
Now
u
pn(En)
FBn i n t o E . Then
x E
E : Tx
E
FBn 1 .
and hence a t l e a s t one Pn ( E 0
meagre. Hence by a theorem of BANACH ([13] onto E t h a t i s
E
T(E)
"0
into F
4.30.
Bn0
c
FB "0
. Then
,
"0
)
i s non
p . 381, pno maps
T i s bounded from E
by t h e c l a s s i c a l c l o s e d graph theorem.
P r o p o s i t i o n : L e t ( E , B ) and ( F , B , ) be complete b o r n o l o g i 
c a l s p a c e s , F of c o u n t a b l e t y p e . Then a l i n e a r mapping T from E i n t o F i s bounded i f and o n l y i f i t h a s a Mackey c l o s e d
graph.
P r o o f : The boundedness of T i s o b t a i n e d by a p p l y i n g t h e Lemma t o t h e r e s t r i c t i o n of T t o t h e Banach s p a c e s {EB : B
4.31.
E
9).
Lemma: L e t X,Y be Hausdorff t o p o l o g i c a l s p a c e s w i t h Y
compact. Then a mapping
f : X+
only i f i t s graph i s c l o s e d .
Y
i s c o n t i n u o u s i f and
I. MIXED TOPOLOGIES
60
P r o o f : see ~ E C HC161, p. 7 9 9 .
4.32.
Proposition: L e t (E,B,T),
( F , B l , ~ l )be a s i n 1 . 4 and
suppose t h a t a)
(E,B)
b)
B 1 has a b a s i s of
i s complete;
compact s e t s .
Then a l i n e a r mapping from E i n t o F i s ycontinuous i f and o n l y i f it h a s a c l o s e d g r a p h .
P r o o f : I f t h e g r a p h of T i s y  c l o s e d t h e n i t i s Mackey c l o s e d f o r t h e bornologies ( B I B 1 ) .
Then i t i s B8,bounded
by 4 . 3 0 .
The r e s u l t now f o l l o w s by a p p l y i n g 4.31 t o t h e r e s t r i c t i o n of T t o t h e sets of B .
4.33.
Remarks:
I.
4 . 2 4 s t a t e s t h a t , under t h e g i v e n h y p o t h e s e s ,
( E , y ) b e l o n g s t o t h e c l a s s of l o c a l l y convex s p a c e s which s a t i s 
f y a c l o s e d g r a p h theorem w i t h a s e p a r a b l e Banach s p a c e as r a n g e s p a c e . KALTON r29.1 h a s g i v e n t h e f o l l o w i n g i n t e r n a l char a c t e r i s a t i o n of t h i s c l a s s of s p a c e s (which h e d e n o t e s by C ( c B ) ) : a l o c a l l y convex s p a c e E b e l o n g s t o C ( G B ) i f and o n l y
i f e a c h o ( E ' , E ) bounded, m e t r i s a b l e b a l l i n E ' i s equicontinuous. Using r e s u l t s of V A L D I V I A , MARQUINA [ 3 6 ] h a s shown f u r t h e r t h a t t h e s e s p a c e s s a t i s f y a c l o s e d q r a p h theorem where t h e range s p a c e i s an wWCG Banach s p a c e ( t h a t i s , a Banach s p a c e which i s t h e union of c o u n t a b l y many weakly compact sets o r
1.4
SPECIAL RESULTS
61
has a dense subset of this form), in particular, where the range space is reflexive. In addition the words "Banach space" in the above formulation can be replaced by "Brcomplete space" (SCHAEFER [ 6 1 ] , p. 1 6 2 ) . 11.
We remark that if E is an infinite dimensional Saks space,
then (E,y) is never nuclear. For we can assume that (E,y) is complete (if not, consider the completion which is nuclear). Then (E,y) is semiMonte1 (SCHAEFER [ 6 1 ] , p. 1 0 1 , Cor. 2) and
so E is the dual of a Banach space F and y is the topology 'rc(E,F) ( 4 . 1 ) . Now the latter topology is never nuclear if E is infinitedimensional (private communication of HOGBENLEND

this fact follows from Lemma 1 in SHAPIRO [ 681 1 .
The class of nuclear spaces of the form
(E,y[B,.rI) coincides
with the class of those spaces which are the strong duals of
.
Fr'echet nuclear spaces (i.e. the (DFN)spaces) Hence they are uninteresting from the point of view of mixed topologies. 111.
If (E,I( 1 1 , ~ ) is a Saks space and F is a subspace of E, \
we say (following SHAPIRO) that F
is weakly normal if B
F
is normal if B
11
/IF
is a(E,E')compact;
Y
is ycompact. /IF Then it follows easily from the results of this section that 1)
11
every normal subspace is weakly normal and every
weakly normal subspace is normclosed; 2)
F is weakly normal if and only if it is isometric
to the dual of E'/Fo (under the canonical mapping):
Y
I. MIXED TOPOLOGIES
62
3)
'IF
F is normal if and only if it is weakly normal and
coincides with T~(F,E'/FO). m
These results are obtained, for subspaces of (C ( S ), 11 ( K the family of compact subsets of S

11,~~)
cf. I.2.D) by SHAPIRO
[691.
IV.
One of the features of many of the concrete Saks spaces
whichwe shall examine in the following chapters is the fact that they possess bases (in contrast to the associated Banach spaces which are usually not separable). The classical theory of bases for locally convex spaces employs some condition like barrelledness and so is not applicable to Saks spaces. It is curious to note that DE GRANDEDE KIMPE [25], in extending these results to nonbarrelled spaces, introduced the classes of Gspaces i.e. locally convex spaces (E,T) which satisfy the following conditions: a)
E has the Mackey topology;
b)
E' is weakly sequentially complete.
i.e. precisely the class of locally convex spaces which satisfy a closed graph theorem with separable Frechet spaces as range. As we shall see, the combination of the above properties will recur with unnatural frequency in the study of concrete Saks spaces. In fact, the determination of whether a given Saks space possesses these properties is often one of the most challenging and fruitful aspects of its theory. In view of these remarks we quote without proof some of DE GRANDEDE KIMPE's results
on Gspaces with bases:
1.5
A.
63
NOTES
If ( E , T ) is a Gspace with a Schauder basis, then
‘I
is the finest locally convex topology on E for which the sequence is a basis:
B.
If E is a Gspace with a Schauder basis then this basis
is equicontinuous (i.e. the associated projection operators are equicontinuous). C.
If (E,T) is a Gspace with a weak Schauder basis, then
this is a strong basis (i.e. a basis for the original topology).
The first statement, applied to the space Lm([ 0,1]) with the mixed topology 8, introduced in 111.1, gives the rather curious is the finest locally convex topology
result that 8,
T
on La)
with the property that the HAAR systems form a basis with respect to
T.
We remark that B and C follow easily from the closed graph theorem (4.23).
1.5. NOTES
1.1 consists essentially of an exposition of the results of WIWEGER [781

[8O]. For the theory of convex bornologies see
BUCHWALTER [ 1 5 ] and HOGBENLEND [27]. 1.5 identifies the mixed topology as a generalised inductive limit in the sense of GARLING [231. Most of the results of 1.1 are contained in his
64
I. MIXED TOPOLOGIES
results. For articles which deal with related topics see NOUREDDINE [ 371

[ 411 I PERROT [ 471 ,[ 481 ,[491
PRECUPANU [531 I ROELCKE [ 571 ,[581
I
,
PERSSON [ 5 0 1 I
RUESS [591. COOK and
DAZORD [19] have studied mixed structures in the context of limit spaces. STROYAN [70] uses nonstandard analysis. The localisation principle involved appears in earlier papers (for examples PTAK's fundamental paper [541 on the closed graph theorem

see also COLLINS [ 181 and WHEELER [ 733 )
.
The completeness theorem of RAIKOV referred to in the paragraph before 1.13 can be found in [ 551
.
The proof used in 1.13 is
taken from DE WILDE and HOUET [ 761 I [ 771
1.2.
.
Examples A,B,C,D,E are classical and can be found at
various levels of generality in the first papers on Saks spaces, twonorm spaces and mixed topologies (cf. [ 21 ,[ 421 ,[ 791 for example). Examples F and G appear in GARLING [23]. The mixed topology of Example K has been used to study Banach spaces with bases by SUBRAMANIAN and ROTHMAN (cf. [ 711 ,[ 721 )
1.3.
.
The term "Saks space" was introduced by Orlicz in [42].
We quote his definition (p. 240): "Let X be a Banach space or an incomplete Banach space
11 11,
(fundamental space) with the norm
and let
11 I]
be
another norm defined in X. In the set R of elements x satisfying the inequality llxll
I
1
E
X
we define the distance
1.5
between the elements x,y d(x,y)
=
NOTES E
65
R by the formula
I I x  ~1 "'.
If this metric space is complete, it will be termed a Saks space.
'I
\
We remark that under a norm, Orlicz understands a pseudonorm or Fnorm i.e. the condition of homogeneity is replaced by continuity in the scalar variable. Essentially, this means that the associated topology need not be locally convex. Thus our definition is more general in the sense that the metrisability condition has been dropped, but more restrictive in the sense that local convexity has been demanded. The class of Saks spaces is studied from a categorical point of view by SEMADENI in [ 6 5 1 . Despite this discrepancy between out terminology and that of Orlicz, we have decided to use the term "Saks space" for several reasons. Firstly any reasonable alternative (e.9. "space with mixed topology") is both unwieldy and misleading and secondly the name "Saks space" possesses the same flexibility as the name "Banach space". For example the algebras in the category of Saks space receive automatically the name "Saks algebra".
In 3 . 1 2 we have introduced natural notions of tensor product within the category of Saks spaces. This suggests the question: are the associated locally convex spaces the same (algebraically and topologically) as the injective resp. projective tensor product of the spaces, regarded as locally convex spaces with the mixed topologies? The same question has been attacked "from
I. MIXED TOPOLOGIES
66
the other end" by NOUREDDINE who showed (in [ 41 1
)
that if E and
and F are semiMonte1 Dbspaces then so is the projective tensor product E $ F. RUESS [59] has shown that the condition
of semiMontelness can be dropped.
The Saks algebras.iintroduced in 3.13 are new. For the theory of Saks spaces, see LABUDA ORLICZ
([
([
421 ,[ 431 ,[ 441 ,[ 451 )
331 ,[ 341 1, LABUDA and ORLICZ [ 351
, ORLICZ
,
and PTAK ( [ 461 1 . The
closely related theory of twonorm spaces is developed in ALEXIEWICZ [I] and [ 21 SEMADENI
1.4.
([
631 ,[ 641
For 4.1

,
ALEXIEWICZ and SEMADENI
([
71

[ 101 )
,
.
4.3 cf. BRAUNER [141, for 4.4 ALEXIEWICZ and
SEMADENI [ 81 , PERSSON [ 501
, WIWEGER
[ 791
. 4.5
is due to WIWEGER
[79]. The BanachSteinhaus problem has been studies intensively

see the papers of ALEXIEWICZ, ORLICZ, ORLICZ and PTAK, LABUDA and LABUDA and ORLICZ. The condition C 1 was introduced by ORLICZ in [421 and goes back to the method used by SAKS in [ 601.
In 4.17 we have adapted the method of partitions of unity for locally convex spaces which was introduced by DE WILDE in [75] (see also KEIM [30]). In 4.23 we have reproduced the proof of a closed graph theorem of KALTON [29]. Note that in the proof of 4.24 we have essentially shown that a locally convex space which
has the BanachSteinhaus property can be used as the source space of a closed graph theorem with the range space a separable
1.5
NOTES
Banach s p a c e . I n f a c t , t h e l a t t e r p r o p e r t y i s e q u i v a l e n t t o t h e BanachSteinhaus
p r o p e r t y w i t h s e p a r a b l e Banach s p a c e s
as r a n g e . KALTON h a s g i v e n a n i n t e r n a l c h a r a c t e r i s a t i o n of s u c h s p a c e s . 4.29 i s a c l o s e d g r a p h theorem o f BUCHWALTER ([ 151
,
2.4.6).
67
I. MIXED TOPOLOGIES
68
REFERENCES FOR CHAPTER I.
[l] A. ALEXIEWICZ
On sequences of operators 11, Studia Math. 1 1 (1950) 200236. On the two norm convergence, Studia Math. 14 (1954) 4956. A topology for twonorm spaces, Func. Approx., Comment. Math. 1 (1974) 35.
[3 1
On some twonorm algebras, Func. Approx., Comment. Math. 2 (1976) 334. On some twonorm spaces and algebras, Func. Approx., Comment. Math. 3 (1976) 310. Some twonorm algebras with ycontinuous inverse, Func. Approx., Comment. Math. 3 (1976) 1121. [71
A. ALEXIEWICZ, Z. SEMADENI A generalization of two norm spaces. Linear functionals, Bull. Acad. Pol. SC. 6 (1958) 135139.
181
Linear functionals on two norm spaces, Studia Math. 17 (1958) 121140.
191
The two norm spaces and their conjugate spaces, Studia Math. 18 (1959) 275293.
[lo1
Some properties of two norm spaces and a characterization of reflexivity of Banach spaces, Studia Math. 19 (1960) 1 1 6132.
[ I 1 1 S. ARIMA
Generalized mixedtopologies in dual linear spaces, Yokohama Math. J. 13 (1965) 1291 44.
Generalization of the mixed topology, 1121 S. ARIMA, M. ORIHARA Yokohama Math. J. 12 (1965) 6368.
REFERENCES
69
[13]
S. BANACH
Theorie des op6rations lineaires (New York, 1963).
[141
K. BRAUNER
Duals of Fr6chet spaces and a generalisation of the BanachDieudonn& theorem, Duke Math. J. 40 (1973) 845855.
[I51
H. BUCHWALTER
Espaces vectoriels bornologiques, Publ. D6p. Math. Lyon 21 (1965) 253.
1161
E. 6ECH
Topological spaces (Prague, 1966).
[171
G. CHOQUET
Sur un theor6me du type BanachSteinhaus pour les convexes topologiques, Sem. Choquet (1973/74) Comm. 4.
[ 181
H.S. COLLINS
Completeness and compactness in linear topological spaces, Trans. Amer. Math. SOC. 79 (1955) 256280.
1191
C.H. COOK, J. DAZORD Sur la topologie mixte de Wiweger, Publ. D&p. Math. Lyon 113 (1974) 128.
[20]
P. and S. DIEROLF
On linear topologies determined by a family of subsets of a topological vector space, to appear in "General Top. and Applications".
[21]
R.M. DUDLEY
On sequential convergence, Trans. Amer. Math. SOC. 112 (1964) 483507.
[22]
D. van DULST
(Weakly) compact mappings into (F)spaces, Math. Ann. 224 (1976) 111115.
[23]
D.J.H. GARLING
A generalized form of inductive limit topology for vector spaces, Proc. London Math. SOC. (3) 14 (1964) 128.
[241
I.C. GOHBERG, M.K. ZAMBICKII On the theory of linear operators in spaces with two norms, Transl. Amer. Math. SOC. I1 85 (1969) 1451 64.
I. MIXED TOPOLOGIES
70
[25] N. de GRANDEde KIMPE On a class of locally convex spaces with a Schauder basis, Indag. Math. 79 (1976) 307312. [26]
A. GROTHENDIECK
Sur les espaces (F) et (DF), Summa Brasil. Math. 3 (1954) 57123.
E271
H. HOGBENLEND
Thborie des bornologies et applications (Springer Lecture Notes 213, 1971).
[281
S.O.
E291
N.J. KALTON
Some forms of the closed graph theorem, Proc. Camb. Phil. SOC. 7 0 (1971) 401408.
1301
D. KEIM
Induktive und projektive Limiten mit Zerlegung der Einheit, Man. Math. 10 (1973) 191195.
E311
G. KUTHE
Topologische lineare RSume I (Berlin, 1966).
E321
P. K W E
Utilisation de limites inductives gbnbralis6es d’espaces localement convexes, Sem. Paul Kree (1974/75) Exp. 1.
[331
I. LABUDA
Continuity of operators on Saks spaces, Studia Math. 51 (1974) 1121.
IYAHEN, J.O. POPOOLA A generalized inductive limit topology for linear spaces, Glasgow Math. J. 14 (1973) 105110.
On the existence of nontrivial Saks sets and continuity of linear mappings acting on them, Bull. Acad. Pol. Sc. math., astr., phys., 23 (1975) 885890.
1341
E351
I. LABUDA, W. ORLICZ
Some remarks on saks spaces, Bull. Acad. Pol. Sc. math., astr., phys. 22 (1974) 90991 4.
E361
A. MARQUINA
A note on the closed graph theorem (to appear in “Arkiv der Math. ‘I)
.
REFERENCES [ 371
K. NOUREDDINE
71
Nouvelles classes d'espaces localement convexes, C.R. Acad. Sc. 276 (1973) 12091212. Espaces du type Db, C.R. Acad. Sc. 276 (1973) 1301 1 303. Nouvelles classes d'espaces localement convexes, Publ. DQp. Math. Lyon 103 (1973) 259277.
1401
Note sur les espaces Db, Math. Ann. 219 (1976) 97103.
C411
Localisation topologique, espaces Db et topologies strictes (Dissertation, Lyon 1977).
[421
W. ORLICZ
Linear operations in Saks spaces I, Studia Math. 1 1 (1950) 237272.
1431
Linear operations in Saks spaces 11, Studia Math. 15 (1955) 125.
[ 441
Contributions to the theory of Saks spaces, Fund. Math. 44 (1957) 270294. On the continuity of linear operators in Saks spaces with an application to the theory of summability, Studia Math. 16 (1957) 6973.
451
1461
W. ORLICZ, V. PTAK
Some remarks on Saks spaces, Studia Math. 16 (1957) 5668.
[471
B. PERROT
Sur la comparaison de certaines topologies mixtes dans les espaces binormQs, Colloqu. Math. 34 (1975) 8190. Rgflexivith dans les espaces mixtes. Application d la reflexivith des espaces bornologiques convexes, C.R. Acad. Sci. 278 (1974) A 10331035.
I. MIXED TOPOLOGIES
72 [49]
B. PERROT
Topologies mixtes dans le cas de structures vectorielles non necessairement convexes, Publ. D6p. Math. Lyon 104
( 1 9 7 3 ) 359370.
[SO]
A. PERSSON
A generalization of two norm Spaces, Ark. f. Math. 5 ( 1 9 6 3 ) 2736.
[51]
H. PFISTER
Uber eine Art von gemischter Topologie und einen Satz von Grothendieck iiber (DF)Raume, Man. Math. 1 0 ( 1 9 7 3 ) 273287.
Uber das Gewicht und den uberdeckungstyp von uniformen Rkiumen und einige Formen des Satzes von BanachSteinhaus, Man. Math. 2 0 ( 1 9 7 7 ) 5172. [53]
T. PRECUPANU
Remarques sur les topologies mixtes, An. Sti. Univ. “Al. I. CUZA” Iasi Sec. Ia Math. 13 ( 1 9 6 7 ) 277284.
[54]
V. PTAK
Completeness and the open mapping theorem, Bull. SOC. Math. France 86 ( 1 9 5 8 ) 4174.
[551
D. RAIKOV
On completeness in locally convex spaces (Russian), Uspehi Mat. Nauk. 14.1
[561
57 1
( 1 9 5 9 ) 223229.
A. ROBERTSON, W. ROBERTSON Topologische Vektorraume (Mannheim, 1 9 6 7 )
.
W. ROELCKE
On the finest locally convex topology agreeing with a given topology on a sequence of absolutely convex sets, Math. Ann. 1 9 8 ( 1 9 7 2 ) 5780. On the behaviour of linear mappings on absolutely convex sets and A. Grothendieck’s completion of locally convex spaces, 111. J. Math. 17 ( 1 9 7 3 ) 311316.
[ 581
[59]
(85)
W. RUESS
HalbnormDualitat und induktive Limestopo logien in der Theorie lokalkonvexer RBume (Habilitationsschrift, Bonn 1 9 7 6 ) .
REFERENCES [601
S.
[61]
H. SCHAEFER
SAKS
73
On some functionals, Trans. h e r . Math. SOC. 3 5 ( 1 9 3 3 ) 5 4 9  5 5 6 . Topological vector spaces (New York, 1966).
[621
L. SCHWARTZ
Probabilitbs cylindriques et applications radonifiantes, J. Fac. Sci. Univ. Tokyo, Sect. I A 1 8 ( 1 9 7 1 ) 1 3 9  1 8 6 .
1631
Z. SEMADENI
Extensions of linear functionals in two norm spaces, Bull. Pol. Acad. Sci. 8 (1960) 427432.
Embedding of two norm spaces into the space of bounded continuous functions on a half straight line, Bull. Pol. Acad. Sci. 8 ( 1 9 6 0 ) 4 2 1  4 2 6 .
641
[ 651
Projectivity, injectivity and duality, Dissertationes Math. 3 5 ( 1 9 6 3 ) 1  4 7 .
[ 661
Banach spaces of continuous functions I, (Warsaw, 1 9 7 1 ) .
[671
R. SERAFIN
[. 681
J.H.
SHAPIRO
On some class of locally convex spaces connected with Saks spaces and twonorm spaces, Bull. Acad. Polon. Sci. 2 2 ( 1 9 7 4 ) 11211 1 2 7 . Noncoincidence of the strict and strlmg operator topologies, Proc. Amer. Matt.. SOC. 3 5 ( 1 9 7 2 ) 8 1  8 7 . Weak topologies on subspaces of C ( S ) , Trans. Amer. Math. SOC. . I 5 7 ( 1 9 7 1 )
1691
471479. [7OI
K.D.
STROYAN
A non standard characterization of mixed topologies (in Springer Lecture Notes 369, 1974).
I. MIXED TOPOLOGIES
74 1711
P.K. SUBRAMANIAN
Twonorm spaces and decompositions of Banach spaces, I. Studia Math. 43 ( 1 9 7 2 ) 179194.
[72]
P.K. SUBRAMANIAN,
1731
R.F. WHEELER
ROTHMAN Twonorm spaces and decompositions of Banach spaces, 11. Trans. h e r . Math. SOC. 1 8 1 ( 1 9 7 3 ) 3 1 3  3 2 7 .
S.
The equicontinuous weak A topology and semireflexivity, Studia Math. 41 ( 1 9 7 2 ) 243256.
1741
M. DE WILDE
Limites inductives d’espaces linhaires a seminormes, Bull. SOC. Roy. Sc. Lidge 3 2 ( 1 9 6 3 ) 476484.
1751
Inductive limits and partitions of unity, Man. Math. 5 ( 1 9 7 1 ) 4558.
1761
Various types of barrelledness and increasing sequences of balanced and convex sets in locally convex spaces (in:”Summer School on Topological Vector Spaces“ Springer Lecture Notes 3 3 1 , Berlin 1 9 7 3 , pp. 21 1  2 1 7 ) .
1771
M. DE WILDE, C. HOUET On increasing sequences of absolutely convex sets in locally convex spaces, Math. Ann. 192 ( 1 9 7 1 ) 257261.
1781
A. WIWEGER
A topologisation of Saks spaces, Bull. Pol. Acad. Sci. 5 ( 1 9 5 7 ) 7 7 3  7 7 7 .
[791
Linear spaces with mixed topology, Studia Math. 2 0 ( 1 9 6 1 ) 4768.
1801
Some applications of the mixed topology to two normed spaces, Bull. Pol. Acad. Sci. 9 ( 1 9 6 1 ) 571574.
CHAPTER I1

SPACES OF BOUNDED, CONTINUOUS FUNCTIONS
Introduction: Chapter I1 is devoted to the most important and welldeveloped application of mixed topologies

the theory
of strict topologies on spaces of bounded, continuous functions. Since BUCK'S original paper (1958) the literature on this topic has grown rapidly. We have tried to give a fairly complete account of this theory from the point of view of mixed topologies. This approach often allows greater generality and simpler and clearer proofs than the original methods.
In section 1, we consider the basic properties of strict topologies. The original strict topology on C"(X) is the mixed topology
y[ll
I.2.D) l l , ~ ~ (see ]
where
T~
is the topology of
compact convergence. However, no particular difficulties arise when we replace the topology of compact convergence by that of convergence on an (almost) arbitrary family S of subsets of X. The first part is devoted to relating the elementary properties of the associated mixed topology with properties of X (resp. S ) . In 1.11 we show how these mixed topologies can be defined by weighted seminorms thus establishing the relation with Buck's topology. In 1.13 we give a very general StoneWeierstraB theorem based on a result of NEL. We use this to attack the problem of the separability of C"(X)
. Using partitions of unity we
give a new proof of a result of CONWAYLE CAM.
75
11. BOUNDED CONTINUOUS FUNCTIONS
76
In
5
2 we study the algebraic structure of C"(X)
. We
identi
fy its spectrum and show that the topological properties of X are faithfully reflected in the topologicalalgebraic
. We obtain a theory of GELFANDNAIMARK type and characterise the 6closed ideals of C"(X) . We then
properties of C"(X)
sketch a functionalanalytic approach to the real compactification of X.
In
5
3 we characterise the dual of C"(X)
as the space of bounded
Radon measures on X, thus extending a result of Buck for locally compact spaces and establishing the basis for applications of strict topologies to measure theory. Tight sets of measures are shown to coincide with the 6equicontinuous sets and the results are used to give a proof of PROHOROV's theorem on the existence of projective limits of compatible families of measures.
The fourth section is devoted to spaces of vectorvalued continuous functions. The main results are a tensorproduct representation (generalising the classical representation for Banach space valued functions on compact spaces), an exponential law and a tensorproduct representation of spaces of functions on product spaces. We also characterise the 8closed Cm (X)submodules

thus generalising the StoneWeierstraR theorem.
In the last section we define new strict topologies on C"(X) which have the spaces of aadditive (resp. radditive) measures
11.1
77
THE STRICT TOPOLOGIES
on X as dual. We use the imbedding of X in its StoneEechcompactification. We then give some results relating to the problem of establishing a connection between the topological and measuretheoretical properties of X.
11.1. THE STRICT TOPOLOGIES
1.1.
In this chapter, X will always denote a completely regular,
Hausdorff topological space, S a saturated family of closed subsets of X (that is,
u S is dense in X and S is closed under
the formation of finite unions and of closed subsets). Such a family is of countable type if there is a countable subfamily
S1 so that each B
E
S is contained in some B1
E
S.
Examples of saturated families are:
F : the finite subsets of X; K : the compact subsets of X;
8 : the bounded, closed subsets of X (recall that a subset B E X is bounded if each continuous, realvalued function on X is bounded on B). We denote by C(X)

the space of continuous, complexvalued €unctions on X;
Cm (X)

the space of bounded, continuous, complexvalued
functions on X.
11
IL denotes the supremum norm on C
m
(X). If B E X then

11. BOUNDED CONTINUOUS FUNCTIONS
78
pB : x
SUP
{lx(t)l : t
E
B}
.
is a seminorm on Coo(X) If B is bounded, we can regard it as a seminorm on C(X). If S is a saturated family, then
' I ~ denotes
the locally convex topology defined by the seminorms (p, : B Then (C"(X),
~ L , T is ~ )a
11
E
S).
Saks space and we denote by BS the
associated mixed topology.
1.2.
Proposition: 1)
C BS
C_ T
II
IIm
and
T~ =
BS
on the norm
bounded sets: 2) a subset of C"(X) is BSbounded if and only if it is norm
bounded: 3 ) a sequence
(x,)
in C"(x)
if it is norm bounded and 4 ) a subset of C"(X)
is BSconvergent to zero if and only
'I
S
convergent to zero;
is relatively BScompact if and only if
it is norm bounded and relatively 5) (C"(X) ,BS)
BS
= 'I
II
compact;
is barrelled or bornological if and only if
IIm;
6) a linear operator T from C"(X) into a locally convex space
is BScontinuous if and only if its restriction to the unit ball of
c"(x)
is
continuous;
7) if S is of countable type, then the unit ball of C"(X) is 'ISmetrisable and so a linear mapping T from C"(X) into a locally convex space is BScontinuous if and only if it is sequentially continuous.
1.3. Corollary: A subset of C"(X)
is BKprecompact if and only
if it is norm bounded and equicontinuous.
THE STRICT TOPOLOGIES
11.1
79
1.4. Definition: Let S , and S2 be saturated families in X. X is S 1  % normal if for every pair A , B of disjoint, closed
subsets of X with x :X
A E
+ [O,l]
S1, B
S2 there is a continuous function
E
with
xIA = 0
and
xIB
= 1.
We say that X is Snormal if X is SP
normal where P is the
family of all closed subsets of X.
Note that every X is Knormal (in fact, for a Hausdorff space, Knormality is equivalent to complete regularity  see BUCHWALTER 1211, Prop. 2.1.5). The following Proposition is a generalisation of URYSOHN's theorem and can be proved in exactly the same way (cf. BUCHWALTER 1211, Th6ordme 2.1.6).
1.5. Proposition: X is S  S
A
normal if and only if for each
S and each continuous mapping
E
u
is a continuous x : X
4
1.6. Proposition: If X is S  S
and only if X
[0,1I
x
:
A +
so that
normal then
(B
E
S)
,
xiA = x. *S =
if
TII II,
4
E
8 is trivial.
S. Then we can find, for each seminorm pB
an x in the unit ball of C"(X) so that pBWx)
and IIx]L = 1.
there
S.
E
Proof: The sufficiency of the condition S Suppose that X
L0,lI
=
0
80
11. BOUNDED CONTINUOUS FUNCTIONS
1.7.
In the following Proposition, we examine the problem
of the completeness of (Cm(X), B s ) .
It is convenient to intro
duce the following concept: a space X is Scomplete mapping
x : X
+
C
continuous for each A
E
if each
is continuous if and only if xIA is S (it suffices to consider bounded
functions). For example, X is Fcomplete if and only if it is discrete. The Kcomplete spaces are precisely the kRspaces (see, for example, MICHAEL [ l o o ] )
.
Locally compact spaces and metrisable spaces are Kcomplete.
To each space X one can associate a Scomplete space in a natural way: we give X the weak topology defined by the family of mappings from X into C which are such that their restrictions to each A
E
S are continuous. We denote X with this topology
by Xs. Then XS is Scomplete and X is Scomplete if and only if X = Xs. Cm(Xs) is precisely the space of bounded mappings from X into Q: whose restrictions to the sets of S are continuous.
1.8.
Proposition: Suppose that U S = X. Then (Cm(X),S)
is
complete if X is Scomplete. The converse is true if X is S

S normal.
Proof: Suppose that X is Scomplete. net in B
II 11,'
the unit ball of Cm(X)
Let (xaIaEI be a .rSCauchy
. Then
(x,) converges point
wise to a function x from X into C and the restriction of x to
A
E
S is continuous, as the uniform limit
x is continuous and
x
a
I x
in BS.
of ( ~ ~ 1 Hence ~ ) ~ ~ ~ .
11.1
THE STRICT TOPOLOGIES
Now suppose that (Cm(X),BS)
81
is complete and that

is such that xIA is continuous for each A
C
x : X
_j
C
S. We show that
x is continuous when X is S  S normal. It is no loss of generality to suppose that
x : X
For each A
[0,11.
E
S,
let xA be a continuous function from X into [O,ll so that
xA
=
x on A (1.5). Then (xAYAES is a Cauchy net for BS and
so converges to a function in
1.9.
c"(x)
i.e. x
E
c"(x).
Corollary: 1 ) (C" (X), B K ) is complete if and,only if X is
K complete ;
2) (C"(X),BF) is complete if and only if X is discrete.
1.10.
Corollary: Suppose that X is S  S normal and that U S = X. is (C"(XS) , B S ) .
Then the completion of (C"(X) , B S )
These results can be interpreted as follows: under the restriction operators, the family
{Cm(A); A
spaces forms a projective spectrum. If
E
S)
X = US
of Banach then the Saks
space projective limit of this spectrum is naturally identifiable with
(C"(XS),
11 l l m l ~ S ) ,
in particular, with (C"(X1, 11
IIrnrTS)
if X is Scomplete.
Now we give a concrete representation of a family of seminorms which defines B S . From these one can easily deduce that the mixed topology BS reduces, in special cases, to the strict topologies which have been studied on spaces of bounded, con
+
tinuous functions. We denote by LS the set of bounded non
TI. BOUNDED CONTINUOUS FUNCTIONS
82
negative upper semicontinuous functions 4 on X which vanish at infinity with respect to S i.e. which satisfy the condition: for each If 4
E
Ls+
E
>
, P$
It
0,
:
x
m
S : $(t) 2
E
E
E)
S

I14xllm
is a seminorm on C (X). The family of all such seminorms defines a locally convex topology ristic function of each A
1.11.
on C"(X) E
S is in L;
Proposition: If X is S  S
a
y
E
E
S and x
x
=
On
then for x p4 (x) I
E
BII IL' If 9 E B II IIm with
. On
E
is finer than
T ~ ) .
f3 = B S .
C"(X) then, by 1 . 5 ,
E
=
there is
pA(x). Let
z
:
x

y.
y + z is a suitable decomposition of x .
We now show that B S is finer than T~
8'
~ L , T satisfies ~) condition
Cm(X) so that y = x on A and Ilyll,
Then pA(z) = 0 and so
than
and so
normal, then
Proof: We note first that (C" (X) ,[I (a) of 1.4.5. If A
(note that the characte
+
LS,
pA(x)
>
E
I
8' 0
i.e. that and
is coarser
A := It
E
S : $(t) t € 1
IEsu l@(t)Il' we have
tER the other hand, if V is a 8neighbourhood of zero,
then, by 1.4.5, it contains a set of the form {x
E
C"(X) : PAn(X) I An)
where (An) is an increasing sequence in S and (A,)
is a strictly
increasing sequence of positive numbers which converges to infinity. Then if
11.1
A;’ +:t0 @
83
THE STRICT TOPOLOGIES (t
E
A,)
(t
E
An \ Anl)
(t
E
X \
U
An)
+
is in LS and V contains the unit ball of p
+ *
We now give a StoneWeierstraB theorem for B S . For convenience, m
we consider the space CIR (X) of real valued functions on X as a real vector space. The results can be extended to complexvalued functions using standard methods. Since the sets of S are not necessarily compact, we need a refinement of the classical StoneWeierstraR theorem due to NEL. Recall that a subset of X is a zeroset if it has the form x’(O) for some x
m
E
.
CIR (X) A subset M of C i (X) separates disjoint zerosets
if for each pair A,B of disjoint zero sets in X, there is an x
E


M so that x(A) and x(B) are disjoint. The following Lemma
follows from the fact that the points of the Stonezech compactification f 3 X of X are limits of zultrafilters in X (cf. GILLMAN and JERISON [551, Ch. 6). Its proof can be found in NEL [ 1081
. m
1.12. Lemma: Let M be a subset of CIR (X) which separates disjoint zero sets in X. Then M, regarded as a subset of C i (8 XI, separates the points of B X . m
1.13. Proposition: Suppose that M is a subalgebra of Cn (X) so that for each A
E
S, MA, the restriction of M to A , separates
11. BOUNDED CONTINUOUS FUNCTIONS
84
disjoint zerosets in A and contains a function which is m
bounded away from zero on A. Then M is 6s dense in Cn (X)
.
Proof: We can assume that M is BSclosed. Then it is norm closed and so is a lattice under the pointwise ordering. We show that if x E C L (XI I 0 < is a y
E
E
< 1
M so that IIylloos 11x11, + 1
and if A and
E
S I then there
pA(xy)
IE.
MA, regarded as an algebra of functions on BA, satisfies the conditions of the classical StoneWeierstraR theorem (for MA seperates the points of f3A by 1.12). Hence MA is normdense m
in Cn (A) and so there is a y 1
E
M with
pA(xyl) 5 E. Then
y := sup{ inf ( y l l llxllm + 1 1 ,  ( I l x l ~+ 1))
is the required function.
1.14. Corollary: Let M'be a subalgebra of C i (X) so that M
separates the points of X and for each t k X there is an x E M with
m
x(t) f 0. Then M is BKdense in Cn (X).
We now consider the problem of characterising those spaces for which (Cm(X),B~)is separable. Since the Banach space Cm(A) ( A
E
S) can only be separable if A is compact (this is a
classical result of M. KREIN and S. KREIN and follows from the fact that the StoneEech compactification of a noncompact space is never metrisable Cm(A) is

see GILLMAN and JERISON 1551, 5 9.6) and
at least when X is S  S normal, a continuous image
of (C"(X , 8 , ) ,
it is natural to impose the condition that S C K
i.e. the sets of S are compact.
11.1 1.15. Proposition: If
THE STRICT TOPOLOGIES
F C
S C K
then
(C"
85
(X),Bs)
is separable
if and only if there is a weaker separable, metrisable topology on X.
Proof: Since we shall use the StoneWeierstraR theorem, it is convenient to restrict attention to C,
m
(XI. m
If M is a countable BSdense subset of C , ( X ) ,
then the weak
topology defined by M satisfies the given conditions. Sufficiency: let
T
be a suitable separable, metrisable topology
on X. Then by Urysohn's metrisation theorem (WILLARD (X,T)
[
can be embedded in a compact, rnetrisable space Y
571
,
23.
For
each positive integer n, we can find a finite open covering (Un) of Y so that maxi diam U : U
E
Un) S l/n
(diam (U) is the
diameter of U). Let (Pn be a partition of unity of Y subordinate to Un and denote by M the subalgebra of C G (XI generated by the restrictions of the elements of m
by 1 .I4 and so
(C, ( X I ,Bs)
;Qn
to X. Then M is BSdense
is separable.
1.16. Remark: It follows from SMIRNOV's metrisation theorem (see WILLARD [1571, 23.G.3) that if X is locally compact and paracompact and S possesses a weaker metrisable topology, then X is metrisable. Hence if X is locally compact, paracompact
and ( C m ( X ) ,Bs)
is separable, then X is metrisable (cf. SUMMERS
11391, Th. 2.5).
)
11. BOUNDED CONTINUOUS FUNCTIONS
86 1.17.
Proposition: If X is discrete, then (C"(X),Bs) is
separable if and only if
card (X) I card (IR)
.
Proof: The necessity follows from 1 . 1 5 and the fact that the cardinality of a separable metrisable space is at most card (XI). On the other hand, XI metrisable topology
1.18.

(and hence any subset) has a separable, the natural topology.
Remark: A more intricate argument shows that the same
result holds for metrisable spaces X (see SUMMERS [ 1 3 9 1 , Th. 3.2).
In the third section of this chapter we shall consider duality for Coo(XI , with strict topologies. However, using the theory of Chapter I we can already provide some information on this duality, without specifically calculating the dual space

in
particular, we can give sufficient conditions for B K to be the Mackey topology and we can characterise the relatively weakly compact subsets of C"(X).
1.19.
Partitions of unity for COD(X): Let X be a locally compact,
paracompact space. Then there exists a partition of unity on X (see BOURBAKI [ 1 5 1 , IX.4.3) so that Now (C"(X) , I ] system
1 1 , ~ ~ ) is
{C(K) : K
E
: K E K )
supp $ I K G K .
the Saks space projective limit of the
K)
of Banach spaces and if we define the
mappings TK :
{$IK
x

(x
11.1
THE STRICT TOPOLOGIES
from C(K) into C.(X) where (x $IK)
A
a7
denotes t h e extension of
x $IK to a function on X obtained by setting it equal to zero off K, then (TK) is a partition of unity in the sense of 1.4.17. Hence we have, by 1.4.19 and 1.4.24:
1.20. Proposition: Let X be locally compact and paracompact. Then (C" (X), B K ) is a Mackey space and has the BanachSteinhaus property. Also a linear mapping from Cm(X) into a separable Frgchet space is BKcontinuous if and only if its graph is closed.
In the next results, the phrase "weak topology on C" (X) will I'
be used to denote the weak topology defined by the dual of (C" (XI ,BK).
1.21. Proposition: A sequence (x,)
in C" (XI converges weakly
to x if and only if {xnl is uniformly bounded and the functions xn converge pointwise to x. 1.22. Proposition: A bounded subset B of C" (XI is weakly precompact if and only if it is precompact for the topology of pointwise convergence on X. Hence if X is Kcomplete, then B is relatively weakly compact if and only if it is relatively compact for the topology of pointwise convergence on X.
Proof: Using 1.1.20, we can reduce 1.21 and 1.22 to the case where X is compact (see, for example, GROTHENDIECK [62], pp. 12 and
209
for this case).
11. BOUNDED CONTINUOUS FUNCTIONS
88
1.23. Remark: Using results from GROTHENDIECK 1601 , one can
strengthen 1.22 as follows: suppose that X is Kcomplete and has a dense subset which is the union of countably many compact sets. Then the following conditions on a bounded subset B of a)
c”(x) are equivalent: B is relatively countable compact for
T
P
(resp. for
the weak topology); b)
B is relatively sequentially compact for
T
P
(resp.
for the weak topology); c) topology 1 Here
T
P
B is relatively compact for
.
T
P
(resp. for the weak
denotes the topology of pointwise convergence on X.
1.24. Remark: One of the main themes of this Chapter will be
that of relating the topological properties of X with the linear (or algebraic) and topological properties of C“ (X) with various mixed topologies (cf. 1.8, 1.9, 1.15 for example). We list here some examples without proofs: 1)
X is hemicompact (i.e. K is of countable type) if
and only if *)
BII
BII II,
11, is
BKmetrisable;
is BKseparable and metrisable if and only if
X is hemicompact and each K 3)
E
K is metrisable;
if X is locally compact, then B
I1 II,
is BKseparable
and metrisable if and only if X is separable and metrisable (alternatively if X is the countable union of compact, metrisable sets);
11.2 4)
ALGEBRAS OF BOUNDED, CONTINUOUS FUNCTIONS
89
the following conditions are equivalent: a) B II11 is BKcompact (i.e. (C" (X),BK) b) (C" (X),BK)
is semiMontel);
is semireflexive;
m
c) (C (X),BK) is a Schwartz space; d) X is discrete. 5)
m
(C (X),BK) is nuclear if and only if X is finite;
B K = ' I ~on C"(X) if and only if the union of countably many compact subsets of X is relatively compact. If this is the 6)
case, then CCm (XI , B K ) is a (DF)space.
1.25. Remark: A number of results given in this section for Cm (X) with the topology B K (e.g. those of 1.24) can be extended
to B,
with the natural changes. We leave the task of carrying
out such extension to the interested reader, mentioning only that 1.20 can be extended to the topology B B by replacing the assumption of local compactness by local boundedness (obvious definition ! ) and that SCHMETS and ZAFARANI have studied the topology B p in I: 1 2 4 1 .
11.2. ALGEBRAS OF BOUNDED, CONTINUOUS FUNCTIONS
In the first part of this section, we work exclusively with the strict topology defined by the family K of compact subsets of X. To simplify the notation, we denote it by 8 . First we note that (Cm(X),11
1 1 , ~ ~ ) is
a preSaks algebra (that is, its
completion is a Saks algebra).
11. BOUNDED CONTINUOUS FUNCTIONS
90
2.1
. Proposition: Multiplication is continuous on
(C" (X),B )
Proof: We use the representation of $ given in 1 . 1 1 . does JI :=
so
If @
.
E
+
LK,
and we have the following inequality
In general, inversion is not continuous on C"(X) and so (Cm(X),B) is not a locally multiplicatively convex algebra in the sense of MICHAEL [ 9 9 1 .
If t
E
X then
6,
:
x
x(t)
.
is an element of the spectrum M (C"(X)) of C"(X) We have thus Y (Cm(X)). 6 t from X into constructed a mapping 6 : t
5
We call it the generalised Dirac transformation. It is injective since
2.2.
c"(x)
separates X. m
proposition: 6 is a homeomorphism from X onto Mv(C (X)).
Proof: Since the topologies on X and M (Cm(X)) are the weak
Y
topologies defined by Cm(X), it is sufficient to show that 6 is surjective. Let f be a $continuous multiplicative functional on C"(X) m cR
x
E
and denote by M the kernel of the restriction of f to
(X). Then there is a to E X so that
M (for otherwise M would satisfy the conditions of 1 . 1 4 and m
SO
x(to) = 0 for each
would be $dense in CR (X) i.e. € would be zero).
11.2
91
ALGEBRAS OF BOUNDED, CONTINUOUS FUNCTIONS

Note that M separates X. For otherwise there would be points s1,s2 in X so that
x(sl) = x(s2)
in the kernel of the linear form
for x x
M. Then M would lie
E
x(sl)
.
so would have codimension at least two

f = 6
If X,X1 are completely regular spaces, $ : X d XI tinuous, then C"($)
:
x
x
0
and
Then M = {x : x(to) = 0 )
(for both these sets have codimension one) and so

x(s2)
tocon
$ m
m
is a @continuous star homomorphism from C (X,) into C (XI. In fact, every such homomorphism has this form as the following result shows:
2.3. Proposition: If 0 is a 8continuous homomorphism from Cm (XI) into
Cm (X)
then
has the form Cm 9 ) for some continuous
mapping $ from X into X I . Proof: If t 
E
X then 6t
Q, is a @continuous multiplicative
form on Cm(Xl) and so is defined by a unique element of X I

we denote this element by $(t). By the construction of this mapping I$ we have @(x) = x
for each x
E
0
9
Cm(Xl). Hence for each x
E
m
C (XI),x o I$
E
Cm(X) and
this property characterises continuity for mappings between completely regular spaces.
11. BOUNDED CONTINUOUS FUNCTIONS
92
2.4. Corollary: X and X, are homeomorphic if and only if
C"(X) and C"(X1) are isomorphic as preSaks algebras.
We remark that the following version of the BanachStone theorem for noncompact spaces can be deduced from 2.4: if there is an isometry from C"(X) onto C"(X,) which is also 8bicontinuous, then X and XI are homeomorphic.
If (A,lI if x
11,~)

is a commutative preSaks algebra with unit and
A, then the mapping
E
2
:
f
f(x)
from M ( A ) into C is an element of Cm(M ( A ) ) . Thus we have con
Y
Y
structed an algebra homomorphism from A into Cm(M (A)). We call
Y
it the generalised GelfandNaimark transform. Note that we can regard My (A) as a subspace of the spectrum M(A) of the normed algebra (A,11
11).
The generalised GelfandNaimark transform is
then the composition of the GelfandNaimark transform for A and the restriction operator from C(M(A)) into Cm(My(A) )
. In
particular, if A is a preSaks C"algebra, then the generalised GelfandNaimark transform is a starhomomorphism (this also follows directly from 2.3).
2.5. Proposition: If (A,II
11,~)
is a commutative Saks (?algebra
then the generalised GelfandNaimark transform is an algebra isomorphism from A onto Cm(M (A)). Y
93
ALGEBRAS OF BOUNDED, CONTINUOUS FUNCTIONS
11.2
P r o o f : We f i r s t n o t e t h a t t h e image of A i n C m ( M y ( A ) ) i s a selfadjoint,
s e p a r h t i n g s u b a l g e b r a which c o n t a i n s t h e c o n s t a n t s
and so i s @  d e n s e by t h e complex v e r s i o n o f 1 . 1 4 . Now l e t P be
a f a m i l y of C"seminorms on A which d e f i n e For each p
E
P I w e d e n o t e by A
'I
(as i n 1 . 3 . 1 3 ) .
t h e a s s o c i a t e d @algebra
P
and
by M ( A ) i t s s p e c t r u m . W e c a n r e g a r d M(A ) as a (compact) s u b s e t P P o f M ( A ) and w e show t h a t M y ( A ) = U M ( A 1 . I f f E My ( A ) , t h e n , Y PEP P by 1 . 3 . 1 0 , t h e r e i s an i n c r e a s i n g s e q u e n c e (p,) i n P and a n fn
E
Ap,l
so t h a t C f n is a b s o l u t e l y summable t o f . W e c a n a l s o
suppose t h a t
11 C
frill
Choose no so t h a t enough
E.
<
1
f
f o r an a r b i t r a r y p o s i t i v e
E
C
IIfnll
4
M(Apno)
n>no
For i f
+
<
E.
Then f
E
E.
.
M(Apno) f o r s m a l l
then t h e r e i s an
x
E
C" (MY ( A ) )
llxll I 1 , x ( f ) = 1 and x = o
on M ( A 1 . By t h e @ Pn0 d e n s i t y of t h e image of A , t h e r e i s a n x1 E A w i t h IIxl 11 S. 1 + so t h a t
and g
lG,(f)/ I 1 Apno'
with

llgll
IIf 
E,
<
],;I 1 +
< E
E
on M ( A ) . Hence, f o r e a c h Pn0 w e have E
gll 2 (1+E)I
Ilf(Xl)
and we o b t a i n a c o n t r a d i c t i o n f o r s m a l l

g(xl)ll 2
E
by t a k i n g
1+E I€ g =
"0
C
n= 1
fn.
To complete t h e p r o o f , w e l e t S be t h e familv of c l o s e d s u b s e t s
of My(A) which are c o n t a i n e d i n some M ( A 1 ( t h e p d e p e n d i n g on
P
*.
t h e s u b s e t ) a n d , a s a t e m p o r a r y n o t a t i o n , A be t h e S a k s s p a c e p r o j e c t i v e l i m i t o f t h e s y s t e m {C(M(A 1 ) lpEp. P
11. BOUNDED CONTINUOUS FUNCTIONS
94
Consider the following diagram
where the vertical arrows are the corresponding GelfandNaimark transforms and so are isomorphisms. Then the general GelfandA
Naimark transform, being the unique arrow from A into A which preserves commutativity, is an isomorphism and so
= Cm(My(A))
and the generalised GelfandNaimark transform is surjective.
Note that the inverse of the generalised GelfandNaimark transform is 8continuous. However, we cannot, in general, expect it to be bicontinuous. For example, if S is a proper subfamily of K which contains F and is such that a function x : X
_j
C
is continuous if and only if its restriction to the sets of S are continuous, then the generalised GelfandNaimark transform for (C"(X), 11 identity
1 1 , ~ ~ ) is (up to the obvious identifications) the from (C"(X), 11 1 1 , ~ ~ ) into (C"(X), 11 1 1 , ~ ~ ) and this is
not continuous, in general (as an example of such an S we could take the family consisting of the ranges of convergent sequences and their limit points in a metrisable space).
We now characterise local compactness for X in terms of properties of C"(X).
Let (A,11
11,~)
be a commutative Saks algebra and let
11.2
ALGEBR3;S OF BOUNDED, CONTINUOUS FUNCTIONS
95
P be a suitable family of submultiplicative seminorms defining If p
E
T.
P I put I := {x P A(Ip) := {y
E
A : p(x)
E
A : YIP = 01.
= 0)
A is perfect if
C A(I ) is ydense in A. Obviously this PEP P property is preserved if we refine the topology T. As an example,
if p is the seminorm pK (K
A ( I ~ )= {x Hence
E
E
K ) on C" (X), then
cm(x)
:
x(t) =
o
for
t
E
x
\ K)
) is Cc(X), the space of functions in c"(X) with P compact support.
A(I
2.6. Proposition: A completely regular space is locally compact
1 1 , ~ ~ ) is
if and only if (Cm(X),I1
perfect.
Proof: In view of the above remarks, this is equivalent to the following statement: X is locally compact if and only if Cc(X) is @dense in C" (X). Suppose that X is locally compact. Then Cc(X) separates X and so is @dense by 1 . 1 4 .
Now suppose that Cc(X) is 6dense in C"(X) is an x
E
. If
t
E
X, then there
Cc(X) so that x(t) > 0. Then { s : x(s)
>
0)
is a relatively compact neighbourhood of t.
Using the generalised GelfandNaimark transform, it is easy to see that if A is a perfect, commutative Saks C"ralgebra,
11. BOUNDED CONTINUOUS FUNCTIONS
96
then M (A) is locally compact. The reverse implication is Y not true.
Let I be an ideal in C"(X) and write ~ ( 1 )= where
Z(x) := x'(O)
n
XE I
z(~)
is the zeroset of x. Then we put
I(z(I)) := {y
E
c"(x)
:
y
=
o
on
Then I(Z(1)) is obviously a 8closed ideal and We shall now show that
I = I(Z(1) )
~(1)). I C I(Z(1)).
if and only if I is 8 
closed. This result is wellknown for compact X and we shall use it for the proof of the result in the general case. We sketch briefly how it can be proved. Suppose that x E I(Z(1)). For each
E
>
0 we can find an open neighbourhood U of Z(1) and
a function xE in C(X) so that xE vanishes on U and We shall show that xE
E
IIx

xEI)I E.
I which will finish the proof. By a
...,
compactness argument, there exist x,, xn in I so that n U 2 n Z(xi). Then U contains the zeroset of the element i=l y := I xi[ of I. But then the zeroset of xE is a neigh
*
bourhood of the zeroset of y and so xE is a multiple of y.
2.7. Proposition: Let I be a 8closed ideal of C"(X)
Then
.
I = I(Z(1)).
Proof: I is a normclosed ideal in C(gX) and so there is a closed set KO in BX so that
I = (x E C(f3X) : x = 0
on KO).
11.2
97
ALGEBRAS OF BOUNDED, CONTINUOUS FUNCTIONS
It is obviously sufficient to show that KO = clf3XZ(1) (closure in 8x1 for then if a function vanishes on Z(1) its extension to f3X vanishes on KO and so is in I. If this were not the case, there would be a with yo(to) = 1
to
E
and
We now show that yo
KO \ clgxZ(I). Then there is a yo
y = 0 on a neighbourhood of

yKIE)
C(f3X)
clgxZ(I).
I which gives a contradiction. To do this,
E
we show that for each K that PK(Y,
E
E
E
K,
E
>
0, there is a yK
IE
in I so
and IIYK,EII 5 IIYoII. Then (YK,€) is a
net in I which is BKconvergent to yo. To construct yK
I E
we
proceed as follows: let IK denote the projection of I in C(K). Then IK is an ideal in C(K) and so
TKIits
space C(K) I is a closed ideal in C(K) {y
E
C(K) : y = 0 on
closure in the Banach
. Hence
Z(1) n K ) . Hence
yIK
it has the form E
yK and so Tietze’s
theorem implies the existence of the required yK,€.
The results of this section can be used to give a natural construction of the StoneEech compactification and the realcompactification of a completely regular space. We describe this briefly, firstly to display the connection between mixed topologies and the theory of topological extensions and secondly because it will allow us to give a significant generalisation of 2.2. If X is a completely regular space, (C”(X) ,[I
11)
is a Banach
algebra. Its spectrum M(Cm (XI 1 is a compact space which we denote by BX. The Dirac transformation can be regarded as a (topo
98
11. BOUNDED CONTINUOUS FUNCTIONS
logical) embedding of X into BX. It has the following universal property: if 4 is a continuous mapping from X into a compact space K, then there is a unique continuous extension
7
of I$ to a continuous mapping from BX into K. For consider the operator Cm(I$) : C(K) d C"(X) = C(BX) ,which is
11
1lB continuous and so (by the closed graph theorem
or, more elementarily, by 1.1.11) by 2.3, C"(@)
11 11  11 11
continuous. Hence,
(regarded as a mapping from C(K) into C(BX), has
7:
the form C"(7) for some
BX
___j
K.
7
has the required
property.
Now we denote by I(X) the set of those functions in C"(X) which have no zeros in X (i.e. are invertible in the algebra C(X)). Every x
E
Cm(X) has a unique extension to a function in C(@X)
which we shall continue to denote by x. Then we put
Ux where C (x) = Cs BX in BX.
:=
E
x
n E
I (XI
cBx(x)
gX : x(s)
0 )
is the cozero set of x
UX is the realcompactification of X and X is realcompact if UX = X. The above rather unfamiliar definition is the natural one from the point of view of strict topologies. The following equivalent forms are better known:
11.2
ALGEBRAS OF BOUNDED, CONTINUOUS FUNCTIONS
ux
2.8. Proposition: 1 ) where
2
fl
=
x
E
99
xI (1R)
C,(X)
denotes the extension of x
E
to a function from
C,(X)
BX into the 2point compactification of El. 2)
uX is the completion of X with respect to the C(X)uniformity
on X.
Proof: 1) follows from the simple fact that if x l/lxl
E
C,(X)
E
I(X), then
and the zeros of x in gX are precisely the points
where (l/lxl) is infinite in value. For 2) see GILLMAN and JERISON [551,
5
15.13.
2.9. Corollary: For a completely regular space X I the following
are equivalent: 1)
X is complete for the C(X) uniformity;
2)
X is realcompact;
3)
for each s
E
BX \ X, there is an x
E
I(X) with x(s) = 0.
Now it is clear that the bounded subsets of X are precisely those subsets of X which are precompact in the C(X)uniformity or, by the above result, relatively compact in uX. This remark makes it natural to generalise the definitions of 1.1
to include saturated families S of subsets of UX (for reasons
which will be clear later, it is convenient to drop the assumption that the sets be closed). Hence if S is such a family, we can define the strict topology B s on C"(X).
We shall always assume
11. BOUNDED CONTINUOUS FUNCTIONS
100
that the subsets of S are relatively compact in uX. The following result is a significant generalisation of 2 . 2 .
2.10.
Proposition: The spectrum of the topological algebra
(c" (XI,
B ~ )
is
u
BES
cluX(~) (closure in UX)
Proof: It is clear that any point in
B;
.
cluX(B) defines
an element in the required spectrum. On the other hand, any point in the spectrum is defined by a member of uX (apply 2 . 2 to the space uX). Hence it is sufficient to show that if
4
is not BScontinuous. But this t0 follows easily from the fact that for each B E S there is an to
cluX(B) then 6
BII
II,
so that x(to) = 1
and
x = 0 on B.
(C"(X) ,BB)
is
u
2.11.
Corollary: The spectrum of
2.12.
Remark: The space of 2 . 1 1 has been introduced by BUCH
BEB
clUX(B).
WALTER [ 2 2 ] who denoted it by X" (because of a certain formal analogy with the bidual of a locally convex space) in connection with the concept of a uspace i.e. a completely regular space in which
B
= K (cf. the concept of semiMonte1 locally convex
space). Every space has a "uification" pX which is obtained as the limit of the transfinite series X,X", (X")",... and X is a pspace if and only if
X = ~ J X(or alternatively if X = X").
It is now clear that X is a pspace if and only if B K = B g C"(X).
on
11.3
101
DUALITY THEORY
11.3. DUALITY THEORY
A classical result of BUCK for the space C ” ( X )
(X locally
compact) is that the dual of (C” (X), B ) is the space of bounded Radon measures on X. In this section, we shall extend this result to completely regular spaces. We shall take this opportunity to discuss various equivalent definitions of Radon measures on completely regular spaces.
3.1. Definition: A premeasure on X is a member of the (vector space) projective limit of the system : M(K1)
,K
“K1
+
M(K);
K Z K1
, K,K1
E
K (XI)
In other words, a premeasure is a system p = { p K ) of Radon measures which satisfies the compatibility relations u K1IK  ’K ( K C K 1 ) . If LI = {p,) is a premeasure on X, I L I ~ ~ ” denotes the outer measure on K defined by Thus Il1~1’‘
I uKI
(see BOURBAKI
is defined as follows: if U E K is open,
is defined to be
sup { J f dl uKI 1
I uKl*
IuKI” (u) : u
open in K , A E
Now if C is a subset of X we define
xu.
IuKI”’ (U) For general
Iul”
u).
(C) to be
sup { l u K \ ” ( C
fl
K) : K
A premeasure
u
on X is said to be tight if for each
E
IV. 1 . 4 )
(A) is defined to be
inf {
is a K
5
where f ranges over the family
of positive, continuous functions on K with f I A 5 K,
131
K ( X ) s o that
E
K(X)).
lu]*
(X
\ K) <
E.
E
> 0 there
An equivalent condition
11. BOUNDED CONTINUOUS FUNCTIONS
102
is the existence of an increasing sequence (K,) that I pI" (X \ Kn) +
in K(X) so
0. We denote by Mt (X) the space of
tight measures on X. It is clearly a vector space. If x E C"(X) p E Mt(X)
, then
the limit
lim n
XI
n
dp
Kn
independent of the particular choice of (K,).
exists and is We write
j x dp
for this limit.
If K
E
K (X) and
p
E
M(K) , then l.~ defines a tight measure on X
in a natural way: if K1 E K(X) and K1 E K we define p
K1
to be
to be the K1 measure induced on K, by p (for example, as a linear form on the restriction of
to K1. If K, 2 K we define p

C(K1 1 , VK1 is the mapping
x Then
:= (pK
1
xlK,du
).
1 is a tight measure on X and IFI"(X \ K) = 0.
Hence we can (and do) identify the space of measures on K with the subspace of Mt(X) consisting of those p for which
I PI''
u M(K) K E K(X) the space of measures with compact support. Then p E Mo(X) if (X \ K ) = 0. We denote by Mo(X) the subspace
and only if there is a K
E
K (X) so that

(X \ K) = 0 .
3.2. Proposition: The dual of (c" (x)8't.K) is naturally isomorphic
to Mo(X) under the bilinear form
Proof: (C" (X), T ~ )is a dense subspace of the locally convex
11.3
103
DUALITY THEORY

projective limit of the system
{ P K ~,K : C(Kl)
C(K); K c K 1 , K,K1
E
K(X)}
Now by a standard result on the duals of projective limits (see SCHAEFER, Ch.1 [61] §IV.4.4), the dual of the latter space is the union of the spaces {M(K)}KE K(X) i.e. Mo(X) under the above identification.

3.3. Proposition: The dual of (Cm(X), B ) is isomorphic to the space Mt(X) under the bilinear form (xru)
Proof: Each 11
E
j x d1.I*
Mt(X) defines a linear form on Cm(X) and we
show that it is TKcontinuous on the unit ball of Cm(X). If E
>
0, choose K
x
E
Cm(X) with llxll
and 11.11'" (X)
<
m
E
K(X) so that I p l " ( X \ K) < I1
and
1x1
5 E
E.
Then if
on K
since 1.1 is tight.
Now let f be a 6continuous linear form on C" (X)
.
C fn where fn is a continuous linear form on some C(Kn) and C (1 f,[( < m . (1.3.10) We can n n1 as above. Now {pi}n,, is aboluteregard fn as a premeasure { u K ly summable (in the Banach space M(K)) and so there is a 1~~ E M(K)
Then we can express f as a sum
.
11. BOUNDED CONTINUOUS FUNCTIONS
104
n is easy to see that 11 := IllK : K n n is a premeasure on X. If we let Kn1 := U Kk then
with
p K = C 1 . 1 ~ . It
E
K(X))
k= 1
and so p is tight. One can check that f(x) = x
E
x du
for
C"(X).
3 . 4 . Alternative definitions of Radon measures: There are
several alternative, equivalent definitions for (bounded) Radon measures on a completely regular space and, before continuing, we describe the most important of these. For convenience, we consider only nonnegative measures: A. A
compactregular Borel measure p on X is a 0additive
finite measure on the Borel algebra of X so that for each Borel set A in X p
(A) = sup { p ( K ) : K

E
K (X) , K
C_
A).
B. A Choquet measure on X is a bounded set function p
: K (X)
p (K1)
IR+ which is increasing, additive (i.e.
+ u (K2)
= p (K1
u
K2)
+ u (K1
n K2)
for each pair K1 , K 2 of
compacta) and continuous on the right (i.e. for there is a neighbourhood V of K so that u ( K , ) each K1
E
K (X) with K C_ K,
C V)
E
>
0, K
Ip(K)
+
E E
K(X)
for
.
C. A tight measure on X is a bounded, Borel measure which satisfies the tightness condition: for every compact set K so that
u (X
\ K)
<
E.
E
>
0, there is a
11.3
105
DUALITY THEORY
D. A Radon measure p on g X is concentrated on X if inf {p(U) : U open and
@X \ X E U) = 0.
Then one can show that the above concepts all coincide in a natural way and correspond exactly to the nonnegative elements of Mt(X). A precise discussion can be found in SCHWARTZ [1251.
In the next Proposition, we can characterise the 6equicontinuous
.
subsets of Mt (X)
3.5. Definition: We remark that if 11 = {p,)
is a tight measure
on X then so is the premeasure i lpK1l. We denote it by
1 ~ 1 .
A subset B of Mt(X) is uniformly tight if it is bounded (for the norm) in Mt(X) and satisfies the tightness condition: for every IpI (X \ K)
E
> 0 there is a K
<
E
for each p
E
E
K(X) so that
B.
3.6. Proposition: A subset B of Mt(X) is uniformly tight if and only if it is 8equicontinuous.
Proof: We remark firstly that it follows easily from the characterisation of equicontinuous subsets in the dual of a locally convex projective limit of Banach spaces that a subset B1 of Mo(X) is TKequicontinuous if and only if it is norm bounded and has compact support (i.e. there is a K E 1.1
E
K(X)
so that each
B vanishes on X \ K). The result follows then from this
fact and 1.1.22 as in the proof of 3.3.
11. BOUNDED CONTINUOUS FUNCTIONS
106 3.7.
Corollary: A uniformly tight subset of Mt(X) is relatively
.
compact for the weak topology defined by C" (X)
Note that the converse of this result is not always true. In fact, the truth of the converse is equivalent to (Cm(X), 8 ) being a Mackey space. Hence, the first claim of 1 . 2 0 can be restated as follows:
3.8.
Proposition: Let X be a locally compact, paracompact space.
Then a weakly compact subset of Mt(X) is uniformly tight.
We now consider properties of the linear operator C"($) induced by a continuous mappincj $ : X transposed mapping of C" ( 4
+Y.
We denote by Mt($) the
so that Mt ( $ 1 is a normbounded
linear mapping from Mt(X) into Mt(Y). Note that if we regard a measure p
E
M t (X) as a Borel measure (as in 3 . 4 . A for example)

then Mt($) (11) is the Borel measure A
v ( 4  I (A))
i.e. it coincides with the measure induced by $ in the classical sense.
3.9.
Proposition: 1 )
C"($) is a quotient mapping from the Banch
space C"(Y) onto a normclosed subspace of Cm(X); 2)
an element p in Mt(Y) is in the range of Mt($) if and
only if for each
I I IJ
(Y \
(K))
<
E
E.
>
0 there is a K
E
K(X) so that
M~ ( $ 1 is a quotient mapping from M~ (XI onto
a normclosed subspace of Mt(Y);
11.3 3)
107
DUALITY THEORY
i s an open mapping ( f o r t h e s t r i c t t o p o l o g i e s )
C"(Ip)
from C m ( X ) o n t o i t s r a n g e i f and o n l y i f I$ (X) i s c l o s e d i n Y and f o r e a c h K
E
K ( + ( X ) ) t h e r e i s a K1
E
K ( X ) w i t h + ( K 1 ) 2 K.
3.9 i s proved by means o f a series of Lemmas. To s i m p l i f y t h e n o t a t i o n , we d e n o t e t h e o p e r a t o r s C m ( + ) and M t ( I p )
by U and V
respectively.
3 . 1 0 . Lemma: 3 . 9 . 1 )
holds.
Proof: W e show t h a t i f y
x
E
E
Cm(Y) then t h e r e is a z
Cm ( X ) h a s t h e form x E
Cm(Y) with
Ip f o r some
I I z I I = llyll
and
Ip. But t h i s i s t h e case f o r z d e f i n e d as f o l l o w s :
y = z
a
3.11.
Lemma: Suppose t h a t X and Y are compact and Ip i s s u r j e c t i v e .
Then i f 11
E
Mt(Y)
there is a v
E
Mt(X)
with
IIuII =
IIVl]
and
vv = u .
Proof: Since
+
i s s u r j e c t i v e , U i s an i n j e c t i o n and so a n iso
m e t r y from C(Y) o n t o a c l o s e d s u b s p a c e A o f C ( X ) regard
u
. We
can t h e n
as a c o n t i n u o u s l i n e a r form on A and t h e r e s u l t t h e n
f o l l o w s by t a k i n g v t o b e a HahnBanach functional t o C (X)
.
e x t e n s i o n of t h i s
108
11. BOUNDED CONTINUOUS FUNCTIONS
3.12. Lemma: for each
E
>
0 there is a compact set K in X so that
<
1 1  1 1 (Y \ $(K))
and IIVII =
Mt(Y) is in the range of V if and only if
11 E
>
Mt(X) with l.~ = Vv
suppose that 11 = Vv with v
0, there is a K
I u ] (Y \
E
II!JII.
Proof: Necessity: E
Then there is a v
E .
$(K)) <
E
E
Mt(X). Then for
K (X), so that I v I (X \ K) <
E.
Then clearly
E.
Sufficiency: we can choose an increasing sequence (K,) pacts in X so that 11.11 (Y \ $ (K,) )
A1 and put 11,
:=
(An is a Bore1 set in Y)
'Ian
IIu11
Then one has
< l/n. Let
$(K1), An := $(Kn) \ $(Knl)
:=
= C
of com
(n > 1 )
.
ll~nll.
By applying 3.11 successively to the restrictions of @ to Kn we get a sequence (v,)
IIvnll
and
= llpn]/
of Radon measures where vn
Vvn  pn. Then the series C vn
E
Mt(Kn).
is absolutely
summable in Mt (X) and its sum v is the required measure. In addition, we have
Ilv11
5
c IIvnII
=
E
lIunIl
=
Ilull.
3.13. Lemma: If V(Mt(X)) is weakly closed in Mt(Y), then $(XI is closed in Y.
Proof: If x on
E
Cq(Y) with Ux = 0, then x vanishes on $(X) and so
90. Hence
if s
E
m, then 6 s
is in the polar of the kernel
11.3
of U . B u t t h e l a t t e r s e t i s V ( M t ( X ) ) and so 6
V1.1 f o r some
=
)J
109
DUALITY THEORY
E
Mt ( X )
by t h e b i p o l a r theorem
. Then
1 = 6 s ( c s 3 ) = V p ( C s 3 ) = Il(+l (s))
and so 0l ( { s } ) i s nonempty i . e . s
3.14.
Lemma:
E
0 (X)
. is the
I f e v e r y Bequicontinuous s u b s e t i n V M t ( X ) t h e n each K
image of a f3equicontinuous s e t i n M t ( X ) ,
E
K ( @(XI 1
i s c o n t a i n e d i n t h e image of a compact s u b s e t o f X.
Proof: For such a K , l e t
B :=
16,
:
s
E
K). Then B i s c l e a r l y
uniformly t i g h t and s o f3equicontinuous. Hence it i s t h e image of a f3equicontinuous s u b s e t B1 of M t ( X ) . i n K ( X ) so t h a t and
v
E
1 ~ ( X1
\ K1) < 1/2
B, w i t h VV = 6 s ,
Then t h e r e i s a K 1
f o r each 1.1
E
B1.
If s
E
K
then
1 = 6 s ( I s H = V U ( { S l ) = ll(9l({sm
and so 0l ( { s ) ) $ X \ K1. Hence $(K,) 2 K.
To complete t h e proof of 3 . 9 , w e r e q u i r e t h e f o l l o w i n g s t a n d a r d r e s u l t on l o c a l l y convex s p a c e s :
3.15. Lemma: A continuous l i n e a r o p e r a t o r T from a l o c a l l y convex space E i n t o a l o c a l l y convex space F i s an open mapping i n t o i t s range i f and o n l y i f T ' ( F ' ) i s a ( E ' , E )  c l o s e d i n E ' and each equicontinuous s u b s e t of T ' ( F ' ) i s t h e image of an e q u i c o n t i n u o u s s u b s e t of F '
.
11. BOUNDED CONTINUOUS FUNCTIONS
110
Proof: See GROTHENDIECK [62].
Proof of 3.9: Only 3.9.3) remains to be proved. The necessity of the given condition follows from 3.13, 3.14 and 3.15. Sufficiency: first we note that the polar B of VMt(X) in Cm(Y) is the set
{x
Cm (Y) : x = 0 on
E
$ (X)).
Suppose that P
We show that P E VMt(X) and so VMt (X) = (VMt(X)loo
E
Bo.
is weakly
closed. There are compact sets Kn in X so that 11.11 (X \ Kn) I l/n. We show that
I 1.1 I (Kn \
$ (X)) = 0
11. 1 (x \ (
~
for each n so that
$(XI) ~ n+ o
which implies (by 3.12) that 11
E
VMt(X). If this were not the
case, there would be an n so that 6 :=
I!JI (Kn \
Choose m so that m > 2 / 6 .
$(XI) > 0. There is a continuous function x in
xd!J > 6/2. Kn We can extend x without increasing the norm to a function x in C(Kn) with
llxll
=
1, x = 0 on (Kn \ $(XI) and
C"(Y) which vanishes on $(X) (and so is in the polar of VMt(X)). Then
1 xdu
>
6/2

l/m > 0
which gives a contradiction.
A similar argument, applied to a uniformly tight set C in VMt(X) produces a uniformly tight set in Mt(X) whose image is C. Hence the sufficiency follows from 3.15.
11.3
DUALITY THEORY
111
As an application of the theory developed in this section, we give a functional analytic proof of PROHOROV's theorem
on the existence of projective limits of measures. We suppose that
{ @ B a: X
F Xa, a I 8 , a,@
o
E
A}
is
a projective spectrum of completely regular spaces and that X is a completely regular space with continuous mappings
6,
X+
:
Xa
so that
(a
$ B a $~B =
I
(thus the
8)
system { $ a } corresponds to a continuous mapping from X into the projective limit of the system {Xal). Suppose that : c1
{pa
E
A)
is a compatible system of bounded Radon measures
Mt(Xa) and M ( 4 ) ( p 1 = p a for a < B ) . We t Ba B seek necessary and sufficient conditions for the existence of
on {Xa) (i.e. 11,
a p
E
E
Mt(X) so that Mt($a) (11) = p a for each a .
3.16. Proposition: Such a ~.r exists if and only if 1)
SUP ~llPallMt(xa) : a ~ A ) <  i
2)
for each
E
lua] (X \
$,(K))
>
0 there exists a K
<
E
K(X) so that
for each a .
E
Proof: The necessity of condition 1 ) is trivial and that of 2) follows from 3.9.2). Now suppose that 1) and 2 ) are satisfied. If n Kn
b
K(X) so that B :=
{u
E
l u a l (X \ $,(Kn))
Mt(X) : IIu11
I
sup
111
E
B : Mt($a)
(u)
= !I,}
hl,
choose
< l/n. Let
1 1 ~ ~ 1 and 1 l u ( (X
Then B is weakly compact in Mt(X). Let Ba:=
E
\ Kn) < l/n)
11. BOUNDED CONTINUOUS FUNCTIONS
112
Then B, is weakly compact and nonempty by 3.9.2).
n
B,
@
Hence
by the finite intersection property.
Using 3.3 and the ideas of 2.10 sation of the dual of (C"(X) I f 3 g )

2.11, we can give a characteri
analogous to that of 3 . 4 . C for
Mt(X). We denote this dual by Mg (X) (so that Mt(X)
MB (X)1 .
3.17. Proposition: The space MB(X) can be naturally identified
with the space of Radon measures p on f3X which satisfy the following condition: for each
I PI (8X \ E)
<
E
E
>
0 there is a B
B so that
E
.
(B denotes the closure of B in f3X)
Proof: By applying 3.3 to (C"(uX) ,f3 ) we see that a B8continuous linear form, which can be regarded as a BKcontinuous linear form on C"(UX) for each
E
>
I
is defined by a Radon measure
0, there is a K
E
K(uX) with
u on BX so that
1 ~ (8X 1 \
K)
<
E.
Since
is BBcontinuous, one can show as in the proof of 3.3, that one can even take K to be of the form
(B
E
B(X)).
11.4. VECTORVALUED FUNCTIONS
In this section, we shall assume uniformly that all spaces X are Scomplete. We recall that this means that a function x from X into IR is continuous if its restrictions to the sets
of S are continuous. Then the same condition holds for functions
11.4
VECTORVALUED FUNCTIONS
113
with values in a completely regular space (since such a space has the weak topology defined by the continuous, realvalued functions on it) and so, in particular, for functions with values in a locally convex space. In addition, we shall assume that u S = X.
4.1.
Definition: Let (Ell[1 1 , ~ )
be a complete Saks space. We
denote by C"(X;E) the space of Tcontinuous mappings x from X into E which are normbounded on X. On Cm(X;E) , we consider the structures:
11
]IE 
the norm
x

sup {
11 x
t) 11 : t
E
XI
 the locally convex topology of uniform TconverS' E is a Saks space. gence on the sets of S. (c"(x:E),II I(E'rS) We denote by Bs the associated locally convex topology.
4.2.
Proposition: 1 )
E
T~
E BS C
T
11
IIE
2)
(C"(X:E) , B ~ )is complete;
3)
a subset B G C"(X;E) is BSbounded if and only if it
is normbounded; 4)
a sequence (xn) in c~(x;E)converges to zero for 8,
if
and only if it is normbounded and riconvergent to zero: 5)
a linear operator from c"(x;E) into a locally convex
space is BScontinuous if and only if its restriction to the E is TScontinuous; unit ball of (C" (X;E),11
l E)
6)

if E is a Banach space and X is SS normal, then BS is
defined by the seminorms Ip$ : $
P$
:
x
E
+
Ls 1 where
II$XIIE
11. BOUNDED CONTINUOUS FUNCTIONS
114
C"(X;E), as a Saks space, has a natural projective limit representation which we now display. Suppose that E is the Saks space projective limit of the sequence A
{
T
A
Ept P
: ~EqC~
Prq
CIr
E
PI
where P is a suitable family of seminorms which defines the topology
T.
Then we can order the set S
x
'"
and if (B,p) I ( B , , q ) we denote by
P in a natural way
PB,,B the mapping
x MTqp(XIB) from the Banach space C" (B
*E )
1' 9
{pqrP
B1 ,B
:
Cm(B
1
:6q )
into Col(B;6
__j
P
)
. Then
C"(B;i 1: p I q, B
P
B,)
is a projective system of Banach space and Cm(X;E) can be identified with its Saks space projective limit in a natural way.
Now if X is compact and E is a Banach space, there is a natural A
isometry from C(X) o E 11271,
if
onto C(X;E) (see, for example, SEMADENI
5 20.5.6) which is constructed as follows:
x = C xi o ai
(xi
E
C(X), ai
E
E) we define Jx to be the
function t
C
xi(t)ai
from X into C . Then J is an isometry from C(X)
8
E (with the
inductive tensor product norm) onto a dense subspace of C(X;E) and so extends to the desired isometry. We can now easily obtain the following generalisation of this result.
11.4
VECTORVALUED FUNCTIONS
115
4.3. Proposition: The Saks space Coo(X;E) is naturally isomorphic
to the tensor product CC" (XI
S
c
Y
E, 11
1 1 , ~ 2~
T)
provided that
K.
Proof: Consider the commutative diagram
where the vertical arrows are the natural isometries desribed above (we are using 1.3.12
to identify C"(X)
gY
E with the Saks
space projective limit of the system defined by the spaces EC(B) $ E 1 . The required isomorphism is the unique vertical P E which preserves the commutaarrow from C" (X;E) into C" (X) Y tivity
.
Now let X and XI be completely regular spaces with suitable saturated families S and S 1 where S S define a saturated family S
x
K(X)
S, on the product space X
consisting of those closed subsets of X in a set of the form B need not be S
x
x
and S1 C K(X1). We
B1 (B
E
S , B1
x E
XI,
x
XI which are contained
sl). Note that X
S1complete, in general. We denote its S
x
X,
x
S1
completion (as described in 1.7) by Y. If x is a mapping from X into Cm (XI) then we define the mapping
JX
to be

11. BOUNDED CONTINUOUS FUNCTIONS
116
ix(s)l ( s l )
(s,sl)
on X
x

XI. If we regard C"(X) and Cm(Xl) as Banach spaces (with J : x
the supremum norm), it is wellknown that an isometry (cf. SEMADENI [127],
is
not surjective but this (loc. cit.
5
7.7.5

5
5.3.4)

Jx
is
in general, it is
the case if X and X, are compact
in fact, it suffices to require that X I
be compact). The following result is a natural generalisation of this fact to noncompact spaces:
4.4. Proposition: J induced a Saks space isomorphism from
.
C" (x;c" (xl) 1 onto C" (Y)
Proof: Firstly, it follows from the result for compact spaces
x
E
C"(X;Cm(Xl))
the form B
x
B1 (B
that if
E
S, B1
then Jx is continuous on sets of E
S 1 ) and so Jx lies in C"(Y).
The
image of C" (X;C" (XI) ) is a selfadjoint subalgebra of Cm (Y), containing the constants, and so is
BSxS,
dense in C" (Y) by the
complex version of 1.14. Since C" (X;C" (XI) ) is a complete Saks space it is sufficient to verify that its image is a Sakssubspace of C"(Y) and this follows from the results mentioned above.
4.5. Corollary: If X and X I are locally compact then J induces an isomorphism from (C" (X;C" (XI)),BK)
Combining 4.3 and 4.4 we get
onto (C" (X x X,) , B K ) .
VECTORVALUED FUNCTIONS
11.4
117
4.6. Proposition: There is a natural Saks space isomorphism
from (c"(Y) ,II
In
9
IIl~sxsl
onto (c"(x)
gy c~(x, 1 ,II 1 1 , ~ 6~
T
~
~
)
.
2 we studied Cm(X) as an algebra. Cm(X;E) does not, in
general have the structure of an algebra but it is, in a natural way, a Cm (X)module. In the next Proposition, we characterise the closed submodules of Cm(X;E). Let M be such a submodule i.e. it is a vector subspace with the property that xM C M for each x
E
Cm(X). We then define, for s Ms
:=
{x(s) : x
E
E
XI closure in B )
MI
i.e. Ms is the 8closure of M ( s ) in E. Then N
M := {x
E
Cm(X;E) : x(s)
E
for each s
Ms
E
X 1
.
is obviously a Bclosed submodule of Cm (X;E) We show that N
M = M if (and only if) M is Bclosed.
4.7. Lemma: Suppose that X is compact and E is a Banach space.
Then if M is a (norm)closed submodule of C(X;E) , M
N
= M.
Proof: The result follows from the following formula which holds for any x
c~(x;E):
E
inf IIx YEM

yll
=
sup inf Ilx(s) SEX ZEM,

211
The lefthand side is clearly greater than the righthand side. Now choose
E
> 0. Then for
Ijx(s)

u(s)ll
s E
X there is a u
< sup inf Ilx(s) SEX ZEM,

E
z]I
so that
M
+
E
11. BOUNDED CONTINUOUS FUNCTIONS
118
This inequality holds on an open neighbourhood of s in X. Hence, by compactness, we can cover X by open sets {U,,...,U,)
...,un
to which there correspond functions u l , jlx(s)

.
.
ui(s) 11 I sup inf Ilx(s) SEX Z E M ~
on Ui (i = 1 ,.. ,n) Let (0, subordinate to {Ui} and put
,. . . ,I$,) y := C

in M so that 211
+
E
be a partition of unity Then a routine cal
culation shows that Ilx(s)

y(s)ll I sup inf Ilx(s) SEX Z E M ~

211
+
E
(s E
X)
so that the above equality holds as claimed.
4.8. Proposition: If X is a completely regular space, E a
complete Saks space and M a submodule of Cm(X;E), then M = if and only if M is Bclosed.
Proof: Suppose that M is @closed. We show that M = y E

E l we construct a net (yKla)indexed by K(X)
x
i. For
A
(where A
is the indexing set in a projective limit representation {IT@, : EB
Let
,a
Ea) of E) which is @convergent to y.
be the projection of M in C(K;Ea). Then MK
is a
la
C(K)submodule of C(K;Ea). Now it is clear that if s
E
K, then
a ( M ( s ) ) where ~l~ is the natural mapping from E into + Ea. Hence IT^ (y) E MKIa and so is approximable (uniformly on K
%,a ( s ) =
71
la
with respect to the norm of Ea) by a function in M (by 4.7). By a standard argument, we can ensure that these approximations are uniformly bounded.
We remark that 4.8 contains Proposition 2.7 as a special case.
11.5
GENERALISED STRICT TOPOLOGIES
119
11.5. GENERALISED STRICT TOPOLOGIES
In this section, we consider a completely regular space X as a subspace of its StoneEech compactification BX. K (BX \ X) denotes the collection of compact subsets of BX \ X.
If K
E
K (BX \ X) then BX \ K is a locally compact space con
taining X. The vector spaces C"(X) and Cm(f3X \ K) are naturally isomorphic (under the restriction mapping from Cm(BX \ K) onto Cm(X)). We denote by B K the strict topology on C"(f3X \ K ) and m
we regard it as a locally convex topology on C (X). For reasons which will become clear later, we denote the topology B K on C"(X) by B t in this section. Then B t is coarser than each 6,.
Now suppose that L is a subfamily of K (BX \ X)
. For convenience,
we shall assume that L is closed under the formation of finite unions. If K E K, then B K is finer than 6 : K
.
Hence the family K1 L ) forms a directed system of locally convex topologies
E
on Cm (XI. We can then define the locally convex topology B L
on C"(X) to be the locally convex inductive limit of these topologies i.e. the finest locally convex topology on C" (X) which is coarser than each $ K (K E L). B L is then finer than Bt. The following choices of L will be especially important: a)
L
b)
La

=

then B L is the uniform topology;
the family of sets in K(BX \ X) which are zero
sets of functions in C ( g X ) . Note that K
E
K(BX \ X) is in L U
if and only if BX \ K is acompact. For if K = 2 (x) (x
E
C(BX) 1
11. BOUNDED CONTINUOUS FUNCTIONS
120
then
and each of these sets is compact. On the other hand, if gX \ K is expressible as a union
where
of compact sets nUEN Kn for each n, then a standard construction
Kn E K Z + l
produces a continuous function x on BX \ K so that 1x1 S l/n on (gX \ K) \ Kn and the extension of x to BX has K as zero set. The topology on C"(X) defined by L, will be denoted by 8,; c)
1,

.the family of all compact subsets of BX \ X.
The corresponding topology on C"(X) will be denoted by 8 , .
We note first that each of the topologies B L is a mixed topology. Of course, the unit ball of C"(X) is closed in each B L (since B1 is finer than the topology of pointwise convergence on X).
5.1.
Proposition:
BL
= y[ll
ll,Bll.
Proof: It suffices to show that any linear operator from Cm(X) into a locally convex space F is B1continuous when it is B L continuous on B
. But then it is f3Kcontinuous on B II
II, and so BKcontinuous. Continuity then follows from the universal II Il"
property of inductive limits.
5.2. Proposition: a)
b)
8,
= 8,
8, C 8,
8,;
if X is locally compact.
11.5
GENERALISED STRICT TOPOLOGIES
12 1
Proof: a) is trivial. b) follows from the fact that X is locally compact if and only if it is open in BX i.e. if and only if $ X \ X
E
LT.
One of the most interesting and important.problems regarding strict topologies is that of characterising in purely topological terms those spaces for which given strict topologies coincide. One simple method of obtaining necessary conditions is to characterise the spectrum of (C" (X), B L )
.
5.3. Proposition: The spectrum of (C"(X) ,@L) can be identified
via the generalised Dirac transform with
II KE
Proof: s
E
L
BX \ K.
element of the spectrum has the form 6 , for some
An
BX. Now 6 s is in the BLspectrum if and only if it is 8,
continuous for each K each K
E
L
E
L i.e. if and only if s
E
BX \ K for
(2.2).
5.4. Corollary: a)
The spectrum of (C"(X) ,B,)
is uX, the real
compactification of X; b)
the spectrum of (C"(X) ,B,)
c)
if B,
= B,,
is X;
then X is realcompact.
Proof: a) follows immediately from 5.3 and the definition of uX. b) is trivial and c) follows from a) and b)
.
11. BOUNDED CONTINUOUS FUNCTIONS
122 5.5.
Proposition: The space (Cm(X),B,)
is a Mackey space for
any X.
Proof: By 1 . 2 0 ,
(Cm(X),BK) is Mackey for each K
E
L 0 and the
inductive limit of Mackey spaces is Mackey.
5.6.
Proposition: For any X, the following are equivalent: a)
X is compact;
b)
(Cm(X), B T ) is barrelled (bornological, metrisable or
normable).
Proof: By 1 . 1 . 1 5 ,
any of the conditions of b) is equivalent to
the condition that B, K
E
= T~~
ILi.e.
that
BK
=
T
It Ilm
for each
LT. Clearly, this can only happen if L T = { $ I i.e. X
=
BX.
Recall that a completely regular space X is pseudocompact if Cm(X) (i.e. if X is bounded in itself).
C(X)
=
5.7.
Proposition: For any X, the following conditions are
equivalent: a)
X is pseudocompact;
b)
8, is barrelled (bornological, metrisable, normable)
Proof: Using 
the simple remark that X is pseudocompact if and
o n l y if UX
BX, one can prove this as in 5.6.
=
.
11.5
GENERALISED STRICT TOPOLOGIES
123
We now define, using a rather different method, a fourth on C"(X). We denote by V the family of all
strict topology B,
continuous pseudometrics on X. If d
E
V then we denote by Xd
the associated metric space (i.e. the quotient of X with respect to the equivalence relation x
y
if and only if d(x,y) = 0).
Then there are natural projections
d
x
'
x
7T
d
:
and, dualising these, we obtain injections Crn(7Td) : Crn(Xd)
___, C"(X)
(i.e. Cm(nd) identifies C"(Xd) with the subspace of those
IT^) .
functions in Cm (X) which factorise over
Since Xd is a completely regular space (with the topology induced by the metric) , we can supply Cm(Xd) with the strict topology B,
and so are in a position to topologise C"(X) by defining B,
to be the locally convex inductive limit of these topologies.
5.8. Proposition: a)
b)
8, c 8,;
8, is a mixed topology 1.e. B,
=
Y[
11
11,8,1


Proof: It is easy to show that Cm(nd) is 8,continuous
crn(xd)into c"(x)
(since
This proves a). b) cf. the proof of 5 . 1 .
'rrd
:
x
xd
from
is continuous).
11. BOUNDED CONTINUOUS FUNCTIONS
124 5.9.
Remark: I. WHEELER [ I 5 2 1 has shown that 11.
The topology 8, is, in fact, a
B,,
8,
5
8,.
although for a
rather awkard choice of L . Let (xci)aEAbe a net in Cm(X) which is uniformly bounded, equicontinuous and decreases to zero. If K := { s
E
gX : xci(s)1
E
> 0 put
for all
E
ci E
A)
Then if we define L to be the set of all K which can be described in this manner, 8, 111.
=
BL.
The above construction is rather artificial in the
context of completely regular spaces and depends more on the fine uniform structure of X (i.e. the finest uniform structure compatible with the topology or, alternatively, the projective uniform structure induced by the (vd))
than on its topology.
In fact, we shall see in the Appendix that the space (Cm(X),8,) arises more naturally in the context of a duality theory for uniform spaces.
In the light of this remark, it is not surprising that the spectrum of the algebra (C"(X) ,Bm)
is the topological completion
of X (i.e. the completion of X with respect to the fine uniformity). This will also be proved in the Appendix as a special case of a result on uniform spaces.
The significance of the topologies f3,
and 8, on Cm(X) is that
their dual spaces are identifiable with important spaces of measures on X

the aadditive Baire measures and the .radditive
11.5
GENERALISED STRICT TOPOLOGIES
125
Borel measures respectively. We recall briefly the details. If X is a completely regular space, we denote by

Ba(X)
the family
Baire sets in X i.e. the ualgebra
gf
generated by the zero sets in X;

Bo(X)
the family of Borel sets i.e. the ualgebra generated by the closed
sets.
Mf(X) denotes the space of finite, finitely additive, complexvalued Baire measures on X; Mu(X) denotes the space of finite, aadditive, complexvalued Baire measures on X; M (X) denotes the space of finite, Tadditive, complexvalued 'I
Borel measures on X.
The basis for the identification between measures as linear forms, resp. as set functions, is the following classical result :
5.10. Proposition: a)
Integration establishes an orderpre
serving vector isomorphism from Mf(X) onto the dual space of (C" (XII
II 11)
b)
;

under this bijection, M,(X)
corresponds to the set
of 0additive functionals on C" (XI i.e. those f which satisfy the condition: f(xn)
0 for each sequence (x,)
in Cm(X)
which is uniformly bounded and decreases to zero; c)
under this bijection, MT(X) corresponds to the set
of 'Iadditive functionals on Cm(X) i.e. those f which satisfy
11. BOUNDED CONTINUOUS FUNCTIONS
126
the condition: f(xJ
0
for each net (xc(laEAin
c"(x)
which is uniformly bounded and decreases to zero.
For details, we refer to VARADARAJAN [ 1 4 4 1 .
Note that if p
E
Mf(X) , then p induces a norm continuous linear
functional on C(f3X) (= C"(X)) and so can be regarded as a Baire measure on gX which we denote by
c;
Bore1 measure on gX which we denote by
v;
We denote the inner measures associated to
1111 and
by
,
Iyln resp. Hence, for A S RX, ( F ( n ( A ) = sup{
(K) : K
c
A, K a zero set 1
The following result characterises u  and Tadditivity in terms of the measure of the remainder space f3X \ X with respect to these measures.
5.11.
Proposition: For a measure p E Mf(X), the following are
equivalent: a) b) c)
u E Mu(X); 1L1,k (BX \ X) = 0; 1111 (K) = 0 for each
zero set in gX \ X.
Proof: The equivalence of b) and c) follows immediately from the definition of
I VI *.
11.5
GENERALISED STRICT TOPOLOGIES
127
a) j c) : let K be a zeroset in f3X \ X. One can construct a sequence (x,)
xK,
creases to
in C(f3X) which is uniformly bounded and dethe characteristic function of K. Then (xn)
decreases to zero on X and so, by aadditivity, c) _d a) : let (x,)
(K) = 0.
be a uniformly bounded sequence in C" (X)
which decreases to zero on X. It is no loss of generality to suppose that 0
5
x
5
Kn := { s Then (K,)
1 . Choose E
gx
:
0 and put
? E l
is a sequence of zero sets in @X and nKn is a zero
from aadditivity that X
and so
X Mu (X).
xn dlul
1111 measure. It follows then
1111 (K,) +

j xn dll.rl
E
>
xn(s)
set in f3X \ X and so has zero
1.1
E
=
j
f3X
xn dl111
0
i.e.
5
0.
But
lul (Kn) +
1 ~ E1
Mu(X)
E I C ~(BX) and so
The next result is proved similary:
5.12.
Proposition: For a measure p
E
Mf(X), the following are
equivalent: a)
11
b)
ljlfi
c)
171 (K) =
E
MT(X): (6X \ X )
= 0;
0. for each K
E
K(f3X \ X)
.
Using this result, we can now identify the duals of C"(X) under f3,
and 8 , .
Firstly we characterise the dual of (C" (XI ,BK)
11. BOUNDED CONTINUOUS FUNCTIONS
128 (K
E
K(BX \ XI). NOW BX \ K is an open (and hence locally
compact) subset of BX and it is classical that the Radon measures on BX \ K coincide with the Radon measures on BX which have support in BX \ K (see, for example, BOURBAKI [ 1 4 1 ) .
5.13.
Lemma: Let K
E
K(BX \ X)
. Then the dual of
is naturally identifiable with the set of those which satisfy the condition
5.14.
u
E
Mf(X)
(K) = 0.
Proposition: The dual of (C" (X),B L ) is naturally identi
fiable with the space of those for each K
E
u
E
Mf (X) for which
(K) = 0
L.
Combining 5 . 1 1 , 5 . 1 2 ,
5.15.
(C"(X) ,BK)
and 5 . 1 4 , we get:
Proposition: a) The dual of (C"(X) ,B,)
is naturally
identifiable with Mu (X); b) the dual of (C"(X) ,BT) is naturally identifiable with MT(X).
We now consider some results which establish a relationship between the topological properties of a space and measuretheoretical properties. As a simple example of such a result note that X is pseudocompact if and only if Mf(X) since B,
M,(X).
For
is the Mackey topology ( 5 . 5 ) , the latter equality holds
exactly when (5.7).
=
8, =
'c
II I ,
and this means that X is pseudocompact
11.5
129
GENERALISED STRICT TOPOLOGIES
In the following, we shall concentrate on three measure theoretical properties of X

the Prohorov condition, measure
compactness and strong measure compactness.
5.16. Definition: A completely regular space X is said to be
+
a Tspace if every weakly compact subset of Mt(X) is Btequicontinuous (the superscript
+ denotes the set of nonnegative
measures) (alternative names

a Prohorov space or space which
satisfies Prohorov's condition). The following result is classical (see BOURBAKI
[
141 ,
5
IX.5.5, Th. 2) :
5.17. Proposition: If X is locally compact, then X is a Tspace.
5.18. Corollary: For any suitable family 1 , a weakly compact m
subset of nonnegative measures in (C (X),Bl)
is B1equicon
+
tinuous. In particular, a weakly compact subset of Mu(X) (resp.
+
MT (XI) is Buequicontinuous (resp. B,equicontinuous)
5.19.
.
Proposition: Let X,Y be completely regular spaces and
suppose that there exists a continuous mapping with the property that if K
E
K (Y), then
I$'
I$ : X 4 Y
(K)
E
K (XI.
Then if Y is a Tspace, so is X.
+
Proof: Suppose that B EMt(X) is weakly compact. Then so is
+
C"(I$) (B) (C_Mt(Y)) and so, for Cm(r$)
E
> 0, there is a K
( P I (Y \ K) <
E
E
K(Y) with
11. BOUNDED CONTINUOUS FUNCTIONS
130
for each l~
E
B. But then l~ (X \ @' (K)) <
f o r each p and
E
so B is equicontinuous.
5.20. Corollary: A closed subspace of a Tspace is a Tspace.
5.21. Proposition: Let (Xn) be a sequence of Tspaces.
Then
X :=
11
n~ N
Xn
is a Tspace.
+
Proof: Let B be a weakly compact subset of Mt(X). We can suppose that B lies in the unit ball of Mt(X). If n
E
N , there is a Kn
E
E
>
0,
(11
E
B)
then for each
K(Xn) so that
Cm(~n)(!J) (Xn \ Kn) < €/2"
(rn denotes the natural projection from X onto Xn). Then if K :=
11 Kn, a simple calculation shows that u(X \ K) < naM
for each u
E
E
B.
5.22. Corollary: a) Let (Xn) be a sequence of Tsubspaces of
a space X. Then n X n is a Tspace; b) a G6subspace of a compact space is a Tspace; c) a complete metrisable space is a Tspace.
Proof: a) follows from the simple fact that n X n is homeomorphic to a closed subset of 11 Xn (via the diagonal mapping). We can then apply 5.20 and 5.21. b) follows from a) and the fact that an open subset of a compact
11.5
131
GENERALISED STRICT TOPOLOGIES
space, being locally compact, is a Tspace (5.17). c) follows from b) and the fact that a metrisable space possesses a compatible, complete metric if and only if it is a G6subset of its Stoneeech cornpactification (see, for example, COMFORT and NEGREPONTIS [351, Theorem 3.6).
5.23. Remark: We mention without proof some further results on Tspaces. An example of a nonTspace (which is even Lusinian i.e. possesses a finer separable, complete metrisable topology) is an infinite dimensional separable Hilbert space, provided with the weak topology (this example is due to FERNIQUE

see
BADRIKIAN [21, Exp. 8). The proof of 5.22.a) shows that every
X which is a G6 in BX is a Tspace. Such spaces are said to be topologically complete (or complete in the sense of tech). PREISS El151 has shown that a separable, coanalytic metric space is a Tspace if and only if it is topologically complete. In particular, Q is not a Tspace (we remark that in 11141, VARADARAJAN had claimed to prove that a metric space is always a Tspace). MOSIMAN and WHEELER [lo51 have shown that a space X is a Tspace provided a) it is the locallyfinite union of closed Tsubspaces or
b) it has an open cover consisting of Tspaces.
(see also HOFFMANNJBRGENSEN [ 801)
.
5.24. Definition: A completely regular space X is said to be measure compact if
M,(X)
= MT(X).
132
11. BOUNDED CONTINUOUS FUNCTIONS
5.25. Proposition: a) If X is measurecompact, then it is rea1 compact; b) X is measurecompact if and only if
8,  6,.
Proof: a) follows immediately from 5.4 and b) from the fact that f3,
is the Mackey topology.
5.26. Remark: The converse of 5.25.a) does not hold (despite several claims to the contrary which have been published). In fact, M O M
[lo13 has shown that the Sorgenfrey plane is a
realcompact space which is not measurecompact. He has also shown that paracompactness is not sufficient for measurecompact ness.
For the next result, we recall that a space X is Lindelof if every open cover of X has a countable subcover. This property can be characterised in terms of the embedding of X in f3X (cf. COMFORT and NEGREPONTIS [351, Th. 2.3).
5.27. Proposition: X is Lindelof if and only if for every compact K
f3X \ X there is a zeroset Z of f3X so that
K E Z C B X \ X .
Proof: Suppose that X is Lindelaf and K as above. For each s there is a continuous xs : B X
[0,11
E
X
which vanishes on
K and takes on the value 1 at s . We can find a sequence (sn) in X
so that
X E
U x1 (]1/2, 1 1 ) . Then n e w sn
x := C 2"x n sn
is
11.5
GENERALISED STRICT TOPOLOGIES
133
continuous on BX and its zeroset satisfies the required condition. On the other hand, if U is an open cover of X and if, for each U
E
U , U' is an open set in
U { U' : U
x
E
E
gx with U
=
U'
n
BX, then
contains a set of the form coz x ( 2X) for some
U)
C(t3X). Now cozx is acompact and so is covered by countably X is then covered by the corresponding U's.
many of the {U'}.
5.28. Proposition: If X is LindeliSf, then
Ba= B,
and so X
is measure compact.
Proof: This follows immediately from the fact that if X is LindeliSf then
{C"(BX \ K) : K
EC"(BX \ K) : K
E
E
La)
is cofinal in
L,}.
5.29. Example: We sketch briefly an example of a nonmeasure
compact space. Let X be the space of ordinals less than the first uncountable ordinal, provided with the order topology. It is known that x an so
E
E

C"(X)
is eventually constant i.e. there is
X so that x(s) = x(so) for s 2 s 0' Then the functional x
x(so)
is obviously not in MT(X). However, it is aadditive since no countable subset of X is cofinal (see GILLMAN and JERISON [551,
99
5.115.13
for the properties of
x
that we have used here).
11. BOUNDED CONTINUOUS FUNCTIONS
134 5.30.
Remark: We mention without proof the following stability
properties of measure compact spaces. A closed or Baire subspace of a measure compact space is measure compact. In general, finite products of measure compact spaces need not be measure compact but this does hold (even for countable products) when localcompactness is assumed (for details, see MORAN 11021, MOSIMAN and WHEELER [lo51 and K I R K [ 8 8 1 ) .
The next condition that we consider is the equality M (X) = Mt(X). We can characterise this property as follows: ‘I
5.31.
Proposition: MT(X) = Mt(X)
if and only if X is absolutely
Borel measurable in BX.
Proof: We recall that the second condition means that for any compact regular Borel measure p on gX, X is the union of a Borel set in BX and a pnegligible set. The proof follows from 5.12.
5.32. Corollary: If X is locally compact or complete metrisable,
then
MT (X) = Mt(X).
Proof: For then X is open (resp. a G6set) in BX and so is absolutely Borel measurable.
5.33.
Definition: A space is strongly measure compact if
Mcr(X)= Mt(X).
11.5 5.34.
135
GENERALISED STRICT TOPOLOGIES
P r o p o s i t i o n : A acompact s p a c e o r a P o l i s h s p a c e ( i . e .
s e p a r a b l e , complete m e t r i s a b l e s p a c e ) i s s t r o n g l y measure compact.
P r o o f : Such s p a c e s a r e c l e a r l y L i n d e l o f and a b s o l u t e l y Bore1 measurable i n BX so t h a t w e can apply 5.28 and 5.31.
5.35. Remark: KNOWLES [92], p . 149
h a s shown t h a t t h e s t a n d a r d
i s n o t s t r o n g l y measure
nonmeasurable s u b s e t of [0,1]
compact. Hence t h i s s p a c e , b e i n g s e p a r a b l e m e t r i s a h l e , i s an example of a measure compact s p a c e which i s n o t s t r o n g l y measure compact. MORAN [ 1 0 2 ] ,
Def. 4 . 1
and Prop. 4.4, h a s
g i v e n t h e f o l l o w i n g c h a r a c t e r i s a t i o n of s t r o n g l y measure compact s p a c e s : X i s s t r o n g l y measure compact i f and o n l y i f f o r e a c h p
+
i n M,(X),
0
there is a K
E
K ( X ) with
p(K)
>
0.
Using t h i s r e s u l t , one can show t h a t t h e c l a s s o f s t r o n g l y measure compact s p a c e s i s c l o s e d under t h e f o l l o w i n g operations: a)
countable products;
b)
Baire o r c l o s e d s u b s e t s ;
C)
countable i n t e r s e c t i o n .
5.36. D e f i n i t i o n : A c o m p l e t e l y r e g u l a r s p a c e X i s 6simple i f 6,
=
6,.
The n e x t r e s u l t follows d i r e c t l y from t h e D e f i n i t i o n ,
5.5 and 5.18.
11. BOUNDED CONTINUOUS FUNCTIONS
136 5.37.
Proposition: Let X be a regular space. Then the following
statements are equivalent: a)
X is 8simple;
b)
X is a strongly measure compact Tspace;
c)
X is strongly measure compact and @, is the Mackey
topology ; d)
every weakly compact subset of M,(X)
is Btequicon
tinuous; e)
+
every weakly compact subset of M,(X)
is Btequicon
tinuous.
5.38.
Corollary: If X is a Polish space or a acompact, locally
compact space, then X is @simple.
5.39.
Remark: The class of @simple spaces is closed under the
following operations (see MOSIMAN and WHEELER [ 1 0 5 1 , Th. 6.1): a)
closed subspaces;
b)
countable products and intersection.
11.6. NOTES
11.1.
The study of strict topologies on C" (X) was initiated
almost simultaneously by BUCK [ 2 6 1 and [ 2 7 1 , LE CAM [ 9 4 1 and MAEIK [ 9 8 ] .
Masik and Buck used weighted seminorms and Le Cam
used a mixed topology. The fact that the two approaches were
11.6
NOTES
137
identical was noted later by several authors (see e.g. COOPER [391, DORROH [451, FREMLIN et al. 1501 and WEBB 11471). Almost all of the results in this section are natural generalisations and strengthenings of results of BUCK [27]. For studies of strict topologies see COLLINS and DORROH [32], FREMLIN et al.
[sol, GILES
1541, HOFMANNJPRGENSEN 1791, VAN ROOIJ [1201 and
SENTILLES [132]. For StoneWeierstraB theorems for the strict topology, see BUCK [271, FREMLIN et al. 1501, GILES [541, GLICKSBERG [56], HAYDON [751, HOFMANNJgRGENSEN 1791, TODD [I401 and WELLS [lSO]. This can be regarded as a special case of the weighted approximation problem (see BIERSTEDT [61, MACHADO and PROLLA [971, NACHBIN [lo61 and NACHBIN, MACHADO and PROLLA 11071). Separability of the strict topology has been studied by GULICK and SCHMETS 1681, SCHMETS 11211 and SUMMERS [1391. In the proof
of 1.15 we have used an idea of WARNER [1461 (who considered the essentially identical problem of separability under the topology of compact convergence).
The question of when the strict topology is a Mackey topology was posed by BUCK and has motivated a great deal of research. The result given here is due to CONWAY 1361 although it had already been obtained by LE CAM for 0compact, locally compact spaces (modulo the identification of the strict topology with a mixed topology). The method given here is from COOPER [411. ZAFARANI [I591 has also simplified Conway's rather complicated proof.
138
11. BOUNDED CONTINUOUS FUNCTIONS
For results of the type mentioned in 1.23, see HOFMANNJBRGENSEN [79].
11.2.
The theory of 2.1

2.5 is an attempt to extend the
GelfandNaimark duality theory for compact spaces to completely regular spaces. Unfortunately, a perfect duality cannot be obtained in this context (see Appendix). The idea of a perfect algebra is borrowed from APOSTOL [l].
For detailed accounts of
the StoneEech compactification and other extensions of topological spaces, see GILLMAN and JERISON [58], WALKER [145] and WEIR [148]. For a functionalanalytic approach, see BUCHWALTER [211.
11.3.
One of the most important justifications for the study
of strict topologies is the fact that the dual space is the space of bounded Radon (or tight) measures. This fact was proven by BUCK [271 for locally compact spaces. For completely regular spaces, see FREMLIN et al. [501 , GILES [ 541 , GULICK 1661 , HIRSCHFELD [77], HOFMANNJgRGENSEN [79] and SENTILLES [1321. For a detailed treatment of Radon measures on completely regular spaces, see BOURBAKI [I41 and SCHWARTZ [125]. The results and proofs of 3.9

11.4.
3.16 are from FREMLIN et al. [501.
Strict topologies on spaces of vectorvalued continuous
functions were studied already by BUCK in [27]. See also FONTENOT [491 and WELLS [1501.
11.6
NOTES
139
The exponential law for noncompact spaces is discussed by SEMADENI [1271. See also NOBLE [log].
The f3dual of Cm(X;E) can be identified with a certain space of Elvalued measures on X. See KATSARAS [83]

1851. 4.7 is a
result of BUCK [281.
For a comprehensive study of very general locally convex spaces of vectorvalued continuous functions, see BIERSTEDT C61 and 171.
11.5.
For detailed studies of aadditive and Tadditive measures,
see VARADARAJAN [I441 and KNOWLES [92]. The topologies f3, f3,
and
were introduced in the form given here by SENTILLES [132].
Equivalent topologies were defined by FREMLIN et al. [SO] using order theoretical methods. The general topology f 3 L was studied by MOSIMAN [1041. Similar methods have been used by BIN2 to define convergence structures on C(X) (see [9] and [lo]). f3,
was introduced by WHEELER [1521. The corresponding dual space
has been studied in various contexts by DUDLEY 1461, LEGER and SOURY [95] and ROME [119]. The characterisation of 6,
as a
is due to MOSIMAN [104]. For 5.10 see VARADARAJAN [1441. 5.11 and 5.12 are due to KNOWLES. Measure compactness and strong measure
compactness have been studied by KIRK 1871, 1881 and [891, M O W
[loll,
11021, [lo31 and, from a functional analytic view point,
by ‘SENTILLES [132] and MOSIMAN and WHEELER [1051.
140
11. BOUNDED CONTINUOUS FUNCTIONS
We mention that the topics of 11.5 have been covered in the following survey articles: BUCHWALTER [23], COLLINS [31], GULICK [67] and HIRSCHFELD [ 7 8 ] .
A detailed study of topological measure theory can be found in TOPSgE 11421. Nonbounded measures are studied by SONDERMANN [135]. For functionalanalytic treatment of vectorvalued measures
see BUCCHIONI [ I 81 I DEBIEVE 1431
.
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K. NOUREDDINE, W. HABRE Topologies Pstrictes, S6minaire d'analyse fonctionelle, Fac. Sc. Univ. Libanaise, 19741975.
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J. GARAY DE PABLO Integracion en espacios topologicos, Zaragoza 22 (1967) 83124.
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D. POLLARD
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D. POLLARD, F. TOPSrdE A unified approach to Riesz type representation theorems, Studia Math. 54 (1975) 173190.
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.
The strict topology in a completely regular setting: relations to topological measure theory, Can. J. Math. 24 (1972) 873890. On the priority of algebras of continuous functions in weighted approximation, Symp. Math. XVII (1976) 169183.
Compact sets of tight measures, Studia Math. 56 (1976) 6367.
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D. PREISS
Metric spaces in which Prohorov's theorem is not valid, 2 . Wahrsch. verw. Geb. 27 (1973) 109116.
11161
J.B. PROLLA
Bishop's generalized StoneWeierstraB theorem for weighted spaces, Math. Ann. 191 (1971) 283289.
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J.B. PROLLA
Weighted approximation and slice products of modules of continuous functions, Ann. Sc. Norm. Sup. Pisa (3) 26 (1972) 563571.
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R. PUPIER
Methodes fonctorielles en topologie g6n6rale (Thdse Science MathGmatiques, Lyon, 1971).
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L'espace Mm(T) , Publ. D6p. Math. Lyon 91 (1972) 3760.
11201
A.C.M. VAN ROOIJ
Tight functionals and the strict topology, Kyungpook Math. J. 7 (1967) 4143.
[ 121 1
J. SCHMETS
Separability for seminorms on spaces of bounded continuous functions, Jour. Lond. Math. SOC. (2) 1 1 (1975) 245248.
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Espaces associds d un espace lingaire d seminormes: applications aux espaces de fonctions continues, Sem. Lidge, 19721973.
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Espaces de fonctions continues (Springer Lecture Notes 519, Berlin 1976).
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J. SCHMETS, J. ZAFARANI Topologie strict faible et mesures discrdtes, Bull. SOC. Roy. Sci. Lidge 43 (1974) 405418.
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L. 'SCHWZWTZ
Radon measures (Oxford, 1973).
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Z. SEMADENI
Inverse limits of compact spaces and direct limits of spaces of continuous functions, Studia Math. 31 (1968) 373382.
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Z. SEMADENI
Banach spaces of continuous functions, I (Warsaw, 1971).
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F.D. SENTILLES
Compactness and convergence in the space of measures, 111, J. Math. 13 (1969) 761768.
Compact and weakly compact operators on C(S)@, 111. J. Math. 13 (1969) 769776.
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The strict topology on bounded sets, Pac. J. Math. 34 (1970) 529540.
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Conditions for equality of the Mackey and strict topologies, Bull. Amer. Math. SOC. 76 (1970) 107112.
Bounded continuous functions on a completely regular space, Trans. Amer. Math. SOC.
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F.D. SENTILLES, R.F. WHEELER Additivity of functionals and the strict tolology (Preprint). Linear functionals and partitions of unity in cb(x), Duke J. Math. 41 (1974) 483496.
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D. SONDERMANN
Masse auf lokalbeschrhkten Rzumen, Ann. Inst. Math. Fourier 19 (1970) 33113.
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P.D. STRATIGOS
Relative compactness in the vague topology, Boll. Un. Mat. Ital. (4) 8 (1973) 198202.
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W.H. SUMMERS
Weighted spaces and weighted approximation, Sherbrooke, Sgminaire d'analyse moderne 1970.
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The general complex bounded case of the strict weighted approximation problem, Math. Ann. 192 (1971) 9098.
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W.H. SUMMERS
Separability in the strict and substrict topology, Proc. Amer. Math. SOC. 35 (1970) 507514.
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C. TODD
StoneWeierstraR theorems for the strict topologies, Proc. Amer. Math. SOC. 16 (1965) 654659.
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Weak and norm sequential convergence in M(S), J. Aust. Math. SOC. 15 (1973) 16.
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Topology and measure (Berlin, 1970). Compactness and tightness in a space of measures with the topology of weak convergence, Math. Scand. 34 (1974) 187210.
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Measures on topological spaces , Amer. Math. S O C . Transl. (2) 48 (1965) 161228.
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R.C. WALKER
The Stoneeech compactification (Berlin, 1974).
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S. WARNER
The topology of compact convergence on continuous function spaces, Duke Math. J. 25 (1958) 265282.
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J.H. WEBB
The strict topology is mixed.
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M.D. WEIR
HewittNachbin spaces (Amsterdam, 1975).
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B.W. WELLS Jr.
Weak compactness of measures, Proc. Amer. Math. SOC. 20 (1969) 124130.
[I501
J. WELLS
Bounded continuous vector valued functions on a locally compact space, Mich. Math. J. 12 (1965) 119126.
Cl5ll
R.F.
WHEELER
The equicontinuous weak 9: topology and semireflexivity, Studia Math. 41 (1972) 243256.
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11. BOUNDED CONTINUOUS FUNCTIONS R.F. WHEELER
The strict topology, separable measures, and paracompactness, Pac. J. Math. 47 (1973) 287302. The strict topology for Pspaces, Proc. Amer. Math. SOC. 41 (1973) 466472.
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A locally compact non paracompact space for which the strict topology is Mackey, Proc. Amer. Math. SOC. 51 (1975) 8690.
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Wellbehaved and totally bounded approximate identities for C o ( X ) , Pac. J. Math. 65 (1974) 261269.
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The Mackey problem for the compact open topology, Trans. Amer. Math. SOC. 222 (1976) 255265.
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S. WILLARD
General topology (Addison Wesley , 1970)
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R.G. WOODS
Topological extension properties, Trans. Amer. Math. SOC. 210 (1975) 365385.
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A sufficient condition for the strict topology to be Mackey, Bull. SOC. Roy. Sc. Liege 44 (1975) 569571.
CHAPTER I11

SPACES OF BOUNDED, MEASURABLE FUNCTIONS
Introduction: In this chapter, we consider the LODspace associated with a positive Radon measure on a locally compact space. We define two supplementary topologies which give it the structure of a Saks space. In 1.1

1.5 we discuss the basic properties
of the corresponding mixed topologies f3 U and 8 , . In 1.6

1.8 we
show that both mixed topologies are topologies of the dual pair (Lw,L1
and that 8, is, in fact, the Mackey topology. The latter
result is equivalent to the DUNFORDPETTIS theorem (on relatively weakly compact subsets of L') but this formulation sheds new light on the result and the theory of Chapter I allows us to deduce some important consequences. In 1.18 we use it to prove that C(K) has the DunfordPettis property (following GROTHENDIECK). The assumption that p is a Radon measure is actually unnecessary and we close the section with some remarks on how the results can be extended to LODspaces associated with abstract measures.
In the second section we consider 8,continuous linear operators on Lm and show that they are induced by vectorvalued measures.
In section 3 , we consider the theory of measurable functions with values in a Saks space. In order to cover this theme properly, it is necessary to use much heavier machinery than we are prepared to use in this book so that this section is no more than a sketch of a possible theory. However, several facts indicate that the development of such a theory would be worthwhile. For
155
111.
156
BOUNDED MEASURABLE FUNCTIONS
example, it would allow a synthesis of various concepts of vector valued integration (Bochner integral, Pettis and Gelfand integrals and the lower star integral spaces introduced by L. SCHWARTZ). We end the Chapter with some remarks on the RadonNikodym property for Banach spaces.
111.1. THE MIXED TOPOLOGIES
In this Chapter we study spaces of bounded measurable functions on a measure space. We begin with some remarks on the simplest case

that of a discrete measure space i.e. the space l"(S)
for some indexing set S. We consider this as a Saks space with the supremum norm and the topology
T
P
of pointwise convergence
(see I.2.B). Note that
where we regard S as a topological space with the discrete topology. We also have
where each E
is the canonical onedimensional Saks space. We
now list some properties of t"(S) and the associated mixed topology
@
which follows immediately from the theory of the earlier
chapters: 1)
(l"(S) ,6 ) is a complete, semiMonte1 locally convex algebra
in which multiplication is jointly continuous (11.2.1)
111.1 2)
THE MIXED TOPOLOGIES
157
the dual of (Lm(S), B ) is canonically identifiable with
l 1 (S) and
B
=
~ ~ ( L " , l l=) T(L~,L') in particular, (L"(S),B)
is a Mackey space (1.4.13 and 1.4.14); 3)
(em( S ) ,B )
has the BanachSteinhaus property, is Brcom
plete and satisfies a closed graph theorem with range space a separable Frhchet space (1.4.16); 4)
if S is countable, the unit ball of L m ( S ) is 8metrisable
and so a subset A of l"(S) is 8closed if and only if it is sequentially closed; 5)
is separable if and only if
(k!"(S) , B )
Card S)
I Card
(11.1.17).
Now let M be a locally compact space, p a positive Radon measure on M. Lp(M;p) (1
5
p
I m)
denotes the space of measurable, com
plex functions x and M so that l l x l b :=
I
M
Ix(t) Ipdp}"p
(1
I I x I h := inf Ml > 0 : Ix(t) I
I,
p <
m)
s M, locally palmost everywhere)
respectively is finite where as usual functions x and y are identified if they agree locally palmost everywhere (i.e. if, for each compact K G MI p(It E K
:
x(t)
+ y(t)})
= 0). Then
(LpIII Ib) is a Banach space for each p. (Lm,L1 ) is a dual pair under the bilinear form
111.
158
BOUNDED MEASURABLE FUNCTIONS
and under this duality, Lm is identifiable with the dual of the Banach space (L1 , 11 topology
Ill).
u := u(Lm,L1
Hence we can define on Lm the
and
(Lrn,II
ILru)
is a Saks space. We denote by Bu the mixed topology y [ l l l L , u ] . We resume its properties in the following Proposition:
1 . I . Proposition: 1)
8,
is a topology of the dual pair (Lm,L1 )
and is, in fact, the topology T,(L m ,L1 ) of uniform convergence on the compact subsets of (L1
,/I I l l )
;
2) 6, has the same convergent sequences and compact subsets as
u (L",L~ 1; m
is a complete, semiMonte1 space;
3)
(L ,Bu)
4)
if M is ucompact and metrisable, then B
II Ib,
is Bumetrisab$e
and so a linear operator from Lm into a locally convex space F is B,continuous if and only if it is sequentially continuous.
Proof: We remark only that 4 ) follows from the fact that under these conditions, (L1,)I
)I1)
is separable.
We shall now introduce a more interesting mixed topology on Lm. We require two results on measurable functions which we now state. First recall the following definition: a subset H of Lp(M;u) is pequiintegrable (1 following conditions are satisfied:
S
p <
m)
if the
111.1 a)
for every
>
E
0 there is a 6
integrable subset of M with
1
A
b)
Ix(t)lP du I
for each
E
>
159
THE MIXED TOPOLOGIES
u(A)
I
E
> 0 so that if
A
is an
then
6
for each x
E
H;
0 there is a compact subset K of M so that
We remark that if M is compact (so that Lm(M;~) C Lp(M;p)) then the unit ball of (Lm,ll
18)
is pequiintegrable for 1 I p <
m.
1.2. Proposition: If H is a pequiintegrable subset of Lp(M;p) then on H the uniformity induced by the norm
1 1 lb
coincides
with the uniformity of convergence in measure on compact sets.
Proof: If
E
>
0 is given, choose 6
>
0 and K as in a) and b).
We show that if x,y in H are such that Ix(t) for t
E

y(t) I I E'/P(~(K))
K \ A where p ( A ) I
/P
6, then IlxyIE I (2p+1 + 1 ) ~
For we can estimate as follows:
Similarly
1
111.
160
BOUNDED MEASURABLE FUNCTIONS
Hence
1.3. Corollary: If M is compact, then the uniformity induced On
B I I Ii
the unit ball of ( L m , l l
la),
by the norm
11
IIp
(1
p < m)
I
coincides with that of convergence in measure.
1.4.
Proposition: Let
Ex, : K
E
K(M))
be a family where xK is
a measurable, complexvalued function on K so that if K E K 1 = xK almost everywhere on K. Then there is a measurable 1 function x on M so that x = xK almost everywhere on K for each
then
xK
K E K(M).
Proof: Consider the family of pairs (x,U) where U is an open subset of M and x is a measurable function on U so that
xK almost everywhere on K ( K E U ) . We order this family by putting (X,U) if U
C
x
=
s (y,V)
V and y is an extension of x. Its is clear that this
ordering is inductive and so there exists a maximal element (x,W) by Zorn's Lemma. Then W = M by maximality and so x is the required function.
We now consider on L m ( M ; p ) the locally convex topology
(1
5
p
<
m ) defined by the seminorms
~y~~
THE MIXED TOPOLOGIES
111.1
1 1 1:
:
x
( (
K
161
Ix(t)lP dU)l’p
as K runs through K ( M ) . Then (Lm,[I[L,Tyoc) is a complete Saks space. To prove this, we consider first the case where M is compact (so that If (xn) is a
.yOc
is just the norm topology induces by
11 11PCauchy
sequence in B
then it has an Lp
II II,’
limit x. We can extract a subsequence of (x,)
which converges
pointwise Ualmost everywhere to x (cf. BOURBAKI [ 2 ] , Thborsme 3 ) and so x
E
B
1 1 lb).
IV.3.4.,
II II’
Now in the general case (i.e. where M is not necessarily compact). {Lm(K) : K
E
K(M)} forms a projective system of Saks spaces
(under the restriction mappings) and, by 1 . 4 , Saks space projective limit
L m ( M ; ~ )is its
S  E m { Lm(K)} (for an element of
the projective limit is just a thread {x,}
of the type described
in the Proposition and so defines an element of Lm(M)1 . Hence we can define the mixed topologies
NOW by 1.3, ryOc =
* loc
On
BII
II,
and so these topologies coincide.
We shall denote this topology by B1. The following result follows directly from the theory of Chapter I.
I .5. Proposition: 1 ) 2)
a sequence (x,)
if it is
11
(L~,B,)is complete; in Lm is B1convergent to zero if and only
ILbounded and convergent to zero in some
even in measure on compact sets;
~y~~
or
111.
162
BOUNDED MEASURABLE FUNCTIONS m
a linear operator from L
3)
into a locally convex space F
is B1continuous if and only if its restriction to B
TP loccontinuous for some p
E
II Ib,
is
[I,[;
metrisable and so a is linear operator from Lm into F is B1continuous if and only if if M is acompact, then B
4)
II IL
it is sequentially continuous.
1.6. Proposition: The dual of (LOD, 6,) is identifiable with (L1 ,11 11,) under the bilinear form
Proof: Since (Lm,r~oc)is a dense subspace of the locally convex projective limit of the spaces {L1 (K) : K
E
K(M)}
(i.e. the
space of locally integrable functions), we can identify its dual with the subspace of L 1 (M) consisting of those functions in Lm which have compact support. The closure of the latter space in the L1norm is clearly L1 and this is the required dual space by 1.1 18.(ii).
We now come to the main result of this section, namely, that m
(L ,B1)
is a Mackey space.
1.7. Lemma: Let H be a subset of L 1 (M;p) that is not lequiintegrable. Then there exists an a > 0, a sequence (xn) in H and a disjoint sequence (A,)
I( An
xn dpl
1
a
for each n.
of measurable sets so that
THE MIXED TOPOLOGIES
111.1
Proof: Firstly, we note that if x
L1 and
E
is a 6 > 0 so that if p(A) < 6 then Mn
:=
{t
E
M : Ix(t)l 1 nl
exists an n E PI so that 1 Let 6 := ~(2x1)
.
IA Ixldp 5
then flMn =
theorem on dominated convergence,
I
E
d,
lxldp
Mn
163
> 0 then there E.
For if
and soI by Lebesgue's
+
0. i.e. there
Ixldp 2 ~ / 2 . Mn Then if p(A) < 6 we have
We shall suppose that condition a) in the definition of equiintegrability is not verified (the proof for the case where b) is violated is simpler). Then there is a E > 0 so that for each 6 > 0 there is a measurable set A and an x
E
H with P(A) < 6
Ixldp 2 E . We construct inductively a sequence (B,) of A measurable sets and a sequence (x,) in H as follows: choose x1 and
I
and B1 so that
IxlIdp 2 E. Now suppose that
B1
B1l...lBn are chosen. There is 6, and p (A) < 6,
>
x1 I
. .
.,xn
0 so that if A is measurable
we have
Then we can choose Bn+l and x ~ so + that ~ p(Bn+,) < 6n
I
IXn+1ldP 2 Bn Now if we put
and
and
E.
'I
Bkt one can verify that the An are k>n Ixnldp 1 ~ / 2 .The result then follows An by a standard argument (reduce to the realvalued case and then
An := Bn disjoint. Also we have:
consider positive and negative parts).
111.
164
BOUNDED MEASURABLE FUNCTIONS
1.8. Proposition: Let H be a subset of L 1 (M:p). Then the following conditions are equivalent: 1)
H is B1equicontinuous:
2)
H is relatively o ( L ,Lm) ~ compact:
3)
H is normbounded and Iequiintegrable.
Proof: 1)
2)
follows from the ALAOGLUBOURBAKI theorem.
3 ) : it is clear that H is normbounded. If it were not
2)
1equiintegrable, then we could choose (xn) and (A,)
as in 1.7.
Now consider the mapping T
xdpInEN An which is continuous from L1 (M:p) into L 1 (N) Then the sequence (Tx,) is not relatively weakly compact in L 1 (N) by 1.4.14 :
x 
((
.
contradiction. 3 ) W 1 ) : let H be a normbounded, equiintegrable subset of I?.
We use the criterium of 1.1.22 to show that H is Blequicontinuous.
>
0 we choose K
For
E
x
H. Choose n
E
for each x
E
E
E
K (MI so that
H where
Then p ( K ' ) s N1nl :=
x

[ xi dp
M\K N so that p(A) I N1.n1 N1 := sup{[lxllI: x
K' := It
x2
(
E
K : Ix(t)l
and so, putting
x1
I~
/ 2for each
implies
A
lxldp
E
HI. Then if x
2
n}.
:=
X'X~\Kl
E
and
x l , we have x1
E
HI := {y
E
L1 : supp y C K
and
I l y l k I n}
s ~ / 2
H I put
THE MIXED TOPOLOGIES
111.1
Hence
H E EBII
1,
165
' equicontinuous. + HI andHI is T loc
Corollary: (Lm,gl)is a Mackey space.
1.9.
1.10.
Corollary: Let T be a continuous linear mapping from 1
(Lm,a(Lm,L ) ) into a locally convex space F with its weak topology o(F,F')
. Then T
is B1continuous and so transforms bounded
sequences in Lm which converge in mean on compact sets into strongly convergent sequences in F.
As an application of 1.10 consider a weakly summable mapping Q

from M into a locally convex space F. That is, for each f t
f (@(t) 1
is in L 1 (M;p). Then for each x
E
F',
Lm, the
E
integral
/
M

f (@(t))x(t)dp
exists and the mapping
f
of the algebraic dual (F')''
x
E
/
M
f ( @(t))x (t)dp
is an element
of F'. The mapping which takes
Lm into this form is then u(Lm,L 1 )u"F')''',F')
Suppose that the range of this mapping lies in F
continuous. (@
is then said
to be strictly weakly summable or Pettis integrable). Then it is glcontinuous by 1.10.
We now consider BanachSteinhaus theorems and closed graph theorems for Lm. We first show that (Lm,ll IL,T:~~) satisfies condition
C 1 of 1.4.9.
111.
166 1.11.
BOUNDED MEASURABLE FUNCTIONS
Lemma: The Saks space (Lml11
dition
lLI~iOc) satisfies con
C1.
Proof: It is convenient to consider realvalued functions. Suppose that xo be an element of B
B I I IL' ll II,
composition x = x 1 +x2 and
1 1 xox2 11
i
E.
E
> 0 and K
with 11x11:
E
K(M) are given. Let x We
5 E/2.
of x with x1,x2
E
BII
shall find a de
II, and
IIxoxlII
5 E
Denote by M, the set where xo and x have the
same sign. Then we define x1 by x. := I
and
l[ xX++ xo(l xo(l + x )
on M 1 on M \ M I
x 2 := x  x l .
Then a simple calculation shows that x1,x2 IIXoX1(I 5 E I
IIXoX211 5
BII II, and
E.
In the next results we assume, for simplicity, that M is acompact ( s o that B
II II,
is f31metrisable). In fact, the results hold with
out this assumption (see 1.14).
1.12. Proposition: If M is acompact, then (Lm,B1) has the Banach
Steinhaus property.
Proof: 1.11
and 1.4.11.
111.1
THE MIXED TOPOLOGIES
167
In particular, this implies the wellknown result that L 1 (M;p) is weakly sequentially complete.
1.13. Proposition: Let M be acompact. Then a linear mapping from (Lm,B1) into a separable Frgchet space is continuous if and only if it has a closed graph.
1.14. Remark: We sketch briefly the method of extending 1.12 (and hence 1.13) to general M. Since (Lm,B1) is a Mackey space, it suffices to show that it has the BanachSteinhaus property 1 for functionals. Now each functional is represented by an L

function and this has acompact support. Hence if (f,)
is a
pointwise convergent sequence of functionals on LO3, then there is a acompact subset Mo of M so that the corresponding L 1 functions vanish outside of Mo. We can then deduce that the pointwise limit is continuous by applying 1.12 to Lm(Mo;p ) . MO 1.15. Remark: We note that it follows from 1.4.33.11
that (Lm,B1)
is nuclear if and only if Lm is finite dimensional (i.e. if and only if p has finite support). If p is atomic (so that (Lm,B1) has the form (Sm(S),B) for some set S, then (Lm,B1) is semiMonte1 and even a Schwartz space. The converse is also true (as DAZORD and JOURLIN have remarked) since if p is not atomic then, by a result of GROTHENDIECK, L 1 possesses a weakly convergent sequence which is not convergent in the norm. (Sw,B) is even a universal Schwartz space in the sense that every Schwartz space is a subspace of a product of (L",B)space.
168
111.
BOUNDED MEASURABLE FUNCTIONS
1.16. Remark: It is perhaps interesting to point out that various ORLICZPETTIS theorems on unconditionally summable series (cf, THOMAS [29] ,
5
0) can be obtained from the closed
graph theorem (1.13). The point is that if E is a complete locally convex space, there is a bijective correspondence between the set of B1continuous linear mappings T from t " ( l N ) into E and the unconditionally summable sequences in E, obtained by mapping T into (Ten) where (en) is the standard basis in k?"(N) (see GROTHENDIECK, Ref. [62] to Chapter 11, p.97).
As far as we know, the topology
B1 was first studied (implicitly)
by GROTHENDIECK [12] in his investigation of the DUNFORDPETTIS property, in particular, to show that C(K) has this property. Since we now have sufficient machinery at our disposal to reproduce Grothendieck's proof we do so here. We recall the definition.
1.17. Definition: A locally convex space E has the strict DunfordPettis property (abbreviated to SDPP) if every a(E,E')Cauchy sequence is Cauchy for the topology of uniform convergence on the equicontinuous, ~~(E',E")compact discs on E' (this is equivalent to the fact that every weakly compact, continuous linear operator from E into a Banach space maps weakly Cauchy sequences into strongly Cauchy sequences). If E is a Banach space, this implies the DunfordPettis property for E (i.e. every weakly compact operator from E into a Banach space maps weakly compact sets into relatiuely compact sets).
111.1
THE MIXED TOPOLOGIES
169
1.18. Proposition: Let K be compact. Then the Banach space C(K) has the SDPP.
Proof: Let (xn) be a weak Cauchy sequence in C(K)
. Then
(x,)
is
uniformly bounded and pointwise Cauchy (cf. 11.1.21) and so has a bounded, pointwise limit x (which is then measurable for any Radon measure on K). We must show that
(
(K
xn dp) converges
uniformly on weakly compact sets of Radon measures. If this were not the case, we could find a relatively weakly compact sequence (pk) of Radon measures on which convergence is not uniform. By
a standard construction, there is a positive Radon measure p on K so that each pk is absolutely continuous with respect to p (take 11 := C 2kl pkl 1 . Then we can regard {pk) as a subset of k L 1 ( K ; p ) and it is relatively weakly compact there. Now by Egoroff's theorem, the sequence (x,) in Lm(K;p)) converges to x in
B1
(regarded as a sequence
and so uniformly on weakly com
pact subsets of L' ( K i p ) by 1.9. This gives a contradiction.
1.19. Remark: I.
Since a Banach space predua1,of a space with
SDPP has SDPP and the dual of L1 is a C ( K ) , it can be deduced from this result that every L1space has the SDPP. 11.
The same proof shows that ( C m ( S ) , B )
has the SDPP f o r any
locally compact space S.
1.20. Remark: We have chosen to develop the theory of Lm spaces in the context of Radon measures on locally compact spaces.
111.
170
BOUNDED MEASURABLE FUNCTIONS
In fact, the theory given here can be developed in the more general setting of abstract measure theory and we indicate briefly how this can be done. Let (M,C) be a pair where
C
is a
aalgebra on the set M. A positive regular measure is a uadditive function p from C into
IR'U
{m)
which satisfies the
(regularity) condition:
for each A
E
C.
Then we can develop as usual a theory of integration with respect to p for complexvalued functions on M and, in particular, we can define the spaces L 1 (M;p) and L 1 (M;p) of absolutely integrable functions respectively of equivalence classes of such functions. L1 (M;u) is a Banach space under the norm
1 1 Ill
: x
++
(MIxldp
. Note that the regularity of 1.1
can be
1
expressed in functional analytic terms as follows: L (M;u) is the Banach space inductive limit (cf. the introductory remarks to 5 1.3) of the spectrum {L' (A;vA) : A
E
Co 1
where Co denotes the family of sets of C with finite Umeasure.
For the time being, we suppose that LOD
u
is finite (i.e. p(M) <
m).
(M;p ) denotes the space of bounded , measurable complexvalued
functions on M and Lm(M;~)denotes the corresponding quotient space. Lm(M;~)is a Banach space under the usual norm
11 IL.
111.1
THE MIXED TOPOLOGIES

171
Then L1 and Lm are in duality with respect to the bilinear form
I
(x,y)
XY du
and this establishes Lm as the dual of L 1 a Saks space with the structure
. We
can regard Lm as
1 1 lLtII [ I I )
(Lm,
We now return to the general case (i.e. where M is not necessarily of finite measure). In order to preserve the duality between
L1 and Lm it is necessary to replace the classical Lmspace by a more complicated one. Recall that L ’(M;p) is the Banach space inductive limit of the spaces {L1 ( A ) ; A
E
Co}.
Hence its dual
is the Banach space projective limit of the Banach spaces {Lm(A); A
E
Co}.
We define the space Xm(M;p) to be the set of
threads x where xA
E
=
{x,
: A
E
Co)
Lm(A;pA) and
 X A ~for At E A and
Then Xm(M;p) is a Banach space and can be identified (naturally and isometrically) with the dual of L 1 (M;p). We note that Xm(M;p) coincides with the classical space Lm(M;p) in the following two special cases: 1)
where p is a Radon measure on a locally compact space
(this is the content of GODEMENT’S Lemma (1.4)): 2)
M is afinite.
172
111.
BOUNDED MEASURABLE FUNCTIONS
Now we can provide i?(M;p)
with a locally convex topology
T
1 CO
defined by the seminorms
cisely the Saks space projective limit of the system
I: (L"(A)I
I1 1L1 II I l l ) :
A
E
co).
We are thus in the position to define mixed topologies B,
:=
YCll
B 1 := Y [ l l
on
im and Propositions 1 . 1 ,
l~,o(~m,L1)l
lL+ 101 1.5,
1.6
,
1.8,
1.9,
1.10,
1.12
can be carried over, mutatis mutandis, to these topologies
1. I 3
In
particular, 8 , is the Mackey topology of the duality ( t m , L
111.2. LINEAR OPERATORS AND VECTOR MEASURES
In this section, we characterise the B1continuous linear mapping from Lm (M) into a Banach space. This characterisation displays the close connection between the mixed topology on La and vector measures and is one of the bases of applications of mixed topologies to such measures. In this section we use abstract measures rather than Radon measures (cf. the discussion in 1 . 2 0 )
since
the latter would involve an unnecessary restriction of generality and would complicate the presentation rather than simplify it.
111.2
LINEAR OPERATORS AND VECTOR MEASURES
173
We recall some definitions and simple results on Banach space valued measures.
2.1. Definition: Let ( M ; C ) be a measure space (i.e. C is a u
algebra of subsets of M). A measure on (M;C) with values in a Banach space E is a mapping
which is additive (i.e. such that
p(A
U
B) =p(A)
+
p(B)
for
disjoint A and B in C ) . p is bounded if (in addition) sup
aadditive if
1 Ilp(A) 11
p ( uAn) = C p (A,)
n each disjoint sequence (A,)
n in C .
:
A
E
Cl <
m;
(convergence in norm) for
Note that the boundedness of p is equivalent to the condition that IIpII ( M I <
m
where IIpII
,
the semivariation of p, is the
positive set function on C defined by
(If
0
variation of the (complexvalued) measure
f o p
If ( M ; C )
is as above, we denote by B ( M ; C )
the vector space of
bounded, Cmeasurable complexvalued functions on M. B(M;C) is a Banach space with respect to the supremum norm and the subspace of stepfunctions (i.e. the linear span of the characteristic functions of sets in C ) is dense. n If ( A k E C, Ak E C ) = k 1 A k xAk
is a stepfunction and p is
174
111.
BOUNDED MEASURABLE FUNCTIONS
an Evalued measure, then we can define (xdp := Then x
( x dp
n C Akp(Ak). k= 1
is a linear mapping from the space of
stepfunctions into E and we have the estimate:
Hence this mapping extends in a unique manner to a continuous linear mapping from B(M;C) into E and we continue to denote the action of this mapping by an integral sign. In fact, all normcontinuous linear mappings from B(M;C) have this form:
2.2.

Proposition: The mapping 1.1
(x
(x dv)
establishes a natural isometry between L(B(M;C) ;E) and Mf (C;E), the space of bounded, Evalued measures on (M;C) when the latter space is equipped with the norm
Proof: If T
E
p
cj
IIpII (MI.
L(B(M;C);E), we can define an Emeasure pT on M by
putting
for each A
E
C. The mapping
T
pT
is an inverse to the
mapping defined above. This establishes a vector space isomorphism between the spaces in the statement of the Proposition. That it is an isometry follows from standard estimates.
111.2
LINEAR OPERATORS AND VECTOR MEASURES
17 5
Now suppose that v is a positive, regular measure on (M;C) and 1.1 is an Emeasure. We say that 1.1 is vcontinuous if and only if, for each A
E
C,
We denote the family of vcontinuous measures by
Muf(C;E).
2 . 3 . Proposition: The isometry
L(B(M;C) iE)
Mf (C;E)
induces an isometry
Proof: The result follows from the simple remark that an operator T lifts from B(M;C) to the quotient space Lm(M;v) if and only if it vanishes on the characteristic functions of vnegligible sets.
We now come to the main result of this section

the identifi
cation of 8, continuous linear operators on Lm with vectorvalued measures. We say that an Evalued measure u is vregular if for each A
E
C
and we denote by Mv(C;E) the space of all uadditive measures U in Mvf(Z;E) which are vregular.
111. BOUNDED MEASURABLE FUNCTIONS
176 2.4.
Proposition: The above isometries induce a natural iso
metry from L(L
m
(M;V)
V ,Bl);E) onto Ma(C;E).
Proof: First we note that if (A,) is a disjoint sequence in C 03 n An then C xAkxA in 8 , (since v is and A := n=1 k= 1 regular). Hence, if T is B1continuous, the induced measure is
u
aadditive. A similar argument shows that it is regular. On the other hand, suppose that p is aadditive and vregular. We must show that the integral operator x
( x dp
is B1continuous. Since 8 , is the Mackey topology, we can assume that E = C
and then the result follows from the RadonNikodym
theorem ( [ 3 1 , Th. 2.2.4).

If we combine 2.4 with the remarks after 1.10 we obtain the fact that a Pettisintegrable function # induces a aadditive measure via the formula A
0 dp
(cf. DIESTEL and UHL
[el,
Ch.2).
111.3. VECTORVALUED MEASURABLE FUNCTIONS
In this section we consider measurable functions with values in a Saks space. We recall some facts on Banach space valued functions. In order to simplify the notation and avoid technicalities, we shall work with a positive Radon measure on a
111.3
VECTORVALUED MEASURABLE FUNCTIONS

177
0compact, locally compact space M. If F is a Banach space, then a function x : M for each K K1
C_
K,
E
p(K
F
K(M) and each \ K,) <
E
and
>
E
XI
K1
is (Lusin) measurable if
0 there is a K1
E
K(M) with
continuous. We recall the fol
lowing basic results on measurable functions (see BOURBAKI [ 2 ]
,
9 IV.5 for a detailed study of Lusin measurability). 3.1. Proposition: 1 )

The pointwise limit of a sequence of
measurable functions is measurable; if E is separable, then
2)
x : M
E
is measurable pro
vided that it is Borelmeasurable (i.e. for each Bore1 subset A of E l xl ( A ) is measurable).
We denote by Lm(M;F) the space of measurable functions x from M into F so that Mm(x) MoD(x) := inf
K
>
<
m
0 :
where Ilx(t) 11 < K

locally palmost everywhere).
M, is a seminorm on Lm(M;F) and the corresponding quotient space is denoted by (Lm(M;F),11 a
F
E
IIX
@
then
x
IS a :
aII, = 11x11, llall.
t
11,) 
Now suppose that (F,II 1 1 , ~ )
it is a Banach space. If x
x(t)a
E
is in Lm(M;F) and
is a complete Saks space. Once again,
for simplicity, we shall always assume that

T
is metrisable. Then
F has a representation as a projective limit of a spectrum {rn : Fn+l
Lm(M) ,
Fn; n
E
nr)
111.
178
BOUNDED MEASURABLE FUNCTIONS
of Banach spaces. A function x : M if for each K
K(M) ,
E
K1 C K, p ( K \ K 1 ) <
E
> 0, n
and
E
1~~
o
E
N
P F
,
is measurable
there is a K
E
K, (M) with
xiKl continuous. we denote by
Lm(M;F) the space of measurable functions x from M into F for
<
which M,(x)
11 11.
where Mm is defined as above, using the norm
m
Once again, Lm(M;F) denotes the associated normed space.
We consider the following structures on LOD(M;F):

11 Ik T
11
the seminorms through K(M)
loc n
[IK
:
x

the above norm; the locally convex topology defined by
jK 1
1 1 ~ ~ x(t)
Ik
du
as K runs
and n through N.
Now if M is compact and F is a Banach space then (Lm(M;F),II
lL,ll l K)
is a complete Saks space (this follows once
again from BOURBAKI [ 2 ] ,
IV.3.4, Thhoreme 3 and 3.3.2)
(Egoroff's
theorem)). Now returning to the general situation, {Lm(K;Fn) : n
E
N, K
E
K(M))
forms a projective system of Saks
spaces and its projective limit is naturally identifiable with the Saks space
(Lm(M;F),11
IL,T:~~).For any function in the
latter space clearly defines a thread in the natural way. On the : n E N , K E K) is a thread. n,K First hold n fixed. By the localisation principle, the thread
other hand, suppose that { x
(xnIK
:
K
E
K (MI1 defines a measurable function from M into F
(BOURBAKI [2]
,
IV.5.2, Proposition 2). Now as n varies, the
resulting functions can be patched together to form a measurable function from M into F. The Saks space structure on Lm(MiF) was
so defined that this vector space isomorphism be a Saks space
111.3
VECTORVALUED MEASURABLE FUNCTIONS
179
isomorphism. Hence we have:
3.2. Proposition: (Lm(M;F), 11
IL,
~ l ~ ~ )
is a complete Saks space.
The corresponding mixed topology is denoted by f3,.
Then
is a complete locally convex space.
(L~(M;F) ,B,)
3.3. Lemma: Suppose that M is compact and F is a Banach space. Then the unit ball of C(M;F) is
T
locdense in the unit ball
of L~(M;F).
Proof: If
E
>
0 and
x is in the unit ball of Lm(M;F), we can
choose K1 compact in M so that ?J(M \ K 1 ) tinuous. Now we can extend
XI
K1
I E
and x i K l is con
to a continuous function? from
M into F without increasing its supremum norm (generalised Tietze theorem). Then ff is in the unit ball of C(M;F) and Ilx%Ill 5
E.
3.4. Proposition: Let M be compact, F a Banach space. Then there is a natural Saks space isomorphism from (Lm(M) onto (Lm(M;F)I
11
sy
F, 11 IlI.rlOc
1
IIITloc).
Proof: There is a natural linear injection j from the algebraic tensor product Lm(M) o F into Lm(M;F) and j is a contraction for both norms. n C aiXMi is a stepfunction in Lm(M;F) where the Mi's Now if i=l are disjoint, then
'c)
BOUNDED MEASURABLE FUNCTIONS
111.
180
and so j is an isometry for the L’norms. If C xi
Q
ai
is a continuous function in Lm(M)
8
F
then
= sup
tEM = sup tEM and so j is an isometry on C(M)
QD
F for the supremum norms.
Now the unit ball of C(M) e+ F is dense in the unit ball of C(M;F) for the supremum norm (see SEMADENI [26], 20.5.6) and hence is dense in the unit ball of Lm(M;F) in the L 1norm (3.3). The result follows now from the construction of the completion of a Saks space (1.3.6).
3.5. Proposition: Let (M;lJ)be a measure space, F a complete
Saks space. Then there is a natural Saks space isomorphism from
Proof: Consider the commutative diagram
(K$ K1, n
S
m) where the vertical arrows are provided by 3.4.
111.3
VECTORVALUED MEASURABLE FUNCTIONS m
The required isomorphism is the induced arrow from L (MI m
181
6Y F
into L (M;F).
.
We now consider duality theory for Lm(M;F) To avoid technicalities, we shall simplify even further and suppose for the rest of this section thatfM i s compact and E is a Banach space. In fact, the main analytic difficulties occur already in this situation. A first guess might be that the B1dual of Lm(M;F) is L 1 (M;F'). It turns out that this is not true in general and that the true story is rather complicated. In fact, the question of when the dual of (Lm(M;F),B1) is L1 (M;F') is closely related to the RadonNikodym property.
In order to apply 1.1.18 we must first describe the dual space of L 1 (M;F).
3.6. Proposition: For any x
E
L 1 (M;F), f
E
Lm(M;F'), the scalar
function <x,f > is in Lm(M;p) and the bilinear form (x,f)
M
<x,f>du
induces an isometry from Lm(M;F') into L 1 (M;F)I ,
Proof: If K1 and K2 are compact so that p (M \ K1) < c / 2 . p (M \ K2)
<
~ / 2
and x (resp. f) is continuous on K1 (resp. K2)
then <x,f>is continuous on K1 n K2 and p(M \ (K1 II K2)) < Hence this function is measurable. We also have the estimate
E.
182
B O U N D E D MEASURABLE F U N C T I O N S
111.
/
M
Itx,f > I ~ P5
llfll,
M
llxlldcl
=
lIflL llxlll
Hence f defines a functional Tf in L1 ( M i F ) ' and the above inequality shows that
IITflI
5
Ilfll,~
To prove the reverse inequality, we assume first that f is countably valued i.e. that f has the form Cfn
x
where (f,) An is a sequence in F' and (An) is a disjoint sequence of measurable QP
subsets (of nonzero measure). If
E
(for
> 0 is given we choose no such that Ilf 11 + "0
llfll, =
Now let x
"0
SEP E
II fnII)
F be such that
Ifno(xno) 1 1 IlfnoII 1 L (M;E)
~ / 22 Ilfll,

of norm p(An
Then x "0 and
E/2. 0
)
IIxnolI = 1
This shows that IITfII 1 llfll,
'Ano
and is an element of
for countably valued f.
Now the countably valued f are norm dense in L m ( M ; F ' )
and from
this it follows easily that the above inequality is valid for all f in L , ( M ; F I ) .
The interesting question is, of course, when this mapping is surjective (i.e. (L1 ( M ; F ) ' = L m ( M ; F ' ) ) . Now this occurs exactly when F' has the RadonNikodym property (discussed in 3.15 below))
111.3
VECTORVALUED MEASURABLE FUNCTIONS
183
and so the above question leads to problems outside the scope of this book. One can, however, give a fairly elementary proof in the special case where F has a separable dual. We begin with some Lemmas:
3.7. Lemma: Let E be a separable Banach space, F a countable,
normdetermining subset of E'. Then the aalgebra A generated by F on E is precisely the Borel algebra of E (i.e. the oalgebra generated by the norm open sets).
Proof: We recall that F is normdetermining if, for each x IIxII = sup { [f(x)l
:
f
E
F, I[flI
E
E,
1).
Then the closed unit ball of E is
and so is in A . Therefore the same holds for the open unit ball (since it is a countable union of closed unit balls). Now every open set in E is a countable union of open balls and so is in A .
3.8.
Corollary: Let E and F be as above and let @ be a function
from M into E so that for each f
E
F, f
o
@
is measurable. Then
@ is Lusin measurable.
Proof: It follows immediately from 3.7 that The result follows then from 3.1.2).
@
is Borel measurable.
111.
184
BOUNDED MEASURABLE FUNCTIONS

3.9. Definition: If E is a subspace of Lm(M;p), then a lifting
for E is a linear mapping 1)
IT
o
p =
p :
IdE where
1~
E
Lm(M;p)
so that
is the natural projection from
L~ onto L ~ ; 2)
3.10.
Ip
(x)(t)I
5
[IxII, for t
E
M.
Proposition: If E is a norm separable subspace of Lm(M;p),
then E has a lifting.
Proof: E contains a countable dense subset Eo which is a vector space over the rationals Q. Let (xn) be a basis for Eo (over Q). For each n
E
N let
znbe
an element of Lm so that
maps xn into Zn.For each x E Eo, l :(x) a locally negligable set Ax. Then for each x
IX(:
xn.
5 from E0 into Lm which (t)I I I[x[Im except for
Then we construct a Qlinear mapping
the function which is equal to
IT(ZJ=
for t
E
Eo we let
4 u
is zero on the latter set. Then the mapping
XE
Eo
p : x
Ax
>
be
and which
Z
is
Qlinear and satisfies the condition
We can extend p by continuation to obtain the required lifting.
3.11
Proposition: Let E be a Banach space so that El is separable
Then the mapping constructed in 3.6 is surjective (i.e. the dual of L (M;E) is L ~ ( M ; E ~ ) ) .
111.3
VECTORVALUED MEASURABLE FUNCTIONS
185
Proof: Let 9 be an element of L(M;E)'. Then for each x
E
E,
the mapping
is an element of the dual of L 1 (M;u) and so is represented by a function in Lm(M;p). Hence we can regard T
9
as a mapping from
I(@II.
E into Lm(M;p). It is clearly linear and has norm 5
Now
the range of T is a separable subspace of Lm(M;p) (since E is 9 separable) and so there is a lifting p on it.
Now if t
E
M I we define f(t) to be the composition

where the last arrow is evaluation at t. Then f(t) norm is f : M
I
E
E' and its
II$II. Hence we have constructed a bounded function El
and for any x
E
E, < x , f > is measurable.
Hence we can apply 3.8 to deduce that f is Lusin measurable (note that any countable dense subset of E is normdetermining as a subset of the dual of El). To complete the proof, we must show that for any x
E
L 1 (M;E) we
have $(XI =
1 <x,f >dp.
It suffices to show this for the case where x has the form (y
E
L1 (M;p), z
E
El. Then we can calculate:
y
@
z
111.
186
BOUNDED MEASURABLE FUNCTIONS
Using this result, we can obtain immediately from 1.1.18 the following result:
3.12. Proposition: Let F be a Banach space whose dual is sepal rable. Then the natural duality bewteen Lm(M;F) and L (M;F1) establishes an isometry from L 1 (M;F') onto the p1dual of L~(M;F).
Before stating the next result, we remark that the definition of equiintegrability (given before 1.2) can be extended to functions with values in a Banach space simply by replacing absolute values by norms.
3.13. Proposition: When F is as in 3.12 and H is a subset of L1 (M;E
,
)
then the following are equivalent:
1)
H is Blequicontinuous;
2)
H is relatively U(L 1 (M;FI),L~(M;F) )compact;
3)
H is normbounded and Iequiintegrable;
Proof: 1)
2)
and
3)
1 ) can be proved exactly as in
the scalar case. 2)
3): H is clearly normbounded. If H were not lequiinte
grable, then by 1.7 (applied to the set (IlflI : f
E
H}
in L1 (M;p))
there would be a disjoint sequence (An) of measurable sets, a sequence (f,)
in H and an
E
> 0 so that
111.3 For every n by An'
E
VECTORVALUED MEASURABLE FUNCTIONS
N choose yn
IIYnIIm~1
E
187
Lm(M;F) such that yn is supported
and
(this can be done by approximating fn by simple functions). Now consider the mapping
from L1 (M;F') into 11 (IN) weakly compact

. Then T
is continuous but ET fn 1 is not
contradiction.
3.14. Corollary: Under the assumption of 3.12 , (Lm(M;E),B1 ) is a Mackey space.
3.15. Remarks on the Radon Nikodym property: If we compare 2.4 with the scalar result, we are led naturally to pose the following question: when can we identify the space Mv (C;E) of bv vcontinuous measures of bounded variation with the space L ' (M;E) i . e . when does every Evalued vcontinuous measure of bounded
variation have a vderivative? The following example shows that this is not always the case: Example: Our measure space is [0,11 with Lebesgue measure. We define a vectorvalued measure p by defining p(A) to be the sequence
((A
sin(2mt)du(t))i=l. Then p takes its values in co
(RIEMANNLEBESGUE Lemma) and it is routine to show that it is aadditive and vcontinuous. Now if 1.1 has a vderivate, then it must be the function t
I
G
.
(sin(21~nt) ) However, this
188
BOUNDED MEASURABLE FUNCTIONS
111.
function does not take values in co. We formalise the above fact in the following definition: Definition: A Banach space E has the Radon Nikodym property (abbreviated RNP) if for each measure space (M;v) the canonical injection L1 (M;E)
qV(C;E)
is surjective. We remark that it is sufficient to check this condition for the canonical measure space [0,11. The above example shows that co does not have the RNP. As a positive result we have : Let E be a Banach space with a boundedly complete basis (x,). Then E has the RNP. We sketch the proof (as we shall see later, this result follows from 3.12
. It is convenient to renorm E so that n
m
NOW let p be an Evalued measure which is vcontinuous. Then the induced measures Ifn
o
p)
(where (f,)
is the associated biortho
gonal sequence) are vcontinuous and so there are measurable functions yn so that fn
o
p(A) =
1
A
Yndp
(A
E
1).
Now it follows from the fact that the basis is boundedly complete that the E valued series Cynxn converges palmost everywhere to a measurable function (since the simple estimate
111.3
VECTORVALUED MEASURABLE FUNCTIONS
189
bounded almost everywhere). 1111 is the variation of F (i.e. 1111 (A) = Sup CIIF(An)
11
;
the supremum being taken over the finite
partitions of A). The limit is a vderivative for
u.
For a detailed study of the RadonNikodym property, we refer to DIESTEL and UHL [ 8 1 and BUCHWALTER and BUCCHIONI [3]. For the reader's orientation, we quote the following results: I.
Any reflexive space and any separable (or even weakly
compactly generated) dual space has the RNP. In fact, the dual of a separable Banach space has RNP iff it is separable. 11.
A Banach space has the RNP if and only if every separable
subspace has the RNP. 111. The spaces L1 (M;p), C (K) and Lm(M;p) do not have the RNP (except for the trivial exceptions and 11 of finite support resp. )

p atomic, K finite
.
In fact, Proposition 3.12 is a special case of the following result:
the space L'(M';E') is naturally isometric to the 6 , dual
of LOD(M;E) if and only if E' has the RadonNikodym property. Hence we can interpret 3.12 as stating that a separable dual space has the RNP (and this result contains the result that a space with a boundedly complete basis has the RNP of course). In the general case, one can identify the dual of (Lm(M;E),B1) with the lower star space LA (M;E')
introduced by SCHWARTZ
(roughly speaking, this is the space of those functions x which are weakstar measurable and are such that the upper integral
190
111.
BOUNDED MEASURABLE FUNCTIONS
of the norm function llxll is finite). Since a reflexive space has the RNP, 3.12 and its Corollaries hold for such spaces.
111.4. NOTES
The topology 6, has never been studied before (of course, 1.1.1) identifies it with a familiar topology of the dual pair ( L " , L ' ) ) . The study of B 1 was suggested by GROTHENDIECK's paper [I21 where the version of the DUNFORDPETTIS theorem proved here (1.8) can be found. 1.4 is due to GODEMENT. Many of the results of
5
1,
in particular 1.9, were obtained independently by DAZORD and JOURLIN 171. The result that ( L m l B 1 )is a Mackey space has also been obtained by STROYAN (for bounded measures). The remark that (e",B,) is a universal Schwartz space is due to JARCHOW [15].
SENTILLES [271 has also introduced a strict topology for LEOspaces in his study of a measure free approach to L'spaces based on the Boolean algebra of measurable sets. For a complete &
account of the integration theory used in 1.20 and in particular, for the definition of Xm(M;C) see the lecture notes of BUCHWALTER and BUCCHIONI [3].
Our basic reference for
9
2 was DIESTEL and UHL ( [81, Ch. 1)

in
particular, 2.2 and 2.3 can be found there.
As mentioned in the introduction,
9
3 is devoted to the elementary
beginnings of a theory of measurable functions with values in a
111.4
NOTES
19 1
Saks space. We refer the reader to SCHWARTZ [25] for a dis
cussion of the problems involved in the duality theory for Banach space valued LPfunctions.
The final form of this chapter owes a great deal to the assistance of W.SCHACHEFMAYER. In addition to simplifying and improving some proofs he pointed out the important references [31 and [81 to me. Paragraphs 1.20. 3.6

3.13 are based on [231and [241.
192
BOUNDED MEASURABLE FUNCTIONS
111.
REFERENCES FOR CHAPTER 111.
[I] T. AND0
Weakly compact sets in Orlicz spaces, Can. J. Math. 14 (1962) 170176.
[21
N. BOURBAKI
Elbments de Mathgmatique: Intggration, Chs. 14 (Paris, 1965).
[31
H. BUCHWALTER, D. BUCCHIONI Intggration vectorielle et th&or$me de RadonNikodym (Dgpartement de Mathematiques, Lyon, 1975).
[41
A.K. CHILANA
The space of bounded sequences with mixed topology, Pac. J. Math. 48 (1973) 2933.
[51
H . S . COLLINS
On the space L"(S) , with the strict topology, Math. 2. 106 (1968) 361373.
161
J.B. CONWAY
Subspaces of (C(S), B ) , the space ( L " , B ) and (Hm,B) Bull. Amer. Math. SOC. 72 (1966) 7981.
171
J. DAZORD, M. JOURLIN Une topologie mixte sur l'espace LOD, Publ. DBp. Math. Lyon 112 (1974) 118.
[81
J. DIESTEL, J.J. UHL, Jr. The theory of vector measures (to appear in the American Mathematical Society's surveys).
[91
J. DIXMIER
Les algebres d'opgrateurs dans l'espace hilbertien (Paris, 1969)
.
[lo] D.H. FREMLIN
Pointwise compact sets of measurable functions, Man. Math. 15 (1975) 219242.
[I11
A. GOLDMAN
L'espace des fonctions localement Lp sur un espace compactologique st (Preprint).
[I23
A. GROTHENDIECK
Sur les applications linbaires faiblement compactes d'espaces du type C(K), Can. J. Math. 5 (1953) 129173.
REFERENCES
193
[131
E.A. HEARD
Kahane's construction and the weak sequential completeness of L I , Proc. Amer. Math. SOC. 44 (1974) 96100.
1141
A. IONESCU TULCEA
On pointwise convergence and equicontinuity in the lifting topology I;II (Z. Wahrsch. u. verw. Gebiete 26 (1973) 197205 and Adv. in Math. 12 (1974) 171177.
1151
H. JARCHOW
Die Universalitst des Raumes co. Math. Ann.
[161
J.W. JENKINS
On the characterization of abelian W"algebras, Proc. Amer. Math. SOC. 35 (1972)
203 (1973) 211214.
436438.
[171
S.S. KHURANA
A vector form of Phillip's Lemma, J. Math. Anal. Appl. 48 (1974) 666668. Weakly convergent sequences in Lm(Preprint).
[181 [191
V.L. LEVIN
Lebesgue decomposition for functionals on the space LT of vectorvalued functions, Func. Appl. 8 (1974) 314317.
[20]
D.R. LEWIS
Conditional weak compactness in certain inductive tensor products, Math. Ann. 201 (1973) 201209.
[21]
S. SAKAI
C2'algebras and W"algebras (Berlin, 1971
[22]
S. SAKS
On some functionals, Trans. Amer. Math. SOC. 35 (1933) 549556.
1231
W. SCHACHERMAYER Mixed topologies on Lm for abstract measures (written communication).
[241
Mixed topologies in spaces of Banach valued functions (written communication).
.
111.
194 1251
L. SCHWARTZ
[261
2.
[27]
F.D. SENTILLES
SEMADENI
BOUNDED MEASURABLE FUNCTIONS Fonctions mgsurables et fcscalairement mgsurables, mesures banachiques majorges, martingales banachiques et proprigtb de Radon Nikodym, Sgm. MaureySchwartz 1 9 7 4 / 7 5 , EXp. 4 . Banach spaces of continuous functions I, (Warsaw, 1 9 7 1 )
.
1
L space for Boolean algebras and semireflexivity of spaces L ~ ( M , c , ~ ) (Preprint)
An
.
[281
K.D. STROYAN
A characterisation of the Mackey uniformity m(Lm,L1) for finite measures, Pac. J. Math. 49 ( 1 9 7 3 ) 223228.
[291
E. THOMAS
1301
J. VESTERSTRgM, W. WILS On point realization of Lmendomorphisms, Math. Scand. 2 5 ( 1 9 6 9 ) 178180.
1311
C.R. WARNER
The LebesgueNikodym Theorem for vector valued Radon measures, Memoirs of the American Math. SOC. 1 3 9 ( 1 9 7 4 ) .
Weak Jc dense subspaces of Lm(lR) , Math. Ann. 1 9 7 ( 1 9 7 2 ) 180181.
CHAPTER IV

VON NEUMANN ALGEBRAS
Introduction: In this chapter, we study algebras of operators on Hilbert space from the point of view of mixed topologies. On the algebra L(H) (H a Hilbert space) there are three classical topologies

corresponding to uniform, strong and
weak convergence of operators. For most applications (in particular, to spectral theory), the uniform topology is too strong. From a theoretical point of view, the weak and strong topologies are very unsatisfactory. VON NEUMANN introduced rather complicated topologies topologies


the ultraweak and ultrastrong
to overcome these difficulties. We introduce here
three natural mixed topologies on L(H) which are related to (but distinct from) the ultraweak and ultrastrong topologies and have several advantages with respect to these topologies.
There are two distinct approaches to algebras of operators: the direct approach by defining them as selfadjoint, ultraweakly closed subalgebras of L(H) (von Neumann algebras) or
.
the axiomatic approach (W" algebras) Comprehensive treatments can be found in the books of DIXMIER [ 8 1 and SAKAI [I41 respectively. The latter approach is certainly the most elegant. However we have chosen the former since it allows us to give a more elementary treatment. In fact, with the exception of one rather deep theorem of SAKAI, we require only acquaintance with spectral theory of selfadjoint operators and (what is
195
IV.
196
VON NEUMANN ALGEBRAS
essentially the same) the representation theory of commutative von Neumann algebras.
In
9
1 we consider two mixed structures on L(H) and identify
its dual as the space of nuclear operators on H. We then present the corresponding Saks spaces as a projective limit of spaces of finite dimensional operators and deduce that L(H) has the approximation property.
In the second section, we consider three mixed topologies
B,,
Bs and
Bs2b
on a von Neumann algebra. For commutative von
. . Lmspaces) these coincide with the
Neumann algebras (i e
topologies considered in Ch. I11 (in this case, Bs

BsgC
=
8,).
After giving routine properties of these topologies, we use them to give a short proof of KAPLANSKY's density theorem. The main result is that Bsn is a Mackey topology and this has several important consequences. The section ends with some brief remarks on the relation between the mixed topologies and classical topologies on von Neumann and W*algebras.
VI. 1 IV.l.
197
THE ALGEBRA OF OPERATORS
THE ALGEBRA O F OPERATORS I N HILBERT SPACES
L e t L ( H ) d e n o t e t h e a l g e b r a of bounded, l i n e a r o p e r a t o r s
on a H i l b e r t s p a c e H . On L ( H ) w e c o n s i d e r t h e f o l l o w i n g structures:
11 11 'lS 
t h e uniform norm: t h e topology of p o i n t w i s e convergence on H w i t h r e s p e c t t o t h e norm ( t h e s t r o n g o p e r a t o r t o p o l o g y ) ;
T
U

t h e topology of p o i n t w i s e convergence on H w i t h r e s p e c t t o t h e weak topology u ( H , H ' )
( t h e weak
o p e r a t o r topology).
Note t h a t w e can c o n s i d e r L ( H ) as a v e c t o r subspace of C"(B11
I I ; H ) and i t i s a l o c a l l y convex s u b s p a c e f o r t h e
following p a i r s of s t r u c t u r e s :
1 . 1 . P r o p o s i t i o n : The u n i t b a l l of L ( H ) i s complete f o r and compact f o r
'la.
Hence w e have two complete Saks s p a c e s a v a i l a b l e : (L(H),
11 1 1 , ~ ~ )
and
(L(H)
11 1 1 , ~ ~ ) .
'lS
IV.
198
VON NEUMANN ALGEBRAS
We denote the corresponding mixed topologies by B s and B,. Note that the above spaces are Saks space subspaces of
I L , T ~ ) and
IL,T~)
(C"(B11 ~ ~ : H ) , ~ ~ and
(C"(Bl1 ~ ~ : H , , ) , ~ ~
is a locally convex subspace of
(L(H)f B , )
that
( C m ( B I I II;H) f f 3 K )
(1.4.4 and 1.1).
We shall consider the properties of the above locally convex spaces in more detail in the next section. In the following we consider duality for L(H). We require two preliminary results:
1.2.
Lemma: Let E and F be locally convex spaces and let
(L(E;F)
f~
P
)
be the space of continuous linear operators from
E into F with the topology of pointwise convergence on E. Then the dual of (L(E;F)
P
f~
)
under the bilinear form n (T, C fi w xi) i=1 Proof: We regard (L(E;F),T (C(E;F),T~).Then if f
P
is,identifiable with n I+
C
i=1 )
E
b
F'
fi(Txi).
as a topological subspace of
(L(E;F),T~)'there is an extension
E

of f to an element of the dual of C(E;F). Such an element has the form T
n C fi(Txi) i=l
...,fn)
for some finite subsets {x,~...,x~)of E and {fl,
of F'.
IV. 1
THE ALGEBRA OF OPERATORS
1.3. Lemma: An operator A
E
L(H) is compact if and Only if
there exist orthogonal sequences (x,)

a positive element (A,) A :x (i.e. A
=
199
in H, (f,)
in H ’ and
of co(N) so that C
n
Xnfn(x)xn
c Anf, a x,). n
Proof: We shall prove that every compact operator has this form. The operator A”:A : H d H
is positive and compact.
Hence, by the spectral theorem for compact operators, there is an orthogonal sequence (xn) in H and a positive sequence 2 in co(N) so that (1,) A“A
:
x w c n
x ~ 2( x I x ~ ) x ~ .
Let M be the orthogonal complement of A”A(H). Then AIM = 0 since if y
E
M
= ( A ~ I A=~ ( )~ \ A $ C= A ~0. ) 11~y11~

Then a simple calculation shows that the sequence (f,) is orthonormal (where fn : x A = C XnXn
8
fn.
in H ’
(Xi’xlAxn)) and that
1.4. Definition: The above proof shows that the positive sequence (An)
is an invariant of A (i.e. independent of the
partiaular representation of A

up to permutations of COUrSe)l
since it is the sequence of the square roots of the eigenvalues of A*A. Hence we can make the following definition:
IV.
200 A
VON N E U M A " ALGEBRAS
compact operator A : H
+
H
We denote the latter sum then by
is nuclear if
1 1 All
1.5. Proposition: If A = C Anfn o xn
C
An <
m.
. is a nuclear operator
in L(H) then fA : T
C
I+
n
Anfn(Tx,)
is a continuous linear form on (L(H), I 1
11)
and
IIfAll
=
IIAII~.
Proof: We verify the statement about the norms. Firstly we can estimate
On the other hand, there is a partial isometry V on H so that Vx,  yn to (f,)
is a sequence in H, biorthoqonal
for each n where (y,)
. Then
IIVII
fA(V) = C An.
and
5 1
Now it is easily checked that the mapping
A
fA
is
welldefined (i.e. the linear form fA is independent of the representation of A). For if m~
C
Xnfn B xn = 0 then for each
N,
Xmxm= ( Cn Anfn where (y,)
bp
xn) (Y,)
= 0
is biorthoqonal to (fn) and so each
A m = 0.
Hence we have embedded N(H), the space of nuclear mappings in
L (H), isometrically in (L( H I
I
11 11) ' .
IV. 1
1.6.
THE ALGEBRA OF OPERATORS
Proposition: (N(H)
( L ( H I ,B s) and ( L ( H I ,B,
1
, I / ),1
i s t h e d u a l of t h e s p a c e s
. H o
P r o o f : W e can i d e n t i f y (L(H),r,)
20 1
H I ,
t h e d u a l of
( L ( H ) , T ~ )and
(1.3), w i t h t h e s p a c e of o p e r a t o r s i n L ( H ) of
f i n i t e r a n k . Now t h e s e are o b v i o u s l y 11 IINdense i n N ( H ) ( f o r m m the operators C Xnf, e xn approximate C hnfn Xn in n= 1 n= 1
.
t h e n u c l e a r norm) and so t h e r e s u l t f o l l o w s from 1.1 :18. ( i i )
1.7.
Remark: I f A
i . e . t h e sum
E
L ( H ) w e d e n o t e by T r ( A ) t h e t r a c e of A
C (Ax,Ix,)
i s an o r t h o n o r m a l b a s i s
where
,€A
f o r H i f t h i s sum is a b s o l u t e l y c o n v e r g e n t . Then T r ( A ) i s i n dependent of t h e b a s i s (x,)
and N ( H ) i s p r e c i s e l y t h e c l a s s
of o p e r a t o r s i n L ( H ) f o r which t h e trace e x i s t s (hence t h e n u c l e a r o p e r a t o r s are sometimes c a l l e d o p e r a t o r s of trace c l a s s ) . One can t h e n v e r i f y t h a t i f T
E
A = C h f
B
xn
n n L ( H ) t h e n T r ( A T ) e x i s t s and i s g i v e n by
E
N(H)
and
T r ( A T ) = C Xnfn(Txn) = f A ( T )
n

and so t h e d u a l i t y between N ( H ) and L(H) i s g i v e n by t h e b i l i n e a r form (A,T)
Tr(AT).
W e now g i v e a n a t u r a l p r o j e c t i v e l i m i t r e p r e s e n t a t i o n of t h e
Saks s p a c e ( L ( H ) , \ l
1 1 , ~ ~ ) and
(L(H)
,\I
I\,Ts).
W e o r d e r t h e family
202
VON N E U M A "
IV.
of p a i r s ( M , N ) (M,N)
ALGEBRAS
of c l o s e d subspaces of H by d e f i n i n g 5
(M1 , N 1 ) cj M E M1
and
N E N,.
I f ( M , N ) I ( M 1 , N 1 ) t h e r e i s a n a t u r a l c o n t r a c t i o n from L(M, , N 1 ) i n t o L ( M , N ) d e f i n e d by T
where TI
IT
i : M 4 M1 N
: N,+
0
T o i
i s t h e n a t u r a l i n j e c t i o n and
t h e o r t h o g o n a l p r o j e c t i o n . Then i f F d e n o t e s
t h e f a m i l y of a l l f i n i t e d i m e n s i o n a l s u b s p a c e s of H , (L(M,N) : M , N
E
F)
{L(M,H) : M E F ) form p r o j e c t i v e systems of Banach s p a c e s under t h e s e mappings. T h e i r Banach s p a c e p r o j e c t i v e l i m i t s are i d e n t i f i a b l e w i t h L ( H ) and t h e i r Saks s p a c e p r o j e c t i v e l i m i t s a r e
(L(H)
,I1 1 1 , ~ ~ )
and
(L(H)
,I[ 1 1 , ~ ~ )
respectively. Now t h e above mapping from L(M1 , N 1 )
i n t o L(M,N) has a c o n t r a c t i v e
r i g h t i n v e r s e , namely t h e mapping T
i O T
Hence c o n d i t i o n b ) of P r o p o s i t i o n 1 . 4 . 2 0 i
tisfied. If M
and N a r e subspaces of H and M i s f i n i t e d i m e n s i o n a l t h e n t h e Banach s p a c e L ( M , N ) h a s t h e metric approximation p r o p e r t y . Hence w e can a p p l y 1 . 4 . 2 0 t o o b t a i n t h e f o l l o w i n g r e s u l t :
IV.2
VON NEUMANN ALGEBRAS
1.8. Proposition: (L(H),B,)
and
(L(H), B s )
293
have the approxi
mation property.
We remark here that it has apparently been shown recently that (L(H),
1 1 11)
dbes
not have the approximation property (SZANKOWSKI).
IV.2. VON NEUMANN ALGEBRAS
2.1. Proposition: Let E be a subspace of L(H). Then the
following are equivalent: 1
E is B,closed;
2)
E is
3)
the unit ball of E is Bucompact.
fj
U
closed;
Proof: The equivalence of 1) and 2) follows from the fact that 8, and 8, are topologies of the same duality, that of 2) and 3 ) from 1.4.3.
2.2.
Definition: Let H be a Hilbert space. A von Neumann algebra
on H is a nsubalqebra M of L(H) which satisfies one of the conditions of 2.1 and contains the unit IdH of L(H)
. Then we
can regard M as a Saks space in two natural ways: as a subspace of (L(H),I]
1 1 , ~ ~ ) resp.
of (L(H), 11
ding mixed topologies by B s and B.,
I~,T,).
We denote the correspon
IV.
204
VON NEUMA" ALGEBRAS
At this point it is convenient to introduce a symmetric form We strengthen the topology
of 6,.
T~
by defining
the topology defined by the seminorms {px,pi : x
Then (L(H),I]
I ~ , T ~ is ~ ~ a)
T~~~ E
to be
H) where
complete Saks space and the associated
mixed topology is denoted by B S n . Note that the dual of (L(H),Bs+) is also N(H) (this follows from 1.6 and the fact that the involution is Bacontinuous). The topology B s n the following advantage over 8 ,
:
has
if we denote by M S the set
of selfadjoint elements of a von Neumann algebra M then M S is a vector space over the reals and M can be identified (as a vector space) with the complexification of MS. Now
0, = B s a
on Ms and the locally convex complexification of ( M S , B s ) (M,B,,T~).
is
Hence to verify for example the Bsacontinuity of a
linear mapping on M , it is sufficient to check its Bscontinuity
on M'.
For convenience we collect in one Proposition the basic properties of the above topologies:
2.3. Proposition: Let M be a von Neumann algebra. Then
and (M,Bs,t)
are complete;
1)
(M,B,),
2)
the dual of M under any of the above topologies is the
Banach space
(M,Bs)
M*
:= N(H) /Mo;
IV. 2 3)
a linear operator
continuous if and only if its
restriction to the unit ball BM of M is T
‘ca
(resp.
‘cS,
if M is of countable type (i.e. if H contains a countable
subset S so that for each T
E
MI
TI
s
= 0  j T = 0)I then the
unit ball of M is metrisable for the topologies B,
5)
resp.
continuous;
sf$ 1
4)
205
from M into a locally convex space
resp. B s 9 ;  )
(resp.
is 6,
@
VON NEUMANN ALGEBRAS
(M,B,)
space M,),
a s and f3,,.,;
is semiMonte1 and so is the dual of the Banach with the topology of compact convergence. Hence a
subset A of (M,@a ) is closed if and only if closed for each r > 0;
A n rBM
is T a 
if M is of countable type, a subset A of M is $ a closed if and only if it is sequentially closed; 6)
7)
multiplication is jointly B,+$continuous on BM;
8)
8, and
‘cS
(resp. BS,.,
and
‘cS9$)
have the same convergent
sequences and compact sets; 9)
B,

is defined by the seminorms 1 T sup A n lTr(AnT) 1 ncIN
where (A,)
runs through the sequences in the unit ball of N(H)
and (An) runs through the family of sequences of positive numbers which increase to infinity; 10)
if MI is a von Neumann algebra with MI E M then 8 , induces
8, on M I .
206
IV.
VON NEUMANN ALGEBRAS
I f M i s a von Neumann a l g e b r a , a l i n e a r o p e r a t o r from M i n t o a l o c a l l y convex s p a c e i s normal i f and o n l y i f i t i s 6,continuous
as w e s h a l l see
( i t i s t h e n Bs9:continuous
b e l o w ) . Note t h a t M9< i s t h e s p a c e of normal mappings from M i n t o C. A n  a l g e b r a homomorphism from M i n t o a von Neumann a l g e b r a M,
i s normal i f i t i s B  c o n t i n u o u s U
a u t o m a t i c a l l y a norm c o n t r a c t i o n

(note t h a t it is
see D I X M I E R [8], p. 8 ) .
Once a g a i n , it i s e q u i v a l e n t t o demand Bs9:continuity. 1.4.32,
By
it s u f f i c e s t h a t t h e homomorphism have a 6,closed
g r a p h . Two von Neumann a l q e b r a s are isomorphic i f t h e r e i s a b i j e c t i v e normal 9:algebra homomorphism from M o n t o M,
(its
i n v e r s e i s t h e n normal).
W e r e c a l l now t h a t i f
(M;u) i s a measure s p a c e t h e n L m ( M ; p )
can be r e g a r d e d ( v i a m u l t i p l i c a t i o n ) a s a s u b a l g e b r a of L ( H ) where H i s t h e H i l b e r t s p a c e L L ( M ; u ) and so as a (commutative) von Neumann a l g e b r a . Then t h e t o p o l o g y B, c o i n c i d e s w i t h t h e 8,
of
5
introduced here
1 1 1 . 1 and
Bs = Bsn = B, on L".
The r e p r e s e n t a t i o n theorem f o r commutative von Neumann
a l g e b r a s s t a t e s t h a t e v e r y such a l g e b r a i s isomorphic t o an a l g e b r a of t h e above form.
The s p e c t r a l theorem s t a t e s t h a t i f T
E
L(H) is selfadjoint
t h e n t h e r e i s a p a r t i t i o n of u n i t y { E ( X ) ) on a ( T ) so t h a t
IV.2 Then if
h
E
Lm(u(T))
h
The mapping
we can define

h(T) :=
1
u (TI
h(X)dE(X).
h(T)
from Lm(u(T)) into L(H)
E
M

is a normal $<algebrahomomorphism
. If T
lies in a von Neumann subalgebra
M of L(H) then the range of h p(T)
207
VON NEUMANN ALGEBRAS
h(T)
lies in M (since
for any polynomial p and the polynomials are B , 
dense in L m ( u (T)))
. In particular, the projections (E(X))
in
the spectral representation of T lie in M (since E ( X ) = X]m,A]na(T) (TI1.
The following Proposition allows us to give a short proof of the KAPLANSKY density theorem:
2.4. Proposition: Let h be a bounded continuous function on IR. Then
T
h(T)
is Bscontinuous from BMs into IIhIImBMS.
Proof: It follows from 2.3.7) that the result holds for functions h which are polynomials on a neighbourhood of [  1 , l l . NOW take T
E
and let h(T)
B
MS
[1,11
IlxlL
and
hood V of T in B Then if S
E
v,
x
I Ilhll,,
MS
of
U be a B,neighbourhood
EB C U. Let E be a MS is a polynomial on a neighbourhood of
h(T). Then there is an function so that
>
+
E
0 so that
I[h
 ElL
so that if S
E
5
~ / 3 .There is a Bsneighbour
V then K(T)

x(S)
E
U/3.
IV.
208 2.5.
VON NEUMANN ALGEBRAS
C o r o l l a r y (KAPLANSKY's d e n s i t y t h e o r e m ) : L e t A be a
Bs*dense
9:
. Then
 s u b a l g e b r a o f t h e von Neumann a l g e b r a M
BA , t h e u n i t b a l l o f A
,
i s Bs,dense
P r o o f : W e f i r s t show t h a t B
i n BM '
i s Bsdense
assume t h a t A i s normclosed.
W e can
in B
AS
MS'
If T
E
B
MS
there is a net
i n A w i t h Ta d T i n 8 , . W e can assume, by (Ta a E A 1 . 1 . 1 4 , t h a t (Ta) i s bounded. L e t h b e a c o n t i n u o u s f u n c t i o n on IR so t h a t 2.4,
h(Ta)
llhlL
+
= 1
h(T)
and
h = Idn f o r 8.,
in B
MS
on [  1 , 1 ] .
Then. by
Now h ( T a ) i s t h e u n i f o r m
l i m i t o f p o l y n o m i a l s i n Ta and so i s i n A .
o p e r a t o r s , we use t h e following
To d e a l w i t h n o n  s e l f  a d j o i n t trick: let
b e a H i l b e r t s p a c e d i r e c t sum
fy operators
5
on
?? w i t h 2
[z :"I
t h e c o n d i t i o n s o f t h e C o r o l l a r y . Now i f T then
T is i n
so t h a t i n B,n.

ns and Ta
IIFII
:=
5 1 . Hence t h e r e
T
in
and i d e n t i 
( T ) where e a c h i j be t h e s e t of o p e r a t o r s ( T i j )
E L(H) L e t x (resp. ij whose components are i n A ( r e s p . i n M ) . Then
T
H
x 2 matrices
a)
.
H
Bs.
Then
E
is a n e t a T2,
E
and
satisfy
M with
IlTll s 1 ,
BA
T')=
and
T'.) il a T21>
W e now p r e s e n t one o f t h e most i m p o r t a n t r e s u l t s o f t h i s
C h a p t e r : t h e r e s u l t t h a t (M,Bs,t)
i s a Mackey s p a c e .
i n BAS
T
IV.2
VON NEUMANN ALGEBRAS
209
We require the following characterisation of normal forms on a von Neumann algebra. A fairly accessible proof can be found in RINGROSE [13].
2.6. Proposition: Let M be a von Neumann algebra, f a form in
Mft. Then f is Bucontinuous (i.e. f
E
M9:) if and only if for
each orthogonal family {E } of projections in M Q
f( C EQ) = C f(E,). a
Q
2.7. Corollary: f
E
Ms': is BUcontinuous if and only if f
I MI
is Bacontinuous for each commutative von Neumann subalgebra
M I of M. 2.8. Proposition: Let M be a von Neumann algebra, K a subset of Mi.;. Then the following are equivalent:
K
1 MI
1)
K is relatively u (Mf:,M) compact:
2)
for each commutative von Neumann subalgebra M, of M,
is u(M1g:,M1)compact.
Proof: 1) = > 2) 2)
>
1):
7
is trivial.
Firstly, if K satisfies 2) then K is pointwise
bounded on MS and so pointwise bounded on M. Hence, by the principle of uniform boundedness, K is normbounded and
0
relatively weakly compact in M*. Thus it will suffice to show that the u(M",M)closure of K lies in Mg. Suppose that f lies in this closure. Then f 1
lies in the weak closure of K
M1
IV.
210
VON NEUMANN ALGEBRAS
for each commutative von Neumann subalgebra MI of M . Thus f is normal ( 2 . 7 ) .
Corollary: T ( M , M + ~ ) is the finest locally convex topology
2.9.
~ on each commutative von on M which is coarser than T ( M ,MI+<)
Neumann subalgebra MI of M.
2.10.
Lemma: Let (T,)
be a sequence in B
MS
with
Tn
0
Then for each 6 > 0 there is a sequence (En) of pro
in 8 , .
jections in M so that En
in 8 , and llTnEnll I 6
1
for
each n.
Proof: Let
En
:=
X[S,SI
(T ) . Then since n
2 1 6 2 (I  En) 2 0 Tn
IEn+
2.11.
0 in B.,
Also llTnEnl]5 6
Proposition: Let M be a von Neumann algebra of countable
type. Then
B,s
= T(M,M~).
Proof: We shall work in the real vector space MS and show that B s = T(M',M:)
there. Since M is of countable type,
BMs
is B s 
metrisable and so it will suffice to show that if (Tn) is a sequence in B
MS
so that Tn+
0
in B s then
Tn
0
uniformly on each weakly compact subset K of M Z . If this were not the case, there would be a relatively compact sequence (f,), E
>
0 and a subsequence of (T )
n
(which we assume, for simplicity
an
IV.2
VON NEUMANN ALGEBRAS
211
of notation, to be (Tn) itself) so that
I fn(Tn)l >
each n. Now by EBERLEIN's theorem, (f,)
has a weakly convergent
for
E
subsequence and s o , by passing to a subsequence and by translating, we can assume that {En] so that
En+
f
n
d 0 weakly. Choose projections for some 6 > 0
in 8 , and [[EnTnllI 6
1
to be chosen later (2.10). Then Ifj(Ti)l
for each i,j Now
f> .
I
[f.(EiTi)l 3 + lf.((I 3
5
2M6
+ lf.((I 3
N where
E


Ei)Ti)l
Ei)Ti)l
M := supCIIfll : f
K).
E
0 pointwise on the u (MS,Mf)compact set B
3
MS
so, by Baire's theorem, there is a point To
E
B
and
so that (f.) 7
is equicontinuous at To i.e. there is a o(MS,M.:)neighbourhood U of zero in M S so that Ifj(T) for each T
E

fj(To)l < 6
V := (To + U) n BMs. Choose J
E
N so that
if j 2 J. Then
Ifj(To)l < 6
If.(T)I < 36 7
for j 2 J, T
V. Now if
E
Si := E.T i oE i
then Si
E
B
and
+
(I

Ei)Ti
To. Hence there is an I
Si+
E
N
MS
that Si and EiToEi are in V for i 2 I. Then Ifj((IEi)Til for i 1 I, j
1
J. Then
I
If.(Si)l 3 + Ifj(EiToEi)l
5 66
so
IV.
212
VON NEUMANN ALGEBRAS
for i 2 I, j 2 J and for small 6 this contradicts the inequality
2.12.
I fn(Tn) I >
E.
Remark: It follows from a characterisation of relatively
weakly compact subsets of MB due to AKEMANN (see C33 or AARNES C 2 l ) that 2 . 1 1 holds without the assumption that M be of countable type. However, the extension to general von Neumann algebras requires some more technical results. Nevertheless, in view of this fact, we shall drop the assumption of countable type in the formulation of the following consequences of 2 . 1 1
2.13.
.
Proposition: Let M be a von Neumann algebra, F a locally
convex space. Then a linear mapping @ from M into F is
BSt.t
continuous if and only if its restriction to each commutative von Neumann subalgebra of M is Bscontinuous.
Proof: Since ( M , B ~ $ : ) is a Mackey space, we can reduce to the case when
F =
Q:
and this is precisely 2 . 7 .
2 . 1 3 is equivalent to the fact that
Bsn is the finest locally
convex topology on M which is coarser than $,
on each commutative
von Neumann subalgebra MI of M (cf. 2 . 9 ) .
2.14.
Proposition: If M is a von Neumann algebra, then (M,Bs,+c)
has the BanachSteinhaus property.
IV.2
VON NEUMANN ALGEBRAS
213
Proof: By 2 . 1 3 we can reduce this to the case where M is commutative and this is 1 1 1 . 1 . 1 2 .
2.15.
Corollary: If M is a von Neumann algebra then Mf: is
u (M%,:,M) sequentially complete.
2.16.
Proposition: Let
M and
M 1 be von Neumann algebras, Q a
linear mapping from M into M I .
Then 0 is Bucontinuous if and
only if it is Bsft.continuous.If , in addition, 0 is a
9:
morphism, then these conditions are equivalent to its continuity for 6.,
Proof: If
@
is Bucontinuous, then it is continuous for the
Mackey topologies T ( M , M ~ : ) and ‘r(M1 ,MI*)
i.e. for the B,+:topo
logies. On the other hand, if it is Mackey continuous, then its adjoint (which maps
M1h
into M,:)
is weakly continuous and so
norm continuous. Hence it maps norm compact sets into norm compact sets and so the result follows from 2 . 3 . 5 ) .
If @ is Bscontinuous, then it is #3,gCcontinuous (proof as for Bucontinuity). On the other hand, if @ ( M S ) C MY
and
8,
tinuous, then it is
@
 B S * on MS and M:.
continuous on MS
is starpreserving, then Hence if @ is BsrCOnand the result follows
by considering real and imaginary parts.
214
IV.
VON NEUMANN ALGEBRAS
2.17. Proposition: If M is a von Neumann algebra, then a linear mapping 0 from (M,Bs>ps)
into a separable Frgchet space
is continuous if and only if it has a closed graph.
Proof: By 2.13, it is sufficient to show that the restriction of @ to each commutative von Neumann subalgebra B,continuous. But 0
MI of M is
also has a closed graph and so is
I M1
continuous by 111.1.13.
As mentioned in the introduction, von Neumann defined two locally topologies, the ultraweak and ultrastrong topologies,
on L(H) (and so on any von Neumann algebra) as follows: if
*,...)
X := (x,,x
and
elements in H so that
Y := (y,,y2,.. .) are sequences of < m and C 11 ynll 2 < m then
I I x ~ ~ ~ ~
C
are seminorms on L(H) and the family of all such seminorms (p,)
(resp.
defines a locally convex topology on L(H)

the ultrastronq (resp. ultraweak) topology. We denote these
us
and
uu. Then
C us
and
us = T s
topologies by Similary,
'cS
T,
E uu and uu
= T~ on B
L(H) on BL (HI (see DIXMIER CS] ,
p. 34).
One can also define a symmetric form the seminorms T
cj
pX(T") ) and
on the unit ball. Thus B,,
us*
of
us"' 2
~~g~
us
(by adding
with equality
6 , and Bsfs are stronger than
uu, us
IV.2 and
usB
VON NEUMANN ALGEBRAS
215
respectively. In general, they are strictly stronger
as the following Proposition shows:
2.18. Proposition: Let M be an infinite dimensional von Neumann algebra. Then B,,
8, and
u u , us and u s B are strictly weaker than
respectively.
Bsiy:
Proof: The topology uu on M is exactly o ( M I M A ) and 8,
is the
topology of uniform convergence on the norm compact subsets of Mfe. Hence
uu = 8,
if and only if the norm compact subsets
of Mi,: are finite dimensional and this is the case precisely when
Mg:
(and hence also M ) is finite dimensional.
For the second part, it suffices to show that 8, is strictly finer than
on M S . Since M is infinitedimensional, there
us
is a mutually orthogonal sequence (En) of nonzero projections in M S . Let
{n’/2En}. Then we can claim that 0 lies in
S :=
the usclosure of S since for each X I px(n 1 /2En) 2 1 For if this were not the case, there is an px(n1/2En) > 1
for some n.
X = (xi) so that
i.e. pX(En) > n1’2. But this contradicts the
additivity of the form fx in Mi) defined by X (i.e. in the notation of DIXMIER [ E l ) since then
CuXiryi C fX(En) 2 C l/n.
We shall show that 0 does not lie in the Bsclosure. For each n let xn be a unit vector in H so that

Enxn = xn
and
define fn to be the form
T Then fn
E
Me and
n 1 /2 (TX~~X,).
K := {fnl is relatively compact in
(MS,II
11).
IV.
216
VON NEUMANN ALGEBRAS
Hence U := {T
E
MS : I f n ( T ) I
<
1
f o r each n )
i s a Bsneighbourhood of z e r o i n M S . But
fn(n1/2En) = 1
2.19.
S n U = 4
since
f o r e a c h n.
W"algebras:
A Wf4algebra i s a @  a l g e b r a
(A,
i s , as a Banach s p a c e , t h e d u a l of a Banach s p a c e
11 11)
which
h4.
Two W#:algebras A and B are isomorphic i f t h e r e i s a C?algebra isomorphism between them which i s b i c o n t i n u o u s f o r t h e c o r r e s p o n d i n g weak t o p o l o g i e s u (A,A;.:) and u (B,B,.,)
s e t of p o s i t i v e e l e m e n t s of A;:.
. We
d e n o t e by T t h e
Then, on A , one can d e f i n e t h e
f o l l o w i n g l o c a l l y convex t o p o l o g i e s : o(A,A;:) s(A,A+:)

t h e weak t o p o l o g y ; t h e topology d e f i n e d by t h e seminorms
xt3 s"(A,A$:)

I $ ( x " x ) l 1/ 2
(0 E
TI;

t h e topology d e f i n e d by t h e seminorms
x c, ( 4 (x"tx)1 1 / 2
x
14 ( x x " ) l 1 / 2
(see SAKAI C141, 95 1 . 7 and 1 . 8 ) . Now it i s clear t h a t e v e r y von Neumann a l g e b r a M is a W;':algebra ( s i n c e it i s t h e d u a l of M f c ) . On t h e o t h e r hand, one h a s t h e f o l l o w i n g r e s u l t of SAKAI " 1 4 1 , e v e r y W"algebra
Theorem 1.16.71:
i s isomorphic t o a von Neumann a l g e b r a .
One t h e n h a s t h e f o l l o w i n g r e l a t i o n s h i p between t h e t o p o l o g i e s
VON NEUMANN ALGEBRAS
IV.2
217
of a W"Ialgebra A and the topologies that it possesses by virtue of being a von Neumann algebra:
Thus all our result can be carried over to the context of W"algebras.
2.20. Remark: There are two alternative methods o f defining
mixed topoloqies on W"algebras which we now describe briefly. If M is a von Neumann algebra, a (twosided, normclosed)
ideal
I C M
is essential if Io, the annihilator

of I , is {O}. The family of all essential ideals is denoted by E . E is closed under the operations of sum and intersection i.e.
1,J
E
E
I
+
and
J
I
n J are in E. If
I is an
essential ideal in M I we define the strict topology 6, on M to be that locally convex topology generated by the seminorms:
Then we define generated by
F {B,
to be the inductive locally convex topology : I
E
E}
i.e. it is the finest locally
convex topology on M I which is coarser than each B,, The topology
F
I
E
E.
was introduced by HENRY and TAYLOR in [ l l ] and
their main result can be summarised as follows:

B =
Bs2b
(M,Mft)
if and only if
is a topology of the duality
and this is the case if M is a countable decompo
sable type I algebra.
IV. VON NEUMA“ ALGEBRAS
2 18

Hence, in this case, 8 coincides with p s f t .
The second approach uses the theory of noncommutative integration which has been developed by SEGAL. With this notion, one can define a mixed topology on a Wfralgebrausing methods which are exact analogues of those used in Chapter 111. We recall briefly this theory and refer the reader to NELSON [I21 for details. Let M be a Wfcalgebra. A normal faithful
semifinite trace on M is a mapping
+ (M
: M+
CO,~]
denotes the set of positive elements of M S ) which satisfies
the condition: 1)
T(X+Y) =
2)
T(XX)
= xT(X)
(x
E
M+,
3)
T(X”X)
= T(XX”)
(x
E
M’);

if (x
4)
(we write T(X,)
a
)
xaf x
(X,Y
‘C(Y)
T(X)
1
+
E
M 1;
E
[O,m[);
is an increasing net in M
+
and
x
=
sup Ex,)
to describe this situation briefly) then
T(X);
5)
if
~(x= ) 0 then
6)
if
x
E
those y in M+ with
M+
x
= 0;
there is a net (x,) T (y)
<
m)
in M A (the set of
so that x, 1. x.
Note that the existence of such a trace is equivalent to the assumption that M be semifinite (see SAKAI [14], 2.5.4 and 2.5.7).
on
+ M
If M is finite in the sense that
T
takes finite values
then we can define analogues of the LPspaces as follows:
IV.2
VON NEUMANN RLGEBRAS
(where 1x1 is the absolute value of x
E
219
M ) is a norm on M
its completion. Then ( M , l l II,II P Saks space. We denote by B 1 the mixed topology y [[I and we denote by M
In the general case (i.e. where
‘I
]Ip)
is a
II,I[
.
is not necessarily finite
valued) we can regard M as the Banach space projective limit
+
of the ideals EeMe) as e ranges over the projections in Mo. We can then provide M with a Saks space structure as the Saks space projective limit of
11, I[
(eM e, 11
[Ip)
denote the corresponding mixed topology by B,. that, in this case, 6,
and
BSft
1
and we
We conjecture
coincide once again.
We conclude this chapter with some remarks on von Neumann algebras.
2.21.
Remarks: I.
AARNES 121 has given an interesting order
theoretical characterisation of the topology Bsi.:. Recall that
MS has a natural ordering and so we can define the notion of order convergence for a net in MS and so in M (by considering the real and imaginary parts). Then B,a convex topology
T
is the finest locally
on M for which order convergence is stronger
than Tconvergence. 11.
We remark that the finiteness classification of von Neu
mann algebras can be expressed elegantly in terms of the mixed topologies as follows: 1)
M is finite if and only if B s 
f3sfei
IV.
220 2)
VON NEUMANN ALGEBRAS
M is semifinite if and only if it is the inductive
limit (in the sense of Banach spaces) of a directed set of weakly closed ideals for which 3)
8 , = Bs*;
M is purely infinite if and only if
B s f @,n
on
each weakly closed (nonzero) ideal of M . (cf. SAKAI [ 1 4 ] , Th. 2 . 5 . 6 ) . 111.
2 . 1 4 can be strengthened to give the following result
of VITALIHAHNSAKS type. Let (T,)
be a sequence of Bs*con
tinuous linear operators from M into a complete locally convex

space F so that for each projection P in M , lim Tn(P) exists. Then ITn} is B,<:equicontinuous and there is a Bs,v<continuous linear operator T from M into F so that
Tn(P)
T(P)
for
each projection. For, using 2 . 1 1 and 2 . 7 , we can reduce successively to the case where
F
= C
and M is commutative i.e.
to the classical VitaliHahnSaks theorem (cf. AARNES [ I ] ) . IV.
It follows easily from the results of Chapter I that if
M is a von Neumann algebra, then M is barrelled, bornological
or nuclear for any of the @topologies if and only if it is finite dimensional. A l s o if ( M , l l 2.17,
BSs =
11 I]
11)
is separable, then, by
and this also implies that M is finite di
mensional. Also, if M is semiMonte1 under 6,
or 6,s
then these
topologies coincide with @, and this implies, once again, that M is finite dimensional.
V.
If
(A,]]
11)
is a C"algebra, then (A,II ll,a(A,A')) is a
Saks space. We denote its completion by
( i , l l Il,;)
(i.e.
2 is
the bidual of A). Then involution is a Saks space morphism
IV.3
NOTES
22 1
A
and so e x t e n d s t o a n i n v o l u t i o n on A.
Similarly, multiplica
t i o n can b e e x t e n d e d ( i n two s t e p s ) t o C"algebra
i.Then (&,I[ 11)
is a
a n d , s i n c e i t i s a d u a l s p a c e , e v e n a Wfsalgebra.
i s t h e e n v e l o p i n g W*algebra o f p r o p e r t y t h a t e v e r y C"morphism
A.
It has t h e universal
from A i n t o a W"algebra
b e e x t e n d e d t o a W"algebra morphism on
can
i.
1 1 . 3 . NOTES
A r e a d a b l e i n t r o d u c t i o n t o von Neumann a l g e b r a s h a s been g i v e n
by RINGROSE i n [ 1 3 ] .
Some r e s u l t s from t h i s c h a p t e r w e r e announced i n COOPER 151. The r e p r e s e n t a t i o n o f t h e p r e d u a l of L ( H ) a s N ( H ) i s d u e t o D I X M I E R (who, i n f a c t , u s e d t h e u l t r a w e a k t o p o l o g y ) . The re
p r e s e n t a t i o n t h e o r e m f o r commutative a l g e b r a s c a n b e f o u n d i n 181,
5
1.7 or i n 1141,
5
1 . 1 8 . F u l l d e t a i l s on s p e c t r a l t h e o r y
c a n b e found i n DUNFORD and SCHWARTZ 1 9 1 . The claim t h a t t h e f u n c t i o n a l c a l c u l u s d e f i n e s a Bscontinuous morphism f o l l o w s from f o r m u l a X . 2 . 8 ( i i ) p r o o f o f 2.5
(using 2.4)
n  a l g e b r a homo
t h e r e . The i d e a o f t h e
i s s k e t c h e d by SHIELDS i n 1153 ( u s i n g
a d i f f e r e n t t o p o l o g y ) . P r o p o s i t i o n 2.6 i s due t o DIXMIER ( f o r p o s i t i v e f o r m s ) and SAKAI. The r e s u l t 2.8

2.11 are based on
p a p e r s [ 3 ] and 1 2 1 o f AKEMANN and AARNES. I n p a r t i c u l a r , t h e proof of 2.11
i s t a k e n from [ 3 ] . 2.15 i s a r e s u l t o f SAKAI.
IV.
222
VON NEUMA" ALGEBRAS
For a full discussion of the topologies discussed in 2.18 and 2.19
see [ 8 1 ,
0
1.3 and 1 1 4 1 ,
of 2 . 1 8 is due to YEADON
55
1.7,
1.8.
The difficult part
[ 161.
We mention the following related articles: DAUNS [ 7 1 examines the problem of tensor problems for W"algebras: GUICHARDET
[ 101
studies W"algebras from a category theoretical
point of view: DARST 1 6 1 considers a theorem of VitaliHahnSaks type for the norm dual of M.
223
REFERENCES REFERENCES FOR CHAPTER I V .
[l]
J . F . AARNES
The VitaliHahnSaks theorem f o r von Neumann a l g e b r a s , Math. Scand. 18
(1966) 8792.
[21
[31
On t h e Mackey t o p o l o g y f o r a von Neumann a l g e b r a , Math. Scand. 22 (1968) 87102. C.A.
AKEMA"
The d u a l s p a c e of an o p e r a t o r a l g e b r a , Trans. Amer. Math. SOC. 126 (1967)
286302.
l'41
S e q u e n t i a l convergence i n t h e d u a l of a W"algebra, Corn. Math. P h y s i c 7 (1968)
222224.
151
J.B.
COOPER
Topologies s u r l ' e s p a c e d e s o p 6 r a t e u r s d a n s l ' e s p a c e h i l b e r t i e n , C.R. Acad. S c i . P a r i s A 276 (1973) 1509151 1 .
161
R.B.
DARST
On a theorem of Nykodym w i t h a p p l i c a t i o n s t o weak convergence i n von Neumann a l g e b r a s , Pac. J . Math. 23 (1967) 473477.
[71
J. DAUNS
C a t e g o r i c a l W"tensor p r o d u c t s , T r a n s . Amer. Math. SOC. 166 (1972) 439456.
[81
J. DIXMIER
L e s a l g s b r e s d ' o p g r a t e u r s dans l ' e s p a c e h i l b e r t i e n ( P a r i s , 1969)
[9]
N.
.
DUNFORD, J. SCHFIARTZ L i n e a r o p e r a t o r s . P a r t I1 ( N e w York, 1963).
[I01 A. GUICHARDET
Sur l a c a t e g o r i e d e s a l g s b r e s d e von Neumann, B u l l . S c i . Math. (2) 90 (1966)
4146.
[11]
J. HENRY, D.C.
TAYLOR The Ftopology f o r W"algebras, J o u r . Math. 6 0 (1975) 123139.
Pac.
IV.
224
VON NEUMANN ALGEBRAS Notes on noncommutative integration, Jour. Func. Anal. 15 (1974) 103116.
[I21
E. NELSON
[ 1 3I
J R
Lectures on the trace in a finite von Neumann algebra (in "Lectures on Operator Algebras", pp. 313354, Springer Lecture Notes 247, 1972).
[I41
S. SAKAI
C"algebras and W"algebras (Berlin, 1971 )
[I51
P.C. SHIELDS
A new topology for von Neumann algebras, Bull. Amer. Math. SOC. 65 (1959) 267269.
1161
F.J. YEADON
A note on the Mackey topology of a von Neumann algebra, J. Math. Anal. Appl. 45 (1974) 721722.
. . RINGROSE
.
CHAPTER V

SPACES OF BOUNDED HOLOMORPHIC FUNCTIONS
Introduction: In this chapter we consider the algebra Hm(G) of bounded, holomorphic functions on an open domain G in C. This algebra has a natural Banach algebra structure (with the supremum norm). Following BUCK
[lo] we list some unpleasant
features of this Banach algebra (for the special case where
G is the open unit disc U): 1)
The polynomials are not dense in Hm(U) and, in fact,
Hm(U) is not separable; 2)
principal ideals in Hm(U) are not necessarily closed
and closed ideals are not necessarily principal; 3)
there are maximal ideals of H m ( U ) which are not determined
by points of U. In 1957 BUCK introduced the strict topology B on Hm and showed that it possessed several attractive properties. It was then studied in more detail by RUBEL, RYFF and SHIELDS. In this chapter we show that B is an example of a mixed topology. In section 1 , we deduce the basic properties of B using the theory of Chapter I. In V.2 we specialise to the case where G is the open unit disc. We introduce a new, finer mixed topology B 1 which seems to be the appropriate topology for certain applications and study tensor products and vectorvalued functions. In V.3 we consider (Hm(U), B )
as a Saks algebra and obtain a
characterisation of its closed ideals. We conclude with an application of the theory developed to operators in Hilbert space.
225
V.
226
BOUNDED HOLOMORPHIC FUNCTIONS
Most of the results in this Chapter are known. However, the methods are new.
V.1. MIXED TOPOLOGIES ON Ha
In this section we denote by G an open domain in
C
which
supports a nonconstant bounded holomorphis function. Then the spece Hm(G) of bounded holomorphic functions on G separates G. On Hm(G) we consider the following structures:
(1 11 T
P
rK
the supremum norm;

the topology of pointwise convergence on G;

the topology of uniform convergence on the compact subsets of G.
Then (Ha,II IIrTp) and ( H m r I I
1 1 , ~ ~ ) are
Saks spaces and, since the
unit ball of Ha is a normal family of functions,
T
P
and
T~
coin
cide there. Hence
~ [ l Il l
r ~ ~ =l
~ [ l llI r ~ K 1
and we denote this topology by
@.We
list now some simple pro
perties of @ which follow immediately from the theory of Ch. I.
I . I . Proposition: I 1
( H ~ , B ) is a complete locally convex space;
2)
the @bounded sets of Hm are precisely the normbounded sets;
3)
a sequence (x,)
m
in H
converges to zero if and only if it
is norm bounded and converges pointwise to zero;
V.l 4)
(Hm,B)
space (Hm,B 5)
MIXED TOPOLOGIES ON Hm
227
is semiMonte1 and so is the dual of the Banach )
' , with the topology of compact convergence:
a linear mapping from H" into a locally convex space is
Bcontinuous if and only if its restriction to B the unit Hm ball of Hm, is T continuous (and hence if and only if it is
P
.
Bsequentially continuous)
Now (Hm,B) is the dual of the Banach space F
F := (Hm,B)' and
is separable (since it is a subspace of the separable space
C(BBm) of continuous functions on the compact metrisable space B
Hm
1 . Hence the following properties follow easily from 1.4.13: a subset A of Hm is Bclosed if and only if it is sequentially closed (for A is closed if
A
(7
nBHm is closed for each n
and the latter space is metrisable): B is the finest topology on Hm with convergent sequences
those described in 1.1.3) (follows immediately from I.4.2(c) and the fact that each nBHm is metrisable).
We can regard (Hm(G),11
1 1 , ~ ~ ) as
a Saks subspace of (Cm(G),11
By 1.4.4, (HmlB)is a locally convex subspace of (Cm(G) , B )
11.~~).
and
so 8 is defined by the seminorms P$ : x
IIxbll
as $ runs through the space Co(G). Hence (Hm,B) coincides with the space B (G) studied by RUBEL, RYFF and SHIELDS ( [ 3 0 ] m
In particular, (H , B )
,[31]
)
.
is a topological algebra. Note, however,
that inversion is not 6continuous.
V.
228
BOUNDED HOLOMORPHIC FUNCTIONS
We can also regard Hm(G) as a Saks subspace of (Lm(G),11
11,~)
(G is provided with planar Lebesgue measure) and once again (Hm,B) is a locally convex subspace of (Lm(G),B,).
Hence by
111.1.1 we have
1.2. Proposition: (Hm(G), B )
I
is naturally isometric to the
1
quotient spaces L (G)/N1 and M(G) /N2 where N, and N2 are the polars of Ha) in the appropriate dual spaces.
We note that the above results show that if
II fll where
IT
= inf ilIu11 :
u
E
M(G),
IT(U) =
f
E
(Hm,B) then
f)
is the natural projection from M(G) onto the dual of
Hm. This infimum is attained if and only if f is equivalent to a measure of the form av where a
E
v is a positive measure
C and
(see RUBEL and SHIELDS [31] , Prop. 2.10)
.
Note that by identifying L 1 (G) with a subspace of M(G) in the usual way, we can regard this result as one on "balayage": if p
E
M(G),
E
> 0 then there is an absolutely continuous
measure (i.e. absolutely continuous with respect to Lebesgue measure) v in M(G) so that
uv (where u
and
 v means that l.~
IIVII 5

IIuII
+ E
v vanishes on Hm).
In fact, the result is true with the Eterm omitted (cf. RUBEL and SHIELDS [31] , Prop. 4 . 1 )
.
V.l
MIXED TOPOLOGIES ON Hm
229
1.3. Definition: Let S be a subset of G. S is dominating if,
x
for each
E
Hm(G) ,
i.e. if the restriction mapping
:
ps
x
isometry from H ~ ( G )into ~ " ( s ) .
1.4.

xis
is an
Lemma: If S is a countable, dominating set then p, (Hm)
is Bclosed in L m ( S ) .
Proof: By 1.4.3
and the fact that the unit ball of L " ( S ) is
@metrisable, it is sufficient to show that ps(Hm) is sequentially closed. Let (xn) be a sequence in Hm so that ps(xnl is Bconvergent in Lm(S) to y. Then (xn) contains a @convergent subsequence .(x ) and it is clear that y = ps(lim xnk). "k It follows from this Lemma that p,
is an isomorphism from
(Hm(G), B ) onto a closed subspace of (L"(S1
1.5.
Proposition: Let S be a subset of
G.
,@)
(cf. 1.4.32).
Then the following
are equivalent: 1)
S is dominating;
2)
for every 11
(i.e. 11

E
G \
M(G)
there is a v in M(S) so that 11
v
m
v vanishes on H 1.
Proof: 2 ) ho
E
1 ) : if S were not dominating, we could find a S
and
x
E
Hm(G)
so that
BOUNDED HOLOMORPHIC FUNCTIONS
V.
230
x(Xo)
for X
Ix(X)I I k
= 1,
If S satisfies condition 2) there is a v v
 &Ao
1
__7
I
I
xn
61
0
=
S (k < 1).
M(S) so that
(Dirac measure at lo). Now for each n 1 =
and so 1
E
E
E
IN,
1 xndv
k"Ilvll which is impossible.
2): by going over to a countable dense subset, we can
suppose that S is countable. The result then follows immediately from 1 . 4 and duality.
V.2. THE MIXED TOPOLOGIES ON Hm(U)
In this section, we specialise to function spaces on U, the open unit disc
{A
:
1x1
< 1).
and by aU the boundary
We denote by
the closed unit disc
\ U of U. We recall some notation and
results on Hardy spaces see HOFFMAN 1211, Ch. 3 , 4 for a detailed discussion.
We denote by Hp(LJ) (or simply by Hp) (1
I
p <
m)
the space of
functions x which are analytic on U and are such that the norm llxllp := is finite. (Hp,II),1 Then we have
27
sup ( Ix(reie)IPde)l'P O
is a Banach space.
HcoC Hp C_ H1 (1 < p < co
mappings are continuous (when H
m)
and the natural inclusion
has the supremum norm).
V.2
MIXED TOPOLOGIES ON Hm(U)
231
Each function x in H 1 has a nontangential boundary function,


which is in L'(aU). More precisely, there exists a function 1 E L (au) so that, for almost every eie in aU,
x
X(X)
;(eie)
as
x
eie
(in particular, radially). The mapping
nontangentially Y
x>
x
is an iso
metry from Hp(U) onto a closed subspace of LP(aU). The range of this mapping is the space of those 27
J
0
y(e ie )einede = 0
y
E
Lp(aU)
such that
(n = 1,2,...)
i.e. the nonnegative Fourier coefficient of y vanish. If
y
x
Hp(U)
E
2.1.
E
LP(aU)
satisfies these conditions, then
y
=
where
can be recovered from y via the Poisson integral
Proposition: 1 )
(Hm(U), @ ) is a topological subspace of
(Lm(au), B J ; 2) 1
(Hm(U),@)' is naturally isomorphic to the quotient space
L (BU)/HA
where HA denotes the space of functions in H 1 (U)
which vanish at the origin.
Proof: 1 ) follows from 1 . 4 . 4 and the fact that on the unit ball of Hm, the topology a(Lm(au) ,L1 (au)1.
T
P
agrees with the weak topology
2) Since the dual of (Lm(aU),@,) is L 1 (aU), it suffices to identify the polar of Hm in L 1 (aU) with Ho1 and this is clear.
V.
232
BOUNDED HOLOMORPHIC FUNCTIONS
For the next Proposition, we recall the following definition:
(xn)z in a topological vector space E is a Cesaro basis for E if for each x E E there is a unique sequence a sequence
(6,) of scalars so that the partial sums n
sn :=
C
k=O
5,
Xk
... + Sn) = XI.
1 converge to x in Cesaro mean (i.e. lim n+ 1 (so + 2.2.
(Hm,B)
2)
m
The sequence (znlo is a Cesaro basis for
Proposition: 1 )
where zn is the function
( H ~ , B ) is separable; m
Proof: 1) If x
E
H I let
Then if
n :=
s
C
k=O
Ek
C 0
cnX n
be its Taylor expansion.
... +
1
an := n+l (so +
zkI
Sn)
s dx
in T~ and so anx . On the other n hand, we have the following kernel representation of an:
we have
where Kn is the Fejer kernel t 
1 [I

1
cos nt cos t
(cf. HOFFMAN [ 2 1 ] , pp.16,17 and Ch. 3 ) . Then Ilanll I llxll (since Kn is positive and
l I T
2.rr
IT
Kn = 1 ) .
MIXED TOPOLOGIES ON Hm(U)
V.2
Hence
a
n
d x
233
in B by 1.1.10.
2 ) follows immediately from 1 ) .
It is well known that if E is a locally convex space with a basis (xn) such that the corresponding projection operators n
m
are equicontinuous, then E has the approximation property. The following Lemma is proved similarly.
2 . 3 . Lemma: Let E be a locally convex space with a Cesaro basis
(xn) so that the projection mappings (Pn) are equicontinuous, where Pn maps C Skxk
into the Cesaro mean an. Then E has the
approximation property.
2 . 4 . Proposition: (H ,B 1 has the approximation property.
Proof: By 2 . 3 and 1.1.7 jections (P,)
it is sufficient to show that the pro
are equicontinuous for the topology defined by
the seminorms {pr : 0 < r Pr : x
<
1)
where
sup{lx(reie)l :
e
E
[0,27111
But exactly the same argument as used in the proof of 2 . 2 shows that pr(an) I pr(x)

q.e.d.
V.
234
BOUNDED HOLOMORPHIC FUNCTIONS
.
Now we introduce a new mixed topology on Hm (U) We consider Ha as a Saks space with the supremum norm and the auxiliary
Note that T is induce6 by the norm 11 11 IIp P' II IIp finer than the topology T of pointwise convergence on U (as
topology
T
II
P
one can see, for example, from the Cauchy integral formula) closed  in fact, T II upIIp complete. We define the topology 6, to be the mixed topology
and so the unit ball of Hm is
'I
II
~ [ l l l r r l l 11,l2.5. Proposition: 1 ) 2)
~1 =
~ [ l Il l r T l l
W
m
H
(1 I
P <
m ) ;
is B,bounded if and only if it is normbounded;
4)
A
5)
a sequence (x,)
C_
\Ip]
is a complete locally convex space:
(H , B l )
3)
B , is strictly finer than B ;
in HOD is B1convergent to zero if and only
if it is normbounded and converges to zero in some HPspace (1 4 p
<
a)
(or even in measure on 8U).
Proof: 1 ) B1 is finer than B by the above remarks. It is strictly finer since
z
n
d 0 ( 6 ) but
zn&
0 (B1)
(
(zn) the
sequence of 2.2.1) 1 .
2.6. Proposition: $ and B 1 are topologies of the same dual.
Hence (Hm,f.3
)
is semireflexive.
Proof: By I.I.lB(ii) it suffices to show that if E
>
0, then there is an
?
E
f
(Hm,B) so that \IfTI1
E
(Hm,B1)', I E.
V.2
MIXED TOPOLOGIES ON Hm(U)
By the same result, we can write
f = fl
235
+ f2 where fl is
continuous on Hm, with the norm induced from L1 (aU), and
1 1 f2 11 s
E.
By the HahnBanach theorem, fl is representable in
the form
with
y
E
Lm(aU)
and this form is 8continuous (2.1).
2.7. Corollary: (Hm,B) is not a Mackey space.
We now consider vectorvalued functions. We recall the notion of a holomorphic function with values in a locally convex space. If G is an open subset of C, E a locally convex space, a mapping x : G
+E
is holomorphic if for each 1im
X(A)

X(Xo)
x

xo
x. xo
Xo
E
G
the limit
exists in E. The following result is basic (cf. GROTHENDIECK
2.8. Proposition: Let x be a function from G into the complete locally convex space E. Then the following are equivalent: 1)
x is holomorphic;
2)
for each f
E', the complexvalued function f
E
o
x
is holomorphic; 3)
x is (weakly) continuous and
x(X)dX = 0 for each
closed, simple, nullhomotopic, rectifiable curve 4)
€or each X o
E
r
in G;
G there is a neighbourhood V of Xo in G
V.
236
BOUNDED HOLOMORPHIC FUNCTIONS
and'a closed, absolutely convex bounded subset B of E so that x(V) L EB, the subspace spanned by B, and x is holomorphic as a mapping from V into the Banach space (EBt 11
l B).
Now let (Ell/1 1 , ~ ) be a complete Saks space, x a norm bounded function from U into E. Then the following are equivalent: 1)
x is holomorphic as a function with values in (E,11
2)
x is holomorphic as a function with values in (EIy);
3)
x is holomorphic as a function with values in (E,T);
4)
for each f
5)
x is Tcontinuous (or even a(E,E;)continuous
ir x(X)dX
= 0
E
(E,T)', f
o
x
11)
;
is holomorphic; and
r
for each closed, simple, rectifiable curve
The equivalence of 1 ) and 2) follows from 2 . 8 . 4 )
in U.
since the
norm bounded and the ybounded subsets coincide. The equivalence of 2 ) and 3 ) follows trivially from the fact that y and T coincide on the range of x. 3 ) and 4 ) are equivalent by 2 . 8 . 2 ) spite the fact that (E,T) is not necessarily complete

(de
we can
work in the completion). We denote by Hm(U;E) the space of functions which satisfy one (and hence all) of the conditions 1 )

5). On Hm(U;E) we consider
the structures:
]I 1, T~
to
T)

the supremum norm

the topology of uniform convergence (with respect
on the compact sets of U.
x
c.$
sup { Ilx(X) 11
:
X
E
U)
V.2
IE,~,)
(Hm(U;E),11
Then
MIXED TOPOLOGIES ON Hm(U)
237
is a complete Saks space. We denote
by 8, the associated mixed topology
y [ l l IIE,~E]. Then
m
(H (U;E),BE) satisfies the appropriate forms of properties 1 ) , 2 ) , 3 ) and 5) (without the last paranthesis) of Prop. 1 . I .
We shall now give a tensor product representation of Hm(U;E) similar to that given in 5 11.4 for space of vectorvalued continuous functions. For this result we require the following fact (see GROTHENDIECK [IS]): space and
x
: U
+
E
if E is a complete locally convex
is analytic, then there is a sequence
(an) in E so that m
X(X) =
c
n=O
(X
anhn
E
U).
(in fact, an is given by the formula an = n! x(n) (0)). The convergence is uniform on the compact subsets of U.
If x
E
Hm(U;E) we define, just as for the case of scalarvalued
functions, the functions (sn) and sn : X
cj
1 an := n+l ( s o
2.9.
x
E
Proposition: If (E,ll Hm(U;E), then
an 
by
n k C akX k=O
+
11,~) >
(q,)
... t s,). is a complete Saks space and
X
Proof: We have already seen that
in 5,.
a
n
d x
in
suffices to show that (an) is bounded i.e. that
T
E’ Hence it
(j
nEN
u
XEU
un(X)
V.
238
BOUNDED HOLOMORPHIC FUNCTIONS
is bounded in E. By the uniform boundedness theorem, it suffices to show that for each
f
E
E'
u
I!
f
Y' n e N X ~ U
o
an(X)
is bounded in C i.e. we can reduce to the scalar case which we have already treated in 2.2.
2.10.
Corollary: If we identify the algebraic tensor product
Hm(U)
8
E
with a subspace of Hm(U;E) under the natural (vector
space) isomorphism
then Hm(U) ca E
2.11.
is BEdense in Hm(U;E).
Proposition: There is a natural (Saks space) isomorphism
H~(u;E)= H ~ ( u )
Y
E.
Proof: Having identified Hm(U) e E
with a dense subspace of
Hm(U;E), we need only show that the Saks space structure induced on
Hm(U)
8
E
from Hm(U;E) coincides with the tensor
product structure. This is standard.
We finish this section by quoting two important Propositions. The first states that the dual of (Hm(U), @ ) is sequentially complete and is due to MOONEY who thus settled a longstanding conjecture. The second is a characterisation of the weakly compact subsets of the dual of Hm(U) and is due to CHAUMAT. It is equivalent to the fact that B 1 is the Mackey topology on
V.2
MIXED TOPOLOGIES ON Hm(U)
239
Hm(U). Unfortunately, the only existing proofs are too long and technical to be reproduced here.
2.12. Proposition: Let (fn) be a sequence in (Hm(U), 8 )
I
which
converges pointwise to a functional f on Hm. Then f is 8continuous.
2.13. Proposition: Let K be a bounded subset of the dual L1/(Hm)O of (Hm(U), 8 , ) .
Then the following are equivalent:
1) K is weakly relatively compact;
ITXII
: x E CBHm) = 0. lim sup (inf Ilf C fcK (n. is the projection from L' (aU) onto the dual of H" (U)).
2)
2.14. Corollary: (Hm( U ),B1) is a Mackey space.
Proof: The condition 2) means that for
E
>
0 there is a C
> 0
so that K C C B 1 + cB2 where B1 is the unit ball of the dual of H 1 (U) and B2 is the unit ball of L 1 (aU)/ (HOD)'. Hence the result follows from 1.1.22.
Note that 2.12 and 2.14 imply that (Hm(U),B,) satisfies a closed graph theorem with a separable Fr&chet space as range space.
V.
240
BOUNDED HOLOMORPHIC FUNCTIONS
V.3. THE ALGEBRA H m
As we have already remarked, (Hm(U),f3) is a topological algebra. In this section, we describe the closed ideals of
.
Hm (U) It turns out that these have an especially simple

form
in fact, they are all principal ideals. This classifi
cation is, of course, of intrinsic interest but it also has a number of useful applications. Following RUBEL and SHIELDS, we use it to give a description of the invariant subspaces of a class of Hardy spaces. In the next section, we give an application of the ideal theory to operators in Hilbert space.
We begin by describing the f3closed maximal ideals of Hm. Of course, Hm(U) is a Saks algebra and can be represented as the projective limit of the spectrum {A(K) : K compact in U) of Banach algebras where A(K) denotes the set of continuous

functions on K which are holomorphic in the interior of K. If 1
E
u, :

x
x(X)
is an element of M (Hm). Thus we have constructed an injection T :
x
Y
from
4,
u
3.1. Lemma: Suppose that A
Then
x
:=
:
X
E
A}
into M (H~). Y U is not relatively compact in U.
is not f3equicontinuous.
V. 3 Proof: If
were equicontinuous, there would be an r
and an
E
>
x
E
BII
11
with 1x1
n
E
N so that
I
0
0 so that for each f
x : X
Then
(x)I
241
THE ALGEBRA Hm

There is a Xo
E.
An
2 1/2

If(x)I
on the set (X :
I E
rn 5
x,
E
1x1 E
I
<
<
1
1 / 2 whenever
r) (1.1.7). Choose
A so that
21Xoln 1 1 .
satisfies the above conditions and
contradiction.
3 . 2 . Proposition: T is a homeomorphism from U onto M (HOD).
Y
Proof: We first show that T is surjective: if 41
X
<
Then
1. For
@(zn)
( i d u ) n d 0 in
= $,(zn)
A".
X
MY(Hm), let
0 (id,) where idU is the identity function on U. Note that
:=
111
E
for each n
Hence @
=
E
@
and so
A
n
d 0.
N where zn is the function
since the linear span of
(2,)
is
8dense.
Now T is clearly continuous and is a homeomorphism on the compact sets of U. Hence we need only show that every compact set in M (Hm) is the image of a compact set in U. But this is precisely
Y
the content of 3.1.
Using similar techniques, we can characterise the Saks algebra
.
endomorphisms of Hm (U)
3 . 3 . Proposition: Let @ be an algebra homomorphism from Hm(U)
into itself. Then @ is @continuous if and only if it has the form
x
U into itself.
x
+
where C$ is a holomorphic function from
V.
242
BOUNDED HOLOMORPHIC FUNCTIONS
Proof: The sufficiency is clear. Necessity: let 9 := @(id,). we have
4"

Since (idUIn d 0 in (Hm(U), B )
0 and this is only possible if
Then @ coincides with the mapping
x
x
@(U) C U.
on z ,
$I
(with
the notation of the proof of 3.2) and so on Hm(U) (since this is the @closed algebra generated by the unit and
2,).
It is perhaps not inappropriate to give here a classical result of KAKUTANI which can be reinterpreted as stating that an algebra isomorphism from Hm (U) onto itself is automatically I
@continuous.

3 . 4 . Proposition: Let @ be an algebra isomorphism from Hm(U)
onto itself. Then mapping
@
@
is @continuous and so there is a conformal
from U onto itself
Proof: Let x
Hm (U), X
E
E
SO
that
@ :
x
x
o
4.
C. Then X is in the closure of the
range of x if and only if xh is not invertible in Hm and this is equivalent to the noninvertibility of @ ( X I  X . closures of the ranges of x and if
@(XI
Hence the
are the same. In particular,
9 := @(idU), then the range of 9 is an open subset of a:
(a nonconstant holomorphic function is open!) whose closure is
E
i.e. @(u)
Now choose A,
C E
U.
U, x
E
Hm(U). We show that
and the result will follow. We know that r$(Ao)
E
U
and
THE ALGEBRA Hm
V.3
(id,

+(Xo))
is a divisor of
+
Hm(U). Hence Thus
O
(x)

+(Ao)
x(+ (Ao)
x

x(+(Ao))
is a divisor of vanishes at A.
)
243
in the algebra
O(x)

x(+ (A,)).
which is the required
result.
We now recall some notation (cf. HOFFMAN [21], Ch. 5). A function
x
E
is called an inner function if 1x1
Hm(U)
= 1
a.e. on aU. A n inner function without zeros has the form
where
ci E
C with la1 = 1 and 11 is a nonnegative measure on
[ 0 , 2 1 ~ ] which is singular with respect to Lebesgue measures.
is a sequence (with "multiplicities") in U \ { O } so
If (1,)
that C (1

IXkl) <
converges for each X
then the Blaschke pro=
m
E
zeros are precisely (A,)
U and defines an inner function whose (up to multiplicities). On the other
hand, if x is an inner function and (A,) zeros
(+ 0 )
of x then C (1
 I Akl
)
<
m
is the sequence of and x can be expressed
in the form x(A) = aArs(A)B(A) where s has the form (A), B is the Blaschke produkt (B), r and
ci E
C with la1 = 1.
E
N
V.
244
A function
x
E
BOUNDED HOLOMORPHIC FUNCTIONS
H 1 (U) is an outer function if it has the form
where k 2 0, In k
E
L1 ([0,2~]) and \ a \= 1.
Then k(0) =lu(eie)l a.e. x function x
E
E
Hp(U) if and only if k
E
Lp. Every
H 1 (U) has a canonical factorisation x = x.x 1 0
where xi is inner and xo is outer. xo is the outer function defined by the formula (C) with
k : 0 ++lu(eie)l.
In preparation for our main result on closed ideals, we prove some results on principal ideals generated by inner and outer functions.
3.5. Proposition: If x is an inner function, then xHa is a
Bclosed ideal of Ha(U).
Proof: We show that xHa is 8closed. It suffices to show that if (xn) is a sequence in Ha so that
xxn
xo
for 8 then
x0 is divisible by x. Now {xxn) is norm bounded and hence so is {xn) (since 1x1 = 1
on aU)
. Then
(xn ) has a pointwise con
vergent subsequence (x 1. "k Let y := lim x Then x x n k d xy "k' m x0 = x y E x H .
for 8 and so
3.6. Proposition: Let x E Ha(U). Then xHm is Bdense if and
only if x is an outer function.
V.3
THE ALGEBRA Hw
245
Proof: Sufficiency: let x be an outer function with llxll lnlx(eie)I
Then
h
Let
hn : 0
: 0
= 1.
is integrable and
min (h(0),n)
and (hhn) is a sequence of nonnegative functions which is L I convergent to zero. Hence IIxynII 2 1
and
x
y
n
I 1
pointwise. Thus the constant function 1 lies in the @closure

of xHW and so xHm = HOD. Necessity: if x has a nontrivial inner factor xo, we have xHm C _ xOHW
+ Hw
and so
xH”E
= xOHQ
+ Hm.
We can now prove the main result of this section. We use the following simple Lemma (see HOFFMAN [ 2 1 ] , p. 85):
3.7. Lemma: Let
F be a nonempty family of inner functions.
Then there is a unique inner function x which divides each function of F and is such that each divisor of F is a divisor of x (x is called the greatest common divisor of F).
V.
246
BOUNDED HOLOMORPHIC FUNCTIONS
Proof (sketch): We use the factorisation of an inner function as a product of a Blaschke product and a function of the form (A). The greatest common divisor is the product of the Blaschke
product formed from the common zeros of the functions of F and the function (A) defined by the measure p which is the supremum of the corresponding measures for the functions of F. m
3.8. Proposition: Let J be a nonzero closed ideal of H
and
let xo be the greatest common divisor of the inner parts of .the nonzero functions of J. Then
Proof: We have by
5, the
J
c xoH
m
J = xoH
01
.
by definition of xo. By replacing J
set of functions of the form
reduce to the case xo = 1
x/xo (x
E
J) we can
i.e. we can assume that the greatest
common divisor of the inner parts of the functions of J is 1. We must then show that Let
y
E
J = Hm.
L1 (aU) be such that
We show that
y
E
au
xy = 0 for each x
E
J, we have
au for each n
as a subspace of

(Loo(au) 8 8,)
x
J.
HA = (Hm)O and this will suffice by the bi
polar theorem (we are regarding ( H m , B )
For any
E
F
Xnx(X)y(A)dA = 0
N. Hence, by the Riesz Lemma (cf. HOFFMAN [Zl], p . 4 7 1 ,
there is an HIfunction Hx (vanishing at 0) so that xy = Hx
THE ALGEBRA Hm
V. 3
247
on aU. Hence y agrees (a.e.1 with the nontangential limit of the meromorphic function Hx/x on aU. In particular, Hx/x is independent of x. Hence it is, in fact, analytic (since the x's of J do not have a common zero in U). Thus y is the nontangential limit of an analytic function (which vanishes at zero since each Hx does)

q.e.d.
We now show that the above result can be used to characterise invariant subspaces of several important spaces of analytic functions on the disc. We consider a topological vector space E whose elements are holomorphic functions on U (with the usual algebraic operations) and are of bounded characteristic i.e. expressible as a quotient x/y of two functions x,y in Hm(U). In addition, we suppose that
1)
if El is a closed subspace of E then
closed in 2)
HOD
(U):
if x
E
E l " Hm
is 6

Hm, the mapping x
xy
maps E continuously into itself;
E has a representation ylz where z is outer y2 and y, and y2 are inner functions without a nontrivial common 3)
if x
E
inner factor, then 4)
z/y2
E
Ei
Hm(U) is dense in E.
The Hardy spaces Hp ( 1 5 p < RUBEL and SHIELDS [ 3 1 1 ,
9
m)
5.9).
satisfy these conditions (cf.
V.
248
BOUNDED HOLOMORF’HIC FUNCTIONS
3.9. Proposition: Let E satisfy the above conditions and let El be a closed subspace of E which is invariant (i.e. xE1 C El for each x
E
Hm). Then there is an inner function xo so that
El = xoE.
Proof: We first note that exactly a5 in the proof of 3.6, one can (using condition 1 ) above) show that if xo is an outer function in E l then the constant function 1 lies in the Eclosure of x ~ H ~ .
Now let x be an element of El form

x has a representation of the
y1z/y2 where z is outer and y l I y 2are inner functions
without common factor. Then y,
E
El. For there is a sequence
(x,) of bounded analytic functions so that zxnY1
Then, by 2) 1 YIZXn = XY2Xn
Let X l
=
El n Hm

then
1
in E and so y1
E
in E.
El.
Zl is a 6closed ideal in Hm(U) (by
There is an inner function xo so that
El
m
= xoH
. We
1)).
claim that
El = xoE. El C xoE : choose x
”’
where z is outer y2 and ylIy2 are inner functions without a common factor. Then a)
E
E, and let x =
N
y1 E El (see above) and so y1 E El. Thus there is an inner function y3 so that y, = xoy3. Then x = x0y3z. Now z/y2 E E y2 (by 3)) and so E E. Hence x E xoE. y2 in Hm(U) which b) xoE G E l : if x E E l there is a net

v. 4
converges (in E) to x. Then xox
E,
E
since
xOxc t

249
H~FUNCTIONALCALCULUS
E
El.
x x
oct
xox
and so
3.10. Corollary: The nonzero closed invariant subspace of
Hp ( 1
s
p <
a)
are precisely those of the form xoHP (xo an
inner function).
3.11. Remark: Note that 3.9 does not state that a subspace of the form xoE is closed in E. However, it is easy to see that this is true in the case of the Hardy spaces Hp.
v. 4 .
THE H~FUNCTIONALCALCULUS FOR COMPLETELY NON UNITARY CONTRACTIONS
If T is a contraction in a Banach space (i.e. IlTll 5 1) then the classical functional calculus can be developed for functions analytic on a neighbourhood of the closed unit disc. SZ. NAGY and FOIAH have shown how this can be improved to give a functional calculus for functions in H ~ ( u )for certain contractions in Hilbert space. In this section we give a brief description of some of their results from the point of view of the strict topology. This allows a simpler and more direct approach. We begin with some preliminary results and definitions. A detailed account can be found in Chapter I .of [25].
BOUNDED HOLOMORPHIC FUNCTIONS
V.
250
4.1. Definition: A contraction T in a Hilbert space H is
completely non unitary if there is no nontrivial subspace M of H which is invariant under T and T" and is such that TIM is unitary. We write c.n.u. contraction for a completely non unitary.contraction. For every contraction T there is a unique orthogonal decomposition H = Ho e H,
of H so that Ho and H1 reduce T (i.e. are
invariant under T and T9') and are such that TI and T I
HI
is c.n.u.
(see [251, Th. 1.3.2).
HO
is unitary
Ho is characterised
as the space
T is c.n.u. if and only if
Ho
= {O}.
Our construction uses the following important result:
4.2.
Proposition ([25] , Th. 1.4.1) : Let T be a contraction in H.
Then there is a Hilbert space K which contains H as a subspace and a unitary operator U on K so that
v a,
m
u
n=m
1)
K =
n=m
UnH, the closed subspace of K generated by
UnH; 2)
Tnx = PUnx (x
E
HI n
E
N ) where P is the orthogonal
projection from K onto H.
With the notation of 4.2, we introduce the subspaces
L := (UT)H;
L S ~:= ( u *  ~ ) H
v. 4
H~FUNCTIONALCALCULUS
of K. M(L) (resp. M(L")
)
denotes the closed subspace of K
u UnL n=m m
spanned by
Mo(L) :=
4.3. Proposition
( [25]
251
, Th.
m
(resp. Mo(L")
:=
u n=m
UnLfe).
11.1.1) : L and L* are wandering
subspaces for U, that is
4.4. Proposition ([25], Prop. 11.1.4): Let T be a contraction. Then H~ = (M(L) v M(L>''))1
(where Ho is as in 4.1 and
M(L) v M(L")
linear subspace spanned by
M(L) U M(L") )
Hence if T is c.n.u. then
denotes the closed
.
M(L) V M(L*) = K.
We now denote by Pol(U) the space of functions x
E
Hm(U) which
are the restrictions of polynomials to U. Then, of course, Pol(U) is Bdense in Hm(U) , in fact Pol(U) n BHOD is .rKdense in BHm (2.2). Thus if we regard Pol(U) as a Saks space with the structures
(I] l l f ~ K ) , Hm(U)
1.3.6. Now if
p
E
is its completion in the sense of
Pol(U), and T is a contraction on H I we denote

by p(T) the operator obtained by formal substitution of T in p. Then cp : p
p(T)
is an algebra homomorphism from Pol(U) into L(H). In the following, we shall show that if T is completely nonunitary, then
252 @
BOUNDED HOLOMORPHIC FUNCTIONS
V.
is @Bacontinuous (resp. @l@scontinuous) and so can be
extended to a continuous algebra homomorphism from Hm into L(H)

4.5. Lemma (von Neumann's inequality) : If p a contraction then IlpCT)1 1
I
E
Pol(U) and T is
llpll. In other words
@ is norm
bounded (and, in fact, IlcPII 5 1).
Proof: Let U be the unitary operator of 4.2. Then'of course Ilp(LJ)II 5 llpll. On the other hand, p(T)x = Pp(U)x (x E H) (by
4.2.2)) and so IlpCT)11 5 IlpCU)11.

If T is a contraction, U as in 4 . 2 , function
x is analytic on
((1
E
((1

U)
1
XI
Q: \ 101;
the following equality holds IIP(U)Xll = IIPIl2
IIXII.
Proof: 1) We use the expansions m
n=O
H the
u1l xly)
Mo(L) (resp. Mo(L")).
the function X
analytic extension to 2)

E
@.
4.6. Lemma: Suppose that x,y 1)
then for each x,y
m
xln(Unxly) =
c
n=O
xln (xI u"zny)
y)
Then
has an
v. 4
H~FUNCTIONAL CALCULUS
253
which have only a finite number of nonzero terms since L

(resp. L") is a wandering subspace for U. n C akXk, we have 2) If p is the polynomial X k=O
4.7. Lemma: Let Q be a norm bounded linear mapping from Pol(U) into L(H). Suppose that M1 and M2 are subspaces of H which are invariant under Q(p) for each p
E
Pol(U) and are such that
M 1 U M2 is total in H. Then 1)

if for each x,y P
E
M1 (resp. M2) the mappinq
(@(P)XlY)
is 8continuous, @ is BBucontinuous; 2)
if for each x
P
E
M1 (resp. M2) the mapping
@(PIX
is 81continuous, Q is B1Bscontinuous.
Proof: 1 ) The hypotheses imply that Q is BBUcontinuous when regarded as a mapping with values in L(M1) (resp. L(M2)). Let PMl denote the orthogonal projection onto M1. Then if x y
E
M2r p
and so
E
Pol(U), it follows that
E
M,,
(@(p)xly) = (Q(p)xlPy)
V.
254
BOUNDED HOLOMORPHIC FUNCTIONS
is 8continuous. The case
x
and so @ is @B,continuous weak topology y induced by
E
M2, y
E
M1
is handled similarly
on the unit ball of Pol(U) since the
M1u
M2 coincides with the weak topo
logy on BL(H). The proof of 2) is similar. Lemma: Let T be a c.n.u. contraction. Then the mapping
4.8.
CP : p
p(T)
from Pol(U) into L(H) is B@,
and B1f3,
continuous.
Proof: @ is normbounded by 4 . 5 .
, p x,y
be a sequence in Pol(U) so that
continuity: let (p,)
68,
in 8 . Since
0 E
H
(pn(T)xly) = (pn(U)xly)
it will be sufficient to show that pn(U)
in (L(K),B,).
Applying 4 . 7 with
only show that if
+
(p,(U)xly)
~T~(P~(u)xIY)
x,y
E
+
M1 = M(L), M2 = M(L"), we need
0. But then
J
pn(X) ((XU)'xl y)dA
= J
pn(X) ((AU)lxly)dX
=
IXI=Dl
and the latter converges to zero since + n p
uniformly on
0
Mo(L) (resp. Mo(L")) then
I X I=r
(by 4 . 6 . 1 ) )
for each
E h : 111 = r}.
The B1Bscontinuity is proved similarly using 4 . 6 . 2 ) .
0
v. 4
H~FUNCTIONALCALCULUS
255
4.9. Proposition: Let T be a c.n.u. contraction on H. Then
+
there is an algebra homomorphism 0 : Hm(U)
L(H)
so that 1)
O(p)
2)
0 is
3)
if T is normal, @(x)
=
p(T)
for
11 1 1  1 1 11,
p
E
Pol(U);
and BIBscontinuous;
68,
=
x(T)
in the sense of the
functional calculus for normal operators: 4)
@(XI =
P
X(U)
o
on H.

Proof: By 4.8 and the remarks before 4.5, we can extend the operator Q : Pol(U)
,L(H) to a ~  8continuous linear U
operator from Hm(U) into L(H). The properties 2 )

4 ) can be
deduced easily.
4.10. Remark: If x is an element of Hm(U), it is natural to
.
denote cp (x) by x (T) Property 4.9.2) provides two natural methods of calculating x(T). If 0 < r < 1 , denote by x r the function
+
x(rX). Then xr is analytic on a neighbour
hood of is and so xr(T) can be defined by the classical functio

3 x in (Hm(U),B1 1 and so nal calculus. x r x(T) = s  lim xr (TI (the strong limit). This was used as the r+ldefinition of x(T) by S2.NAGY and FOIAB. Alternatively, if n C akhk then x E Hm has the Taylor expansion k=O
x(T) where
n s
n =
C
k=O
= s
akTk
.

1 lim n+l (so n+.m
+
... +
sn)
V.
256
BOUNDED HOLOMORPHIC FUNCTIONS
A c.n.u. contraction T (SO) on H is of class Co if there is
a nonzero x
E
Hm(U) so that x(T)
= 0.
Let
J := {x E Hm(U) : x(T) = 0 ) .
Then J is a nonzero, @closed, proper ideal of Hm. By 3.8 it has the form xoH where xo is an inner function. xo satisfies the following properties: 1)
xo(T) = 0;
2)
if y
E
Hm is such that y(T) = 0 then xo is a divisor
of y.
xo is called the minimal function of T and denoted by mT. It plays a role corresponding to that of the minimal polynomial of a matrix (i.e. finite dimensional linear operator) and contains useful information on T. As examples we give two results which follow easily from the theory developed here.
If x is an inner function with factorisation of the form
(cf. the remarks before 3.5) we define the generalised zero set Z(x) to be the union of the following three sets: 1)
if
the zeros of the Blaschke product (together with 0
r 2 1);
2)
the accumulation points of the sets 1 ) ;
3)
the (closed) support of the singular measure in the
canonical representation of s (see equation ( A ) after Prop. 3.4).
v. 4
H~FUNCTIONALCALCULUS
257
(For some remarks on the functiontheoretical significance of this set see HOFFMAN [ 2 1 ] ,
4.11.
pp. 68,69).
Proposition: If T is a c.n.u. contraction of class Co
with minimal function mT, then the spectrum u ( T ) of T is given
by the equation
Proof: (i)
a(T)
C_
Z(mT) : suppose that .A
E
AA.1 [mT(Ao) 
x : A
f \ Z (mT). Then rnT(~)]
is in Hm and s o , "substituting TI', we have (AoI

T)x(T) = x(T)(AoI = mT(Xo)I


T)
mT(T)
= mT(Ao)I.
Hence
A.
I$
U(T).
(ii) Z(mT) n U
C a(T) :
mT(A) =
let
xA.
A.
E:
U n Z ( x ) . Then we can write
xi ( A )
where xi is inner. Substituting T once again gives 0 = (T
and so
(T

xoI)
a divisor of xi !

AoI)xi(T)
is not injective since xi(T) f 0 ).
(mT is not
V.
258
BOUNDED HOLOMORPHIC FUNCTIONS
(iii) Z(mT n aU E a(T) : suppose that Xo is in p(T) n aU ( p ( T ) := C \ u(T) is the resolvent of T ) . Then by part (ii) of the present proof, Xo is not an accumulation of the zeros of the Blaschke product of mT (since p(T) is open). The proof will then be completed if we show that for any open arc W in p(T)
n aU,
p ( W ) = 0 where is the singular measure involved
in the canonical representation of the singular part of mT. If this is not the case, there is a closed subarc K of W with p(K)
> 0. Then the inner function
is a nonconstant divisor of mT and so is
x2
:= mT/xl, x2 is
inner and x1 (T)x2(TI
=
mT(T) = 0.
xl(T) is thus not injective (for otherwise we would have x2(T) = 0 which would imply that mT is a divisor of x2). Hence the subspace HI := (y
E
H : x1 (T)y = 0)
is a nonzero invariant subspace for T. Let
,
TI := T
'
.
Then HI the minimal function of T I , is a divisor of mT and so has
mT1 the form
where p l is a nonnegative singular measure which is bounded above by p on K and vanishes outside of K . Thus
Z(mT 1 1
K
v. 4
H~FUNCTIONALCALCULUS
259
and so by part (i) of the present proof, o(T1) c Z(mT
)
1
C K.
However, it is easy to see that
K E p(T1) and so
a(T1) C p(T1) a contradiction. Hence
w(W)
= 0
and the proof is finished.
4.12. Proposition: If T is a c.n.u. contraction of class Co and x is an inner divisor of mT then Ho := {y
E
H : x(T)y
= 0)
is invariant for T.
As a Corollary, we obtain the following partial result on the famous invariant subspace problem.
4.13.
Corollary: If
dim(H) > 1
then every c.n.u. contraction
of class Co on H has a nontrivial invariant subspace.
The proof of 4.12 is obvious (in fact, one can show that Ho is hyperinvariant, that is invariant for every operator which commutes with TI. The Corollary follows immediately if mT has a nontrivial factorisation. Otherwise, it has the form
for some la1 T = XoI
= 1,
X0
E
U. An easy calculation shows then that
and so has nontrivial invariant subspaces.
V.
260
BOUNDED HOLOMORPHIC FUNCTIONS
V.5. NOTES
As mentioned in the introduction to this chapter, the topology 6 on Hm was first introduced by BUCK
[lo]. It was systematically
studied by RUBEL and SHIELDS 1 3 1 1 and RUBEL and RYFF [30]. The results of section 1 can be found in these papers (usually with different proofs).
The results and methods of
5
2 are mostly new

in particular,
the topology P 1 seems to be new. The fact that f3 is not the Mackey topology is due to CONWAY [13]. This result has been generalised to more general domains by RUBEL and RYFF. BIERSTEDT [6] and [7] has obtained tensor product representations for
Hm(Ux U) and considers the approximation property using different methods. BIRTEL [8] and BIRTEL, DUBINSKY [9] have considered Banach space tensor products of Hmspaces. MOONEY's theorem (2.12) seems to have been conjectured originally by A.E. TAYLOR
(see PIRANIAN, SHIELDS, WELLS [26]). For proofs see
AMAR 1 1 1 and MOONEY [241. BARBEY 121 and 131 has given a version
of this theorem in the setting of spaces of abstract analytic functions. CHAUMAT's theorem is in [12]. The topology 8, can be defined for domains G more general than U. It is necessary that the boundary of G be rectifiable and regular enough that Hmfunctions have (nontangential) boundary values and be recoverable from then. A nice discussion of this problem can be found in ZALCMAN [371.
V.5
8
NOTES
261
3 is based on the work of RUBEL and SHIELDS. Concerning the
main result (3.9) (which is originally due to SRINAVASAN), they write: "we prove the result here by using Beurling's characteri2 It would be sation of the closed invariant subspaces of H
....
good to find a direct intrinsic proof". The proof given here is adapted from the proof of the corresponding result for the space of functions in Ha which have continuous extension to 3 (a result due to BEURLING and RUDIN

see HOFFMAN [211, Ch. 6). The theorem (3.4) of KAKUTANI was published in [221. we have taken the proof from HOFFMAN 1211, p. 144. a
U, the yspectrum of H (U), is naturally embedded in the spectrum of a Banach slgebra (Ha,11
11) .
The famous problem whether U is
a dense subspace (the "Corona problem") was solved positively by CARLESON in [ I l l .
In
9
4 we have given a brief introduction to part of the theory
of completely nonunitary contractions in Hilbert space. The theory is developed in detail by SZ.NAGY and FOIAF in [251. We have used some ideas of TONTHATLONG 1361.
Further references on (Ha,B) are BARTELT (141 and [51), and SHAPIRO (1331, [341 and [351).
There are three recent survey articles on spaces of bounded analytic functions

GAMELIN [ 1 7 1 , RUBEL [291 and SARASON 1321.
V.
262
BOUNDED HOLOMORPHIC FUNCTIONS
The paper of ZALCMAN [ 3 7 1 also contains an excellent account of current work on algebras of analytic functions.
In
$5
2 and 3 we have considered only functions on the open
unit disc in order to simplify the presentation. However, many of the results can be extended to more general regions by simple methods. For example, 2 . 1 1 holds when U is replaced by a region which is the finite union of disjoint, simply connected domains (as BIERSTEDT has shown). The question of whether these results hold for general regions G can lead to complicated analytic questions. For example, RUBEL and RYFF give a fairly detailed account of extensions of Proposition 3.2. RUBEL and SHIELDS have introduced the notion of inner and outer functions for general regions (even in higher dimensions) but little seems to be known about them.
REFERENCES
263
REFERENCES FOR CHAPTER V.
[I]
E. AMAR
Sur un theoreme de Mooney relatif aux fonctions analytiques bornges, Pac. J. Math. 49 (1973) 311314.
[2]
K. BARBEY
Ein Satz tiber abstrakte analytische Funktionen, Arch. der Math. 26 (1975) 521 527. Z u m Satz von Mooney fur abstrakte analytische Funktionen (Preprint)
I. 31
[ 41
.
M. BARTELT
Approximation in operator algebras on bounded analytic functions, Trans. Amer. Math. SOC. 170 (1972) 7183.
[ 51
161
Multipliers and operator algebras on bounded analytic functions, Pac. J. Math. 37 (1971) 575584.
K.D. BIERSTEDT
Injektive Tensorprodukte und SliceProdukte gewichteter R h n e stetiger Funktionen, Jour. reine angew. Math. 266 (1974) 121131. Gewichtete Raume stetiger vektorwertiger Funktionen und das injektive Tensorprodukt 11, Jour. f. reine angew. Math. 260 (1973) 133146.
171
[81
F. BIRTEL
191
F. BIRTEL, E. DUBINSKY Bounded analytic functions in two complex variables, Math. 2 . 93 (1966) 299310.
[I01
R.C. BUCK
Slice algebras of bounded analytic functions, Math. Scand. 21 (1967) 5460.
Algebraic properties of classes of analytic functions, Seminars on analytic functions 11, 175188 (Princeton, 1957).
264
V.
111
L. CARLESON
Interpolation by bounded analytic functions and the corona problem, Ann. of Math. 76 (1962) 542559.
[I21
J. CHAUMAT
Une g6nsralisation d'un thborsme de DunfordPettis (Sem. Analyse Harmonique d'Orsay No. 85 1974).
[
BOUNDED HOLOMORPHIC FUNCTIONS

[ 131
J.B. CONWAY
Subspages of (C(S),B , the space (Loo,@ and (H , B ) , Bull. Amer. Math. SOC. 72 (1966) 7981.
[ 141
J.B. COOPER
The Hmfunctional calculus for completely nonunitary contractions, Math. Balkanica 4.15 (1975) 8990.
[ 151
F. DELBAEN
Weakly compact sets in H1 , Pac. J. Math. 63 (1976) 367369.
1161
T.W. GAMELIN
Uniform algebras (New Jersey, 1969)
.
The algebra of bounded analytic functions, Bull. Amer. Math. SOC. 79 (1973) 10951 108.
1171
[ 181
A. GROTHENDIECK
Sur certains espaces de fonctions holomorphes, I,II, J. reine angew. Math. 192 (1953) 3564, 7795.
1191
V.P. HAVIN
Weak completeness of the space L 1 /HA (Russian), Vestnik Leningrad Univ. 13 (1973) 7781.
[20]
H. HELSON
Lectures on invariant subspaces (New York, 1964).
[21]
K. HOFFMAN
Banach spaces of analytic functions (New Jersey, 1962).
[22]
S. KAKUTANI
Rings of analytic functions, Lectures on functions of a complex variable, 7183 (AnnArbor, 1955).
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A. KERRLAWSON
A f i l t e r d e s c r i p t i o n of homomorphisms o f H m , Can. J . Math. 17 (1965) 734757.
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M . MOONEY
A theorem on bounded a n a l y t i c f u n c t i o n s , Pac. J. Math. 43 (1972) 457463.
[251
B.
[261
G. P I R A N I A N , A.L.
1271
M . von RENTELN
SZ.Nagy,
C.
Harmonic a n a l y s i s of o p e r a t o r s on H i l b e r t s p a c e (Amsterdam, 1970).
FOIAF
SHIELDS, J . H . WELLS Bounded a n a l y t i c f u n c t i o n s and a b s o l u t e l y c o n t i n u o u s m e a s u r e s , P r o c . Arner. Math. SOC. 18 ( 1967) 81 8826. F i n i t e l y s e n e r a t e d i d e a l s i n t h e Banach a l g e b r a H", C o l l e c t . Math. 26 (1975)
1151 25.
1281
J.R. ROSAY
[291
L.A.
RUBEL
[301
L.A.
RUBEL, J.V. RYFF The bounded w e a k  s t a r t o p o l o g y and t h e bounded a n a l y t i c f u n c t i o n s , J . F u n c t i o n a l Anal. 5 (1970) 167183.
[311
L.A.
RUBEL, A.L.
Une e q u i v a l e n c e du Corona p r o b l s m e e t un probldme d ' i d e a l s dans H m ( D ) , J o u r . Func. A n a l . 7 (1971) 7184. Bounded c o n v e r g e n c e o f a n a l y t i c f u n c t i o n s , B u l l . Arner. Math. SOC. 77 (1971) 1324.
SHIELDS The s p a c e o f bounded a n a l y t i c f u n c t i o n s i n a r e g i o n , Ann. I n s t . F o u r i e r
16 (1966) 235277.
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J.H.
SARASON
SHAPIRO
Algebras of f u n c t i o n s on t h e u n i t circles, B u l l . Amer. Math. SOC. 79 (1973) 286299. Weak t o p o l o g i e s on s u b s p a c e s of C (S), T r a n s . Amer. Math. SOC. 157 (1971)
471 479.
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V. J.H. SHAPIRO
BOUNDED HOLOMORPHIC FUNCTIONS The bounded weak star topology and the general strict topology, J. Func. Anal. 8 ( 1 9 7 1 ) 275286.
Noncoincidence of the strict and strong operator topologies, Proc. h e r . Math. SOC. 35 ( 1 9 7 2 ) 8181.
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TONTHATLONG
Sur le calcul zonctionnel d'une contraction completement non unitaire, Proc. Func. Anal. Week, 2636 (Aarhus, 1 9 6 9 ) .
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L. ZALCMAN
Bounded analytic functions on domains of infinite connectivity, Trans. h e r . Math. SOC. 1 4 4 ( 1 9 6 9 ) 241269.
APPENDIX
Introduction: In this appendix we assemble some results on Saks spaces from a categorytheoretical point of view. It is our opinion that this approach puts a number of the results of the previous Chapters in their proper perspective. The leitmotiv is the establishment of a duality theory for various important categories.
In the first section, we define formally the category of Saks spaces, list some of its simple property and identify its dual category. We then use these categories to establish in
5
2 a
duality theory for uniform spaces and for compactologies. The latter are a generalisation of the compact spaces and allow us to give a symmetrical form of the duality theory for completely regular spaces established in Chapter 11. In
5
3 we consider
some important functors and bifunctors on categories of Saks spaces and their duals.
In the fourth section, we use the duality theory of
5
2 to give
a duality theory for certain classes of semigroups and groups.
In the last section we describe an extension process forcategories which formalises the transition from Banach spaces to Saks spaces and demonstrate thereby that this is a special exapmle of a construction which includes, for example, the
267
268
APPENDIX
generalisation from Banach spaces to locally convex spaces or to convex bornological spaces.
A.1. CATEGORIES OF SAKS SPACES
1.1.
The categories PSS, SS and CSS: We introduce three cate
gories
PSS, SS and CSS
PSS

with the following objects:
triples ( E , l l 11,~)where ( E , I I
11)
is a normed space,
r is a locally convex topology on E and B unit ball of ( E , l l

SS
triples ( E , l l
11,~)
assumption that B CSS

11)
II II'
the
is Tbounded;
as above with the additional
II It
is rclosed and
r
=
v[ll 1 1 , ~ l ;
as for SS with the additional assumption that (E,T) be complete.
In each case, if (E,11 II,r), (F,1 1
Ill
,rl) are objects of one of
the above categories, the morphisms from E into F are the linear contractions T : E
F
so that TiB
I I II
is rrlcontinuous.
1.2. Proposition: SS and CSS are full, reflective subcategories of PSS.
Proof: The reflection from PSS into SS is obtained by first employing the procedure described after 1.3.2 BII
II by
(i.e. replacing
its rclosure) and then replacing r by y [ l l 11,rI. The
reflection from PSS into CSS is obtained by composing this functor with the completion functor (1.3.6).
A.l
CATEGORIES OF SAKS SPACES
269
We remark that all of the above categories are complete and cocomplete. Products and equalisers in PSS are formed as described in 1.3.5 and 1.3.7. The construction of sums and coequalisers is a little more complicated and we shall not describe it explicitely here. The various limits in SS and CSS are constructed by first forming them in PSS and then reflecting down into SS and CSS respectively.
1.3. The categories PMW and MW: An object of PMW is a triple
(E,1 1 I1,B) where (E, 11 E so that each B
E
11)
B is
is a normed space, B is a bornology on
11 11
bounded and, in addition, if B
there is a locally convex topology
‘ I ~ on
normed space EB so that
B)
TBcompact. If
(EB,
11 llB,
E
8,
the corresponding
is a Saks space with B
(E,ll ] 1 , B ) , (E1,II I I l , B 1 )
are objects of PMW,
a PMWmorphism from E into E l is a linear normcontraction T from E into El which satisfies the following condition: For each B T
If
B
B , there is a B1
E
B1 so that T(B)G B, and
is a Saks space morphism from EB into ( E
“B
(€,I1
E
11,?3) :=
B1
.
is an object of PMW, we put
{C C ! E : for each
E
>
0 there is a B
E
B so that
C C B + E B MW denotes the full subcategory of PMW whose objects satisfy the conditions B =
% and
(E,II
11)
II I?
(€,[I
l1,B)
is a Banach space.
In addition, we require the existence of a (separated) locally
270
APPENDIX
convex topology on E which coincides with object of MW is called a
1.4.
Examples: I.
on each B. An
T~
CoSaks space.
If (Ell/1 1 , ~ )
is a Saks space, then its dual
space E' has a natural PMWstructure ( 1 1 l l , B ) where 11 11 is the Y Y dual norm and B is the family of absolutely convex, weakly
Y
closed yequicontinuous subsets of El
Y'
If B
E
8
Y
then
T~
is
defined to be the weak topology a(E',E). Y Then (E;,ll l l , B is even an object of MW by 1.1.22. In particu
Y
lar, if
S
is a completely regular space, then the spaces Mt (S),
MT (S) and M,(S) 11.
of measures on S, are CoSaks spaces.
Let S be a completely regular space. Cm(S) has a natural
) where 11 11 is the supremum norm and 8 equ equ is the set of uniformly bounded, pointwise closed equicontinuous
MWstructure
(11
Il,B
subsets of Cm(S). For B
E
Bequ, the topology
T~
is that of point
wise convergence on S. 111.
We can generalise I1 as follows: let S be a uniform space
and denote by Um(S) the vector space of bounded, uniformly continuous functions from S into C. Then Um(S) has a natural CoSaks space structure defined as above (replacing "equicon
.
tinuous" by "uniformly equicontinuous") If S is a completely regular space and we regard it as a uniform space with the
fine
uniformity (i.e. the finest uniform structure compatible with its topology), then the CoSaks structures defined in this section and in I1 coincide.
CATEGORIES OF SAKS SPACES
A.l IV.
If (E,II
11)
271
is a Banch space, we can regard E as a CoSaks
space by defining 8 to be the family of all absolutely convex, compact subsets of (E,"
11).
This allows us to regard the cate
gory of Banach spaces as a full subcategory of MW.

1.5. The dual of a CoSaks space: If ( E , I I 11,B) is an object of PMW, we define its dual E ' to be the space of all linear functionals f : E
restrictions to each B
which are normbounded and whose
Q:
E
B are TBcontinuous. Then E ' has a
( 1 1 1 1 , ~ ~ ) where 11 11
natural Saks space structure norm and
'cB
is the dual
is the topology of uniform convergence on the sets
of B .
1.6. Lemma: Let f be a linear functional on the CoSaks space (E,"
1 1 , B ) . Then f is normbounded (and so an element of
provided flB is TBcontinuous for each B
Proof: If f were not bounded on B
(x,)
in B
II II
so that
If (x,)
I
II II'
2 2 n
Proposition: If ( E , l l 11,8)
is an object of CSS.
E 8.
we could find a sequence
. Now
in 8 and hence f is bounded on this set
1.7.
E l )
{xn/n) is in

ff and so
contradiction.
is a CoSaks space, then ( E ' , l l
11,~~)
APPENDIX
272
are obviously functorial (if we map morphisms into their adjoints) and so we have defined duality functors P and D where
0 : CSS
1.9.
___j
MW
and
D : MW
j . CSS.
Proposition: CSS and MW are quasidual under the functors
D and D. Proof: The proof that D and D are mutually adjoint on the left is standard. It then suffices to show that the natural transformation from D
o
P
to the identity functor is equivalent to
the identity or, speaking loosely, that
DPE = E
for each
complete Saks space. This follows from Grothendieck's completeness theorem.
A.2. DUALITY FOR COMPACTOLOGICAL AND UNIFORM SPACES
In 5 11.2 we established a form of GelfandNaimark duality for W
completely regular spaces. Because the space (C (S), B K )
is not,
in general, complete, this duality is not perfect and in this section we show how a complete duality can be obtained by replacing the category of completely regular spaces by that of compactological spaces. We also develop a duality theory for uniform spaces using CoSaks algebras.
A.2
DUALITY
273
Definition: A compactology on a set S is a family K of
2.1.
subsets of S, together with a compact (Hausdorff) topology T
K on each K
K so that
E
1)
K is closed under finite unions and covers S;
2)
if K 1 C K where K
E
K then K 1
is TKclosed. If this is the case
T
K~ 
A compactological space is a pair ( S , K )

K if and only if K 1
E
T
~
)* ~
l
where K is a compacto
logy on S. The class of compactological spaces becomes a category when we define morphisms from (S,K) f : S that
S1
f (K) E; K 1
so that for each K and f l K is
into (S1,K1) E
to be mappings
K there is a K1
E
K1
so
T ~  Tcontinuous. ~
1
If S is a Hausdorff topological space, S has a natural compactology K ( S ) (define T~ to be the induced topology). This construction defines a forgetful functor from the category of Hausdorff topological spaces into the category of compactological spaces.
If
(S,K)
is a compactological space, we denote by
Cm(S)
the space
of bounded morphisms from S into 6! (with its natural compactologyl
. Then
(Cm(S), 11
mum norm and
T~
1 1 , ~ ~ ) is
a Saks space where
11 11
is the supre
is the topology of uniform convergence on the
sets of K. In fact, (C"(S),ll
1 1 , ~ ~ ) is
even a Saks C"algebra
(and, in particular, complete).
S is said to be regular if C m ( S ) separated S; of countable type if K possesses a countable
APPENDIX
274
basis K1 (i.e. every K
E
K is subset of some K1
K1).
E
We denote by RCPTOL the full subcategory of regular compactological spaces.
2.2. Proposition: Let (S,K) be a compactological space. Then
the following statements are equivalent: regular;
1)
( S , K ) is
2)
there is a completely regular topology on S so that K
is the induced compactology; the restriction operator
3)
jective for each K
E
K.
Cm(S)

C(K)
is sur
Each of these conditions is satisfied if K is of countable type.
Proof: 1 )
+2 ) :
the weak topology on S induced by Cm (S) satis
fies the required condition. 2) d 3 ) follows from Tietze's theorem. 3)
1 ) : let s,t be distinct points of S. Then {s,t}
E
K.
Let x be the function t1
s0 ,
from {s,t} into
ct.
Then an extension ';i of x in Cm(S) separates
s and t.
Now let S be of countable type

suppose that (K,)
is an in
creasing basis (i.e. KnC_ Kn+l for each n). We verify condition 3 ) .
A.2 Choose K
E
K , x1
suppose that K
=
E
DUALITY
C(K). It is no loss of generality to IIxl1 1
K1 and
5 1.
tively a sequence (xn) where xn IlxnlI I 1
275
E
Then we can choose inducC(K,),
xnflIKn = xn and
for each n. Then the xn define an extension of xl.
The equivalence of 1 ) and 2) of 2.2 might suggest the conclusion that there is no difference between completely regular spaces and regular compactologies. This is not the case, the point being that if
and S1 are completely regular spaces, there are
S
more compactological morphisms from S into S1 as there are topological ones.
2.3. The spectrum of a Saks algebra: If (A,1 1
11,~)
is a commuta
tive Saks algebra with unit, then M (A), the yspectrum of A,
Y
has a natural compactology: the sets of K are yequicontinuous subsets and, for each K topology a(A;,A).
E
K , T~ is the restriction of the weak
Then M (A) is even regular. It is obvious

Y
that the correspondences S
Cm(S)
and
A
My(A)
are functorial i.e. we have constructed functors Cm (from the category of compactological spaces into the category CSC" of commutative, Saks Cfcalgebraswith unit) and M
Y
(from the cate
gory of commutative Saks algebras with unit into RCPTOL).
2.4.
Proposition: CSC" and RCPTOL are quasidual under the pair
of functors
COD
and M Y'
APPENDIX
276
Note that 2.4 means that
COD
and M
Y
are right and left adjoint
to each other and that the corresponding unit and counit (i.e. the generalised GelfandNaimark transform and generalised Dirac transformation) are isomorphisms. The proof is exactly that of 11.2.2 and 11.2.5 with the suitable changes.
We now establish a duality theory for uniform'space. The dual object of a uniform space S will be the CoSaks space UOD(S) described in 1.4.11. We repeat the details. We denote by D the family of all bounded, uniformly continuous pseudometrics on S. Um(S) denotes the vector space of all bounded, uniformly continuous complexvalued functions on
s.
11)
(U"(S),ll
is a Banach
space (even a Cfralgebra). H denotes the family of all uniformly
.
bounded, uniformly equicontinuous subsets of Urn(S) We remark that a set B is in H if and only if there is a
B is majorised by d (i.e. Ix(s) s,t
E

x(t) I
I
d
d(s,t)
E
D
so that
for x
E
B,
S). The sufficiency of this condition is clear. On the
other hand, if B
E
H, then
is a suitable majorant in U.
If we supply each B
E
H with the topology of pointwise (or
compact) convergence, then ( U m ( S ) ,11 II,H)
is a CoSaks space.
In addition, it is a commutative algebra with unit and so we can define the CoSaks spectrum MC(Urn(S)) to be the subset of the dual of U m ( S ) consisting of the unitpreserving multipli
A.2
DUALITY
277
cative functionals. We regard MC(Um(S)) as a uniform space with the structure of uniform convergence on the sets of H. We can define, in the obvious way, a mapping 6 from S into Mc(Um(S) )

the generalised Dirac transform.
2.5. Proposition: MC(Um(S)) is a complete uniform space and 6 is a uniform isomorphism from
S
into MC(Um(S)).
Proof: The statement about the completeness of MC(Um(S) 1 is clear (for example, MC(Um(S) 1 is closed in the dual of the CoSaks space Um(S) and this is a complete locally convex space).
The 6mapping is injective since Um(S) separated S . Now, by its very definition, the uniform structure of MC(Um(S) ) is generated by the pseudometrics
where B runs through H. Now
aB induces the pseudometric
on S and the uniform structure of S is generated by
{d, : B
E
BI.
For, as we have already seen, each dB is in V and, on the other hand, if d
is in H and
E
V then
d = dB.
APPEND IX
278
We shall now show that S is dense in MC(Um(S)) so that we can identify MC (Um(S)1 with the uniform completion of S.
2.6. Lemma: Let
s
E
f
E
MC(Um(S)). Then
a)
f is positive:
b)
if x 2 0, f(x) = 0, then
c)
for each
S so that
x,,
. . .,xn
sup /xi( s ) i=l , ,n
.
0
E
5);
in Urn(S) and

f (xi)I <
>
E
0 there is an
E.
Proof: a) This is standard (for example, we can regard f as an element of the spectrum of the C"algebra Urn(S)1. b) If 1x1 2 c) Put
x
for some
E
:=
C
f(x) = Cf(lxi for each
E
E
> 0, then If(x)l
2 f(E) =
[xi  f(xi) 1 . Then we have: f(x)

f(xi)I) = C If(xi)
> 0, there is an
s
E

E.
= 0
(for
f(xi)l = 0). Hence, by b),
S so that
x(s) <
E.
2.7. Lemma: Let B be a realvalued set in H I f an element of
sup B
Mc(Um(S)) which vanishes on B. Then tinuous and
is uniformly con
f(sup B) = 0.
Proof: We replace B by the set of functions obtained by adding the suprema of finite subsets. It follows easily from the characterisation of uniform equicontinuity given after 2 . 4 that the new set is still in H . A l s o if x,y
E
B, then
f(sup{x,y))
(2.6) and so f vanishes on the extended set. But then
= 0
sup B
a pointwise limit and its uniform continuity follows from a
is
A.2
DUALITY
classical result. Also f vanishes on
279
sup B
because its
restriction to B is continuous for the topology of pointwise convergence.
2 . 8 . Lemma: Let
f
E
S so that
there is an s
E
Proof: If x
B, put
E
Ix(s)
yx := inf (1, 1x Then
B 1 := {yx : x
E
B)


f(x)I <
f (x)1
E
HI
>
0
for each x
E
MC(Um(S)). Then for each B
E
E
B.
)
satisfies the conditions of 2 . 7 and
so f vanishes on sup B. This implies the result.
2 . 9 . Proposition: The Dirac transformation is a uniform iso
morphism from S onto a dense subspace of
MC(Um(S) )
.
2 . 1 0 . Corollary: The Dirac transformation is an isomorphism from
S onto MC (Urn (S)) if and only if S is complete.
Hence, in general, the above method yields a functional analytic construction of the completion of a uniform space. The above results establish a duality between the category of complete uniform spaces and a full subcategory of the category of CoSaks star algebras

namely those which have the form U m ( S ) for some
uniform space S. Unfortunately, in contrast to the situation for compactologies, there is no indication of how one could give an internal characterisation of such algebras.
APPEND IX
280
The analogy between the above results and those of Chapter I1 make it natural to consider the dual of
(Um(S),
11 11,
H) as the
natural space of measures on S (natural in the sense that it is intimately connected with the uniform structure of S). We de
.
note this space by Mm (S) As the dual of a CoSaks space, it has a natural Saks space structure
(11 1 1 , ~ ~ ) .
We denote the
corresponding locally convex structure by @ H' Note that Mm(S) is a subspace of M ( B S ) , the space of Radon measures on BS, the Stoneeech compactification of S (i.e. the Banach algebra spectrum of u ~ ( s )).
2.11. Proposition: 1 )
(M~(s)
is a complete locally convex
space: 2)
the @,,bounded subsets of
Mm(S)
are precisely the norm
bounded sets; 3)
a linear mapping from M ~ ( s ) into a locally convex space is
@,,continuous if and only if its restriction to the unit ball of 4)
MO(S)
is eHcontinuous;
the dual of (Mm(S),f3,,)
2.12.
Remarks: I.
is
Um(S).
Once again, one can relate the uniform pro
perties of S to the lineartopological properties of
Mm(S)
. As
simple examples of this phenomenom, we have: a)
S is discrete if and only if Mm(S) is a Banach space
(and in this case Mm(S) is just L 1 (S)); b)
S is pseudocompact (i.e. every uniformly continuous
function on S is bounded) if and only if
Mm(S)
is semireflexive
A.2
281
DUALITY
(or a Schwartz space); c) S is finite if and only if Mm(S) is nuclear. A s well as the above CoSaks structure on U m ( S ) , there are
11.
natural Saks space structures which can be defined in a manner similar to that used in Chapter 11. For example, we can consider the auxiliary topologies
T~
and
T~
(uniform convergence
on compact, resp. bounded subsets of S resp.). The corresponding dual space will be spaces of measures on S, the corresponding spectra will represent extensions of S.
2.13. Remarks on completely regular spaces: Every completely
regular space has a natural uniform structure

the fine
structure. In fact, we can identify the category of completely regular spaces with the full subcategory of fine uniform spaces (a uniform space is
fine if
it possesses the finest uniformity
compatible with its associated topology). Hence the above constructions can be applied to completely regular spaces so that the theory of Chapter I1 could be regarded as a special case of a theory for uniform spaces. We note that for a completely regular space, the measure space Mm(S)
in
5
introduced here coincides with the space Mm(S) defined 1 1 . 5 . In fact, every CoSaks space E has a natural locally
convex structure

the finest which agrees with
T~
on each B
E
I t is clear that the locally convex dual of E coincides with
its CoSaks dual. In the case of
Cm(S) = U m ( S ) (S a completely
8.
APPENDIX
282
regular space, respectively, a fine uniform space), the topology 8, defined in 11.5 can be shown to be exactly the locally convex structure associated with the CoSaks structure on Ca(S).
2.14. Remark: In the light of the above duality for compacto
logical spaces, it is natural to consider the ydual of Cw(S) (S a regular compactological space). Since the definition of
tight measures on a completely regular space involves only its compactology, the definition of 11.3.1 can be carried over to define Mt(S) and, once again, Mt(S) is identifiable with the ydual of Cm(S). However, it can easily be seen that the space of tight measures on the compactological space S is the same as the space of tight measures on the associated completely regular space (i.e. S with the weak topology defined by Cw (S))
. One
im
portant point to keep in mind is the fact that if S is a completely regular space which we regard also as a compactological space, then, although the corresponding spaces of tight measures are the same, the weak topologies need not coincide (and indeed this will be the case exactly when
S
is a kmspace).
A.3. SOME FUNCTORS
3.1. The functors At
and
Mt: In 2.14 we constructed a functor
Mt from RCPTOL into MW which takes a compactological space S into the space Mt(S) of tight measures on S (Mt(S) is given a CoSaks structure as the dual of Cm (S))
. There is also a natural
A.3
28.3
SOME FUNCTORS
functor from MW into RCPTOL, defined as follows: we associate to the CoSaks space
( € , / I 11,B)
the unit ball B
II II of
(E,II
11)
with the compactology defined by the sets of B which are contained in B
This correspondence is functorial and we denote
I I 11
the functor by At.
3 . 2 . Proposition: Mt is a left adjoint to At.
Proof: Let S be a regular compactological space, E a CoSaks space. We must establish a natural isomorphism between Hom (Mt(S),E) Clearly, any T
E
and
Hom (S,At(E)).
Hom (Mt(S),E) defines an element of Hom ( S I B
It II)
by restriction (we regard S as a subset of Mt(S) by the.6transformation). On the other hand, if x is a CPTOLmorphism from S into B
II I I
we can extend x to a morphism
Tx : Mt (S)_ _ _ _ j
E
by defining Tx(u) = where
lxdp
xdu
is the weak integral defined by the formula
3 . 3 . The functors Am
and
Mm:
Similarly, we can define a pair of
functors Mm : UNIF Am : CSS
A
CSS
+ UNIF
APPEND IX
284 m
where M
ascribes to a uniform space S the measure space
M”(S) described in
9 2 and Am associates to a Saks space its
unit ball with the uniform structure induced by As
T.
for 3.2, one can prove:
3.4. Proposition: Mm is a left adjoint for A”.
3.5. Remarks: I.
We can, as usual, deduce certain continuity
properties of the above functors from the adjointness. 11.
If we compare 3.4 with the corresponding result for Banach
spaces (see BUCHWALTER [S] , 8 1.2.20) we see that Mm is a continuous analogue of the 1’functor. In particular, every complete Saks space is a quotient of an Mm(S)space. 111.
Proposition 3.2 and 3.4 can be interpreted as stating that
the injections S
3
M”(S)
and
S 
3
Mt(S)
have
certain universal properties (which we shall not make explicit here) , in other words, that Mm(S) and Mt(S) are the free Saks (resp. CoSaks) spaces over S.
A.4. DUALITY FOR SEMIGROUPS AND GROUPS
In
5
A.2 we have extended the GelfandNaimark duality for compact
spaces to regular compactologies. HOFMA” has given a duality theory for compact semigroups and groups based on GelfandNaimark duality and we are now in a position to extend this duality
DUALITY FOR SEMIGROUPS AND GROUPS
A.4
285
to a large class of (not necessary abelian) groups and semigroups which includes locally compact groups and semigroups.
4.1.
Definition: A compactological semigroup is a semigroup in
the category of compactological spaces i.e: where S
S
it is a pair (S,m)
is a compactological space and m is a multiplication on
under which
S
is a semigroup so that m is a compactological
morphism from S x
S
into
S.
The compactological semigroups
form a category (the morphisms are the compactological morphisms which preserve multiplication). A compactological grouE is a compactological semigroup (S,m) with
unit so that (S,m) is a group and inversion is a compactological morphism. A compactological group (semigroup) is regular if its underlying compactology is regular. We can thus form three categories SGRCPTOL
SGRCPTOL,
and
GRCPTOL
whose objects are, respectively, regular compactological semigroups, semigroups with unit and groups.
In order to motivate the following definition, we reformulate that of a compactological group in terms of commutative diagrams: a compactological group is a quadruple (S,m,e,i) where S is a compactological space and m :SxS
+
S,
e : 1.1
+ S and
i :
S dS
are CPTOLmorphisms so that the following diagrams commute:
APPENDIX
286
idS m
sxsxs m x idS
I
s i s
{.I
s
x
.s
m
e x idS
s
d
> sxs
sxs
idS x e
idS x i
ix idS
> sxs <
lm
sxss
d
I
( { * I is the onepoint set i.e. the final object in CPTOL, d is the diagonal mapping).
In order to describe the dual objects for the above categories, we simply reverse the arrows in
CSCfr (the category of
commutative Saks Cfcalgebraswith unit).
4.2.

Definition: A Saks C*cogebra is a commutative, complete
Saks Cfralgebrawith unit (E,11 c : E
L
E ray E
so that
Il,.r)
together with a CSC*rnorphism
A.4
DUALITY FOR SEMIGROUPS AND GROUPS
commutes. A counit
for a Saks C"cogebra is a morphism q : E
so that
E commutes.
G
Y
E

idE
287

C
E6 c Y
ra 17
A Saks C"cogroup is a Saks C*cogebra with counit toggther
with a morphism that the diagram
(
a :E
E
(the antipodal mapping) so
(a,idE) denotes the canonical morphism from
E
$ E, the
CO
Y product in CSC" into E, defined by the mappings a and id,).
APPENDIX
288
We denote by
SC"COG, SC"COGC and SC$'COGR
the categories of
Saks C"aogebras , Saks C'icogroups with counit and Saks C"'semigroups resp. We leave to the care of the reader the task
of untangling the suitable morphisms.
4.3.
Examples: If (S,m,e,i) is a compactological group, then
Cm (S) is a Saks C"cogroup under the operations
i.e.
m
c = C (m), 11
m
C"(e), a = C (a). (Here we are reverting
=
to the normal terminology for groups i.e. we have written st for m(s,t), e for e(.), s 1 for i(s)). Similar remarks hold for semigroups (with a unit). It is clear that the functor Cm lifts to a functor from SGRCPTOL
into
SC"COG
SGRCPTOLe
into
SCtkCOGC
GRCPTOL
into
SC"COGJ3.
On the other hand, if (E,c,~,a)is a Saks C*cogroup, then

M (E) is a compactological group under the operations Y m : (f,g) e :
.
> I
a : f Then we can regard M
(x
f
QD
g(c(x)))
TI
(x
Y
as a functor from
f(a(x))).
A.4
289
DUALITY FOR SEMIGROUPS AND GROUPS
SC~~COG into
SGRCPTOL
S C ~ S C O G ~into
SGRCPTOL~
SC"C0GR
into
GRCPTOL.
Combining 2.4 with the above remarks, we get the following duality theorem:
4.4. Proposition: The functors Cm and M SC"COG
and
SGRCPTOL
SC"COGC
and
SGRCPTOL,
SC"C0GR
and
GRCPTOL.
Y
induce a duality between
4.5. Remark: The locally compact groups (semigroups etc.) can be regarded as a full subcategory of GRCPTOL etc. so that 4.4 contains a duality theory for locally compact groups, semigroups and semigroups with unit. The dual objects are recognised by their perfectness (cf. 11.2.6).
Of course, every topological
group has a natural compactology under which it is a regular compactological group. However, one cannot get a duality for topological groups from 4.4 (the problem lies once again in the fact that there are more compactological than topological group morphisms between two topological groups). One
can get
a duality
theory if one is prepared to accept as morphisms those group morphisms which are continuous on compacta but this is of course just another way of saying 4.4 and it seems better to use the language of compactologies from the beginning.
290 4.6.
APPENDIX
The Bohr compactification: As an example of the useful
ness of this duality we give a (conceptually, at least) very simple construction of the Bohr compactification. The reader will observe that it is formally exactly the same as the construction
5
of the Stonetech compactification given in recall, we formed the dual object (C"(S),ll
11.2 where, we
1 1 , ~ ~ ) of
S, forgot
the mixed structure and passed over to the dual object M ( C m ( S ) ) of the Banach algebra Cm(S). In the case of groups we require an intermediary step since it is not quite so easy to forget the mixed structure of a Saks cogroup. Essentially the same construction works for semigroups (with or without unit) or groups. For the sake of simplicity of notation, we consider semigroups. Let (E,c) be a Saks C'"cogebra. In general, the compact space M(E) (i.e. B ( M (E))) is not a semigroup. The problem lies in
Y
the fact that the associated &'algebra
E need not be a cogebra
(since c(E) need not be contained in the C"tensor product E which is, in general, smaller than E
A
6
Y
E)
. We can circumvent
this difficulty with the following Lemma:
4.7.
Lemma: Let (E,c) be a Saks C"cogebra. Then there is a

largest C$:subalgebra i? of E with the property that
 *8 E.
c(i?) C_ E
g, with the induced norm, is a C': cogebra.
The assignment E
E
f
Is functorial.
*
E
A.4
DUALITY FOR SEMIGROUPS AND GROUPS
291
Proof: For each ordinal a we define a subalgebra Ea of E inductively by Eo := E Ea := c1 (EB $ E ) B E, := EB B
+
( a = f3
n
1)
( a a limit ordinal).
Then the family {Ea) is eventually stationary. We denote its limit by
z. Then 5
is a C"subalgebra of E with the desired pro
perties.
The functor
U :
E

E
is now the required forgetful
functor from SC"C0G into C"COG, the category of C"cogebras. We can now define a functor B := M o U
m
o
C
from SGRCPTOL into SGCOMP, the category of compact semigroups. If S is a compactological semigroup, B ( S ) is called the
Bohr
compactification of S. There is a natural morphism js : S
___j
B ( S ) , the adjoint of the embedding from 1
into Cm(S). j s has dense image but need not be injective.
4.8.
Proposition: B ( S ) has the following universal property:
every morphism from S into a compact semigroup T factorises over js.
Proof: If
9
: S
___j
C"(9)

T
: C(T)
is a SC*COGmorphism. Since
is a morphism, then Cm(S)
UC(T) = C(T)
and U is a functor,
APPENDIX
292
acting on morphisms by restricting them, we see that C"(I$)
e
maps C(T) into C (S) and the result follows by dualising.
4.9.
Remarks: I.
Almost the same construction produces Bohr
compactification functors for compactological semigroups with unit or compactological groups. We remark that in the case of groups, in the proof of Lemma 4 . 7 we replace c by the mapping
c
(id,
:=
11.
OD
a)
o
c
to obtain the C"cogroup
C E.
As far as we know, this construction of the Bohr compacti
fication is more general than those which have appeared until now in the literature. If one compares the above construction with the standard ones, then one can see that C
m can usually
be concretely identified with a space of almost periodic functions on
S.
We close this section with some remarks on the convolution algebra of a semigroup and on PONTRYAGIN duality in the light of the above duality theory.
If (E,c) is a Saks C"cogebra then the dual E' of E has a Y natural multiplication, defined as follows: if f,g E E; then f
fr
g
is the form
Then E' is a Banach algebra under this multiplication. Y It is commutative if c is cocommutative i.e. if the diagram
A.4
DUALITY FOR SEMIGROUPS AND GROUPS

293
E
commutes where the horizontal arrow denotes the flip operator (x
8
y
y
Q)
XI.
If E is the dual object Cm(S) of the
semigroup S then E is cocommutative if and only if S is commutative.
NOW if
E is the cogebra
crn(s)(S a
compactological semigroup)
then E' Y is Mt (S), the space of bounded Radon measures on S. The Banach algebra structure of Mt(S) described above is that of ordinary convolution. For if
x
E
Cm(S), .p,v
E
Mt(S) , then
Now suppose that (E,c,q,a) is an object of SC"C0GR. x
E
E is
called strongly primitive if a)
c(x) = x e x;
b)
rl(x) = 1;
c)
a(x) = x'
(the inverse of x in E).
We denote by P(E) the set of strongly primitive elements of E.
APPENDIX
294
4.10. Proposition: P(E), with the topology and multiplicative structure induced from ( E , y ) , is a topological group. It is a subset of the unit sphere of E.
Proof: P(E) is closed under multiplication for if x,y then
c(xy) = c(x)c(y) = (x
b)
n(xy) = r)(x)~(y)= 1.1 = 1. a(xy) = a(x)a(y) = x1 y 1 = (xy)1 (E is commutative).
c)
Q
y) = xy
P(E) ,
a)
8
x) (y
E
8
xy.
The constant function 1 is a unit for P(E). If x
E
P(E) , then a(x) = xl is an inverse of x. Hence P(E) is
a topological group.
=
1Ix
8 XI( =
Ilc(x
we conclude that
4.11. Definition: Let S be a commutative compactological group.
A character on S is a GCPTOLmorphism from S into the circle group T. The set $ of all characters form a group. S l with the topology of uniform convergence on the compacta of S I is a topological group (in fact, a complete topological group). If S is locally compact, then this notion coincides with the classical one.
4.12. Proposition: Let S be a commutative, compactological group. Then
2
= P(Coo(S))
(as topological groups).
A.5
EXTENSIONS OF CATEGORIES
IIx~/~
295 and so x takes
Proof: If x
E
its values
n T. It is trivial to verify that the conditions
a)

c) of
P(Cm(S)), then IIxII = 1
=
he definition before 4.10 ensure that x is a group
morphism. Since y =
T
K On
BII I l l
it follows that the topologies
coincide.
4.13. Remark: Using the duality of 2 . 1 0 we could presumably establish a duality theory for semigroups and groups in the category of uniform spaces (note that a semigroup in UNIF is just a topological semigroup while a group in UNIF is a topological group in which the left and right uniformities coincide). For this purpose it would be necessary to identify a tensor product in MW which defines the coproduct on the category dual of UNIF.
A.5. EXTENSIONS OF CATEGORIES
A central theme of this book could be regarded as the attempt
to extend a classical theory for some restricted category to a larger category (for example, the GelfandNaimark duality between compact spaces and commutative C"algebras with unit was extended to a duality between compactologies and Saks algebras). In each case, the extended category could be regarded as a kind of completion of the original category in the sense that its objects were projective (or inductive) limits of spectra in the smaller category. Of course there is a general construction lurking in the background and we describe this now.
APPENDIX
296
Spectra over categories: Let A be a category, A a directed
5.1.
set regarded as a category in the usual way. A spectrum in A over A is a covariant functor i from A into A . A ) for such a spectrum. If
We write (i : A
will sometimes be convenient to write E,
8 it
c1 5
for i(a) and iaB for
We shall thus sometimes write the 8' spectrum more explicitly in the form
the morphism from E,
If
E = {i
ClB
:
into E
E,+
ED, a,B
E
A)
and
are spectra over A , a pre: F _ _ _ j F6, y:6 E B} Y6 Y mapping from E into F is a pair (r,{T,)) where r is an increasing
F = {j
map from A into B and T, is a morphism from E,
into Fr(,)
so
that, for a I 8 , the diagram
commutes.
If G
E (=
(s,{U
(=
(k :
B
(i : A.
r
+
___3
A ) ) , F (= (j : B
A))
A))
are spectra in A and (r,{T,I)
and (resp.
1 ) ) is a premapping from E into F (resp. from F into G )
then we define the composition of (r,{T,}) the premapping (t,{Val t:=sor
and (s,{UB)) to be
where and
V, := Ur(a)Ta.
A.5
EXTENSIONS OF CATEGORIES
297
We have clearly defined a category whose objects are spectra in A and whose morphisms are premappings. The category that we require is a quotient of this category. Two premappings (r,IT,)
)
and (r
,IT;)
commutes for each a and each y with
)
are equivalent if
y 2 r(a)
and
y 2 r' (a).
It can then easily be seen that this is, in fact, an equivalence relation. A mapping between two spectra is an equivalence class of premappings. One can check that this equivalence relation is compatible with composition so that compositions of mappings are welldefined. Hence we have a new category whose objects are spectra in A and whose morphisms are mappings. We denote it by Spec(A)
.
Dually, we can define cospectra over A as contravariant functors from A into A. These are denoted by explicitly, by {.rrga
: EB

(TI
:
A) or, more
A
Ea, a , @
E
A, a I 8 )
The corresponding category is denoted by Cospec(A).
APPENDIX
298
If, in the above construction, we consider only spectra over a fixed directed set A we obtain categories CospecA(A). In the special case where
A = N with its usual
Specu(A), Cospecu (A).
ordering, we write
We can regard A as a full subcategory of
A
E
Spec(A) by identifying
A with the natural spectrum over a onepoint set.
5.2. Proposition: 1 ) 2)
SpecA(A) and
Spec(A) has inductive limits;
A is dense in Spec(A) (i.e. every object in Spec(A) is the
inductive limit of a spectrum in A); 3)
if A has finite sums and coequalisers, then so does Spec(A).
Hence in this case Spec(A) is complete.
The proof of 5.2 is simple but rather tedious to write out specifically. Of course, there is a dual version for Cospec(A).
5.3.
Examples: I.
Let COMP denote the category of compact (Haus
dorff) spaces. We show that Spec(C0MP) is naturally identifiable with CPTOL, the category of compactological spaces. To do this, we construct functors F : Spec (C0MP)
as follows: if
CPTOL,
M = iia8 : Ka
G : CPTOL +Spec (COMP) K8, a
5
8, a,B
E
A)
is
a spectrum of compacta, we denote by F ( M ) its inductive limit in CPTOL. On the other hand, if ( S , K ) is a compactological space, we define
G(S)
to be the spectrum
A.5
EXTENSIONS OF CATEGORIES
IiK,K, : K
M K1, K , K 1
E
299
K, K L KII
in Spec(C0MP). It is clear that F and G are functorial and that they define an isomorphism from Spec(C0MP) onto CPTOL (i.e. F o G
and
G o F
are naturally equivalent to the appro
priate identity functors). We remark here that Spec(C0MP) is strictly larger than COMP despite the fact that inductive limits exist in COMP. This is generally true and is due to the fact that morphisms between inductive limits of spectra say in COMP do not factorise over the components as in CPTOL. In the following examples we shall identify Spec(A) and Cospec(A) for concrete A without specifying the details as above. In each case, it is clear how the isomorphisms are constructed. 11.
The category CLOCCONV of complete locally convex spaces
can be identified with Cospec(BAN1. This result follows from the well known fact that a complete locally convex space has a canonical representation as a projective limit of a spectrum of Banach spaces so that every morphism into a Banach space factorises over an element of the spectrum. 111.
We have the following identities between categories of
Saks spaces: CSS
=
Cospec(BAN1)
CSA = Cospec (BALG) CSC*= Cospec (cc")
where CSA, BALG, CSC" and CC* denote the categories of complete
APPENDIX
300
Saks algebra, Banach algebras, commutative Saks C"algebras and commutative Cf'algebras respectively. The category of complete convex bornological spaces can be
IV.
identified with Spec(BAN). The category MW can be regarded as Spec(W) where W is the category of Waelbroeck spaces (i.e. the subcategory of CSS consisting of those objects (Ell[1 1 , ~ ) B
II
V.
c compact

see BUCHWALTER [S] )
with
.
The category of complete uniform space is identifiable with
Cospec(M1 where M is the category of metric spaces (with contractions as morphisms).
5.3.

Extensions of functors: Let A and 8 be categories, F a co
variant functor from A into B. If in A over A, then F
o
i
i : A
A
is a spectrum in 8 over A. If (r,{Ta))
A) into a
A is a premapping from a spectrum (i : spectrum (il : Al ____) A) then (r,{F(T
a
from
F
o
i
into
F
0
is a spectrum
)))
is a premapping
il. The association
preserves equivalence of premappings and composition and so F induces a functor, which we denote by Spec(F), from Spec(A) into Spec(B). Similary, F induces a (covariant) functor Cospec(F) from Cospec (A) into Cospec ( B )
.
Of course a similar construction can be carried out for a contravariant functor G from A into 8. Then G induces functors
A.5 Spec (GI and
EXTENSIONS OF CATEGORIES from
Cospec (G) from
Spec (A)
301
into
Cospec (B)
Cospec (A) into
Spec (8).
5.4. Examples: Amongst many possible examples, we mention only
that the duality functors from CPTOL into CSC$: (resp. from CSS into MW) discussed earlier in this Chapter are the extensions of the functors from COMP into CC" (resp. from BAN, into W) in the sense of 5 . 3 .
5.5. Lemma: Let E be an object of A, F
a spectrum in A over A. Then spectrum in SET and
=
{ia B : Fa
{HomA(E,Fa)IaEA forms an inductive
Hom
spec (A ) with its inductive limit.
(E,F) is naturally identifiable
Proof: We remark first that composition by i
induces a mapping
a8
from Hom(E,Fa) into Hom(E,F mappings that a
E
A and
Ta
E
B
FBI
)
and it is under these linking
{HomA(E,Fa)}
forms an inductive spectrum. If
HomA(E,Fa)
then Ta defines a premapping from

E into F and every premapping has this form (recall that we are identifying E with the spectrum
E
over the onepoint
set). Hence there is a natural mapping from the disjoint union onto the set of premappings from E into F. Now clEA HomA(E,Fa) the SETinductive limit is the quotient of this direct union under the equivalence relation: TaUB
if and only if there is a
so that
iay
o
y 2 $,a
Ta = iBY o U B
APPENDIX
302
and for premappings of this special form this is just the equivalence relation which defines morphisms in Spec(A).
5.6.
Lemma: Let IE
.
Spec (A) Then
tiaB : E
=
a
{HornSpec (A) ( E a , F ) )
(IE,IF) Horn Spec (A 1 its projective limit.
a5
:
1 , IF be objects of 5 forms a projective spectrum
is naturally identifiable with
in SET and
Proof: We note that {i
F E
E
a
7 E 1 can be regarded as
5
an inductive spectrum in Spec(A) and IE is its inductive limit there. Hence the result is merely a restatement of the universal property of inductive limits.
5.7. proposition: Let IE and IF be objects of Spec(A1. Then
5.8. Proposition: Let
G : A
___3
B
and
H : 8
A
be an adjoint pair (with G adjoint on the left). Then Spec(G)
.
is leftadjoint to Spec (H)
Proof: We have an identification
for each object E of A and object F of 8, the identification

being natural in E and F. Now we have, for IE = {ias : EaF = tjy6 : F
Y
E5,
a,@ E A}
F A , y,6
E
B)
A.5
EXTENSIONS OF CATEGORIES
303
the equations
Of course, there are numerous variations of 5.7 and 5.8 obtained by permuting Spec and Cospec and introducing contravariant adjointness. They can all be obtained from the above results by noting e.g. that Cospec(A) = (Spec(AoP))OP.
5.9. Remark: Using similar techniques, it would presumably be possible to give an abstract duality result which would include for example the transition from the classica1,GelfandNaimark duality to that given in 2 . 4 i.e. to show that functors which establish a duality between A and 8 extend to a duality between Spec(A) and Cospec(Z3). However, care must be taken since the example CPTOL = Spec(C0MP)
shows that a separating duality for
A need not induce a separating duality on Spec(A). It is presumable necessary to restrict the duality to some reflective subcategory of Spec(A) for which the duality is separating (in our example, to the subcategory RCPTOL of CPTOL).
5.10. Remark: Using the construction of Cospec and Spec, together with those of semigroup in and dual to the category, one can construct from a given category a whole range of new categories. For example, the "family trees" of the categories BAN and BAN,
look like this:
CSALG
The arrows are labelled by words or initials which indicate the process applied to the source to obtain the target. The labels have the following meaning:
Af
D
> B
:
A and
A
SG
>8
:
B is the category of semigroups over A.
8 are dual categories;
A.5
EXTENSIONS OF CATEGORIES
305
A subscript " 1 " means that morphisms are assumed to be contractions.
We have given titles only to those categories which have been mentioned in this monograph. However, many of the nameless categories are of importance.
It is thus remarkable how many of the categories which have proved so important in the recent development of functional analysis can be obtained from (essentially) one category, that of Banach spaces. Similar tables can be constructed with, for example, C*:algebras, metric spaces or compact spaces as the "mother category".
A.6. NOTES
BUCHWALTER
[ a ] , using ideas of WAELBROECK, identified the cate
gory dual to that of Banach spaces. We have extended this result in the natural way to Saks spaces. In [ 9 1 he also established a duality theory for compactologies which is essentially equivalent to that given in
5
2. The latter has the advantage that
we can describe explicitly the sum (i.e. the tensor product) in the dual category so that multiplication is preserved by the duality. 2 . 2 is Th&or$me 1 . 1 . 3 of [ 9 ] . For the duality theory for uniform spaces see, for example, PUPIER 1 3 4 1 . Results analogous to 2 . 1 1 for completely regular spaces can be found in
APPENDIX
306
ROME [351. Measures on uniform spaces have been studied in detail by AZZAM [I], BEREZANSKY [61 and DEAIBES [151. In 8 3 we have extended some parts of section 111.1 in BUCHWALTER [81 to Saks spaces. 5 4 is based on MICHOR [30] and COOPER and MICHOR 1141. The idea of using GelfandNaimark duality as a basis for a duality theory for compact groups was used by HOFMANN [20]. In
8
5 we have followed COOPER [13]. Similar con
structions have appeared elsewhere in the literature (see, for example, MARDESIC
[ 281 )
.
As general references for this chapter we recommend Mac LANE 1271, SCHUBERT 1361 and SEMADENI [371 (category theory) and ISBELL [22] and [23] (uniform spaces). BARR [2]

[4] also considers cate
gories of mixed spaces. The decisive influence of the work of BUCHWALTER on this appendix will be obvious to anyone who is familiar with the contents of [8] and [9].
We take this opportunity to mention two topics which we have not discussed in this monograph. Firstly, there exists a noncommutative version of the theory of chapter I1 based on the concept of the double centraliser of a Banach algebra (or, more generally, of a Banach module). We have omitted this topic, firstly because of the large number of preliminary definitions that it would require and secondly because we have nothing new to say about it. We refer the reader to
A.6
NOTES
307
One of the motivations for the introduction of strict topologies has its origins in harmonic analysis (see BEURLING 1 3 7 1 and HERZ [ 1 9 ] )
and this is potentially one of the most important
fields of application of mixed topologies. The present author does not regard himself as competent to cover this topic. BENEDETTO [5] contains some remarks on applications of the Strict topology to spectral synthesis.
APPENDIX
308
REFERENCES FOR THE APPENDIX
[I]
N. AZZAM
Mesures sur les espaces unifornes (Prepublication No. 2 Universitg de St. Etienne, 1974).
121
M. BARR
Duality of Banach spaces, Cah. de Top. Geom. Diff. 17 (1976) 1532.

[31
Closed categories and topological vector spaces, Cah. de Top. Geom. Diff. 17 (1976) 223234.
141
Closed categories and Banach spaces (Preprint).
[5] J. BENEDETTO
Spectral synthesis (Stuttgart 1975).
161
Ia. BEREZANSKY
Measures on uniform spaces and molecular measures, Transl. Moscow Math. SOC. 19 (1968) 140.
[7] A. BEURLING
Une thgordme sur les fonctions bornges et uniformement continues sur l'axe reel, Acta Math. 77 (1945) 127136.
[8] H. BUCHWALTER
Topologies, bornologies et compactologies, These Doctorat, Lyon 1968. Topologies et compactologies, Publ. Dep. Math. Lyon 62 (1969) 174. Double centralizers and extensions of C"algebras, Trans. h e r . Math. SOC. 132 (1968) 7999.
[lo]
R.C. BUSBY
[111
H.S. COLLINS and R.A. FONTENOT Approximate identities and the strict topology, Pac. J. Math. 43 (1972) 6379.
REFERENCES
309
[121
H.S. COLLINS and W.H. SUMMERS Some applications of Hewitt's factorisation theorem, Proc. Amer. Math. SOC. 21 (1969) 727733.
[131
J.B. COOPER
[141
J.B. COOPER and P. MICHOR Duality of compactological and locally compact groups (Categorical topology  Springer Lecture Notes 540, 188207)
Remarks on applications of category theory to functional analysis (Preprint, 1974).
.
[I51
A. DEAIBES
Espaces uniformes et espaces de mesures, Publ. D8p. Math. Lyon 124 (1975) 1166.
1161
J.W. DAVENPORT
Multipliers on a Banach algebra with a bounded approximate identiy, Pac. J. Math. 63 (1976) 131135.
[17]
R.A. FONTENOT
The double centralizer algebra as a linear space, Proc. Amer. Math. SOC. 53 (1975) 99103. Topological measure thory for double centralizer algebras, Trans. h e r . Math. SOC. 220 (1976) 167184.
1181
[191
C.S. HER2
The spectral theory of bounded functions, Trans. Amer. Math. SOC. 94 (1960) 181232.
1201
K.H. HOFMANN
The duality of compact semigroups and C*bialgebras, (Springer Lecture Notes 129, 1970).
121 1
1221
[ 231
Categorical theoretical methods in topological algebra (Categorical topology Springer Lecture Notes 540, 345403). J.R. ISBELL
Algebras of uniformly continuous functions, Ann. of Math. 6 8 (1958) 96125. Uniform spaces (Providence, 1964).
APPENDIX
310 [241
B.E. JOHNSON
Centralizers on certain topological algebras, Jour. Lond. Math. SOC. 39 ( 1 9 6 4 ) 60361 4.
[25
An introduction to the theory of centralisers, Proc. Lond. Math. SOC. 1 4 ( 1 9 6 4 )
1
299320. [26]
A. LAZAR and D.C. TAYLOR Double centralizers of Pedersen's ideal of a C"algebra, I,II, Bull. h e r . Math. SOC. 7 8 ( 1 9 7 2 ) 992997 and 79 ( 1 9 7 3 ) 361366.
1271
S.
MAC LANE
Categories for the working mathematician, (Springer, 1 9 7 1 ) .
1281
S.
MARDEZId
Precategories and shape theory (in Springer Lecture Notes 5 4 0 ( 1 9 7 6 ) 4 2 5  4 3 4 ) .
[291
K. McKENNON
The strict topology and the Cauchy structure of the spectrum of a Cgalgebra, Gen. Top. and Appl. 5 ( 1 9 7 5 ) 249262.
[301
P. MICHOR
Duality in groups (Unpublished note, 1 9 7 2 ) .
[31]
A. NAZIEV
The realization of dual categories, Soviet Math. Dokl. 1 4 ( 1 9 7 3 ) 14921495.
[32]
J.W. NEGREPONTIS (J.W. PELLETIER) Duality in analysis from the point of view of triples, Jour. Alg. 1 9 ( 1 9 7 1 ) 228253.
1331
G. PEDERSEN
Applications of weak" semicontinuity in C"algebra theory, Duke Math. Jour. 4 0 ( 1 9 7 3 ) 431450.
[341
R. PUPIER
Methods fonctorielles en topologie gBn6rale (UniversitB de Lyon, 1 9 7 1 ) .
1351
M. ROME
L'espace Mm(T), Publ. DQp. Math. Lyon 91 ( 1 9 7 2 ) 3760.
REFERENCES
311
1361
H. SCHUBERT
Kategorien 1,II (Springer, 1970).
1373
Z. SEMADENI
Banach spaces of continuous functions I (Warsaw, 1971 1.
[38] F.D. SENTILLES and D.C. TAYLOR Factorisation in Banach algebras and the general strict topology, Trans. h e r . Math. SOC. 142 (1969) 141152. [39]
D.C. TAYLOR
The strict topology for double centralizer algebras, Trans. Amer. Math. SOC. 150 (1970) 633643.
1401
A general Phillips theorem for C"algebras and some applications, Pac. J. Math. 4 0 (1972) 477488.
~411
Interpolations in algebras of operator fields, Jour. Func. Anal. 10 (1972) 1591 90.
[42]
J.L. TAYLOR
Topological invariants of the maximal ideal space of a Banach algebra, Adv. in Math. 19 (1976) 149206.
[43]
J.K. WANG
Multipliers of commutative Banach algebras, Pac. J. Math. 1 1 (1961) 11311149.
[441
R.F. WHEELER
The strict topology, separable measures and paracompactness, Pac. J. Math. 47 (1973) 287302.
This Page Intentionally Left Blank
CONCLUSION
As mentioned in the introduction, we have attempted in this monograph to present a first synthesis of a theory of mixed topologies and their applications. We would like to conclude with a brief list of some general problems. Some of these topics are the object of current research.
We begin with the closed graph theorems and BanachSteinhaus theorems of Chapter I. At present, there are essentially two approaches to such results

partitions of unity and some kind
of geometric condition on the unit ball such as the C 1 condition (we take this opportunity to inform the reader that there
is a
C 2 condition which was introduced also by ORLICZ but which
we have not treated here). It would be interesting to find some new ideas which lead to new closed graph theorems. In particular, neither of these approaches is applicable to spaces of holomorphic functions (incontrast to continuous or measurable functions) Hence the question: is there a closed graph theorem for abstract Saks spaces from which MOONEY's theorem (V.2.12) can be deduced ? As a second topic we have seen that almost none of the important classes of locally convex spaces (e.g. FrBchet, nuclear or barrelled spaces) have any relevance for Saks spaces. Now most of these classes can be classified within the category of locally convex spaces in a manner which can be formally carried over to the category of Saks spaces. It would be interesting to investigate the corresponding classes of Saks spaces.
313
CONCLUSION
314
We give some examples to indicate more precisely what is meant. The natural analogy of a Frgchet space would be a Saks space projective limit of a sequence of Banach space i.e. a complete Saks space (El1 1
11,~)
where
T
is metrisable. We could
also define a nuclear Saks space as a space for which various Saks space tensor products coincide (in analogy to the condition
E 6 F = E $ F
for arbitrary F which characterises
nuclearity for locally convex spaces). The first question would be to decide if this is a useful notion, the second to give an internal characterisation of these spaces. Of course, this definition depends on which tensor products for Saks spaces we consider to be relevant and in this connection we mention that in defining tensor products for Saks spaces in Chapter I we restricted attention to the two simple definitions which were tailormade for the applications we had in mind duct representations of Cm (S
x
s,)
and Lm (M
x

M, )
the tensor pro
. It would
perhaps be useful to have a more differentiated treatment, with particular regard to universal properties for example.
With regard to Chapter 11, we list, in a rather haphazard way, some problems which seem to us to be relevant. A fruitful branch of modern pointset topology has been the study of coverings of a topological space. We can define strict topologies on Cm(S) which are intimately related with covering properties of S as follows: if U is (for example, an open) covering of a completely regular space S, we can define a Saks space structure on
Cm(S)
by mixing the supremum norm with the topology of uniform con
CONCLUSION
315
vergence on the sets of U. Given a suitable family of coverings, we can define a Saks space structure on Cm(S) by taking the locally convex inductive limit of the structures induced by the individual coverings (possible candidates for the distinguished family of coverings could be all open coverings or all locally finite open coverings). The problem would be to establish relationship between the topology of
S
and the properties of Cm(S)
under the corresponding strict topologies and to characterise the duals as spaces of measures on
S,
in particular to establish
conditions for the equality of these strict topologies with those introduced in Chapter 11. We conclude our remarks on Chapter I1 with two problems: firstly to develop an approach to cylindrical measures on a locally convex space E using a strict topology on Cm(E) and secondly to give a detailed analysis of the space of continuous linear operators from Cm(S) into a Banach space E and its relation with vector measures, corresponding to the theory for compact
S.
J.L. TAYLOR has recently shown how topological
properties of a compact space are reflected in the properties of a Banach algebra which has this space as spectrum

in par
ticular, the cohomology and Ktheory of the space. Can this theory be carried over to noncompact spaces using strict topologies ?
The outstanding question raised in Chapter I11 is that of developing a complete theory of integrable and measurable functions with valued in a Saks space, thus obtaining a synthesis of various notions of vectorvalued integration. It would also be
CONCLUSION
316
useful to make more precise the relation with vector measures, in particular, with regard to the RadonNikodym property.
With regard to Chapter IV it would be particularly interesting to try out the various Saks space tensor products on W*algebras and to compare the results with the tensor products of W*algebras which have been studied in the literature. Another interesting question is that of studying the corresponding mixed topologies on spaces L(E) or even L ( E , F ) ( E and F Banach spaces) to see how much of the theory can be carried over. For example, can the dual of L(E) or L(E;F) be identified with a space of operators (the approximation property will presumably play a role here) ?
As regards Chapter V, we remark that in the two cases (Cm(S) and Hm(U) ) where we have investigated the spectrum of a commutative Saks algebra, we have found that M (A) was dense in the Banach
Y
algebra spectrum M(A). In the first case the result is not very deep (it simply means that S is dense in B S ) . The second case is of course the Corona problem. This leads us to pose the following problem: is M (A) always dense in M(A)? It may be that
Y
there is a simple counterexample. Of course, if the result is true it must be very deep and presumably can only be solved when we have considerable information on the structure of Saks algebras. Another problem is to study the spectrum of HW(G) where G is a domain in higher dimensions or even a more complicated domain in
C.
In general, M (HW(G)) is much larger than G
Y
CONCLUSION
317
e.g. if there is a domain G, properly containing G with the property that each bounded holomorphic function on G has a bounded holomorphic extension to G1. We thus pose the following problem: can M (Hm(G)) be regarded as some kind of bounded en
Y
velope of holomorphy of G; more precisely, does M (Hm(G)) have
Y
a complex analytic structure so that the GelfandNaimark transform of elements of Hm(G) are holomorphic functions on M (HQJ(G) )?
Y
More generally, we can ask for general conditions on a commutative Saks algebra which ensure that its spectrum has such an analytic structure. Another problem which is related indirectly to the theme of Chapter V is that of applications of mixed topologies to spaces of smooth functions and distributions. As an example, we could consider the space of bounded Cmfunctions on IRn (or on a manifold) and mix the supremum norm with the topology of uniform convergence on compacta of all derivatives. The dual is a space of distributions on the manifold. However, there is one problem

the auxiliary topology is not coarser than the
norm topology so that the conditions of Definition 1.1.4 are not satisfied. However, many of the results of Chapter I do not require this condition or can be modified in an obvious way and, in fact, a number of the references given in Chapter I contain results on this more general situation. Just as in measure theory where the mixed topologies are useful in the generalisation of the theory of Radon measures from locally compact to completely regular spaces, so one would expect that such mixed topologies could be helpful in the study of distributions on a Banach space (or on a manifold modelled on a Banach space) where
3 18
CONCLUSION
the classical Schwartz approach breaks down because of the absence of nontrivial test functions on a Banach space (analogous to the nonexistence, in general, of continuous functions with compact support on a nonlocally compact space).
It has been our intention in this monograph to expose the main outlines of the theory of Saks spaces and some of their applications. We hope that we have succeeded in demonstrating that it is the natural tool for some problems which are rather intractable when treated by classical methods. The fact that we have not included deeper applications to various problems on function spaces is, we believe, rather a testimony to the inadequacy of the author than a condemnation of the potential of the theory. We hope that this monograph will induce some experts to explore the possibility of applications of Saks spaces to such problems.
INDEX OF NOTATION
125 125 26 173 26 26 291 5
89,
157,
226
CC"OG
291
82
C (K)
33
270
Cospec(A)
28 0
CK(S)
23
21 7
co ( S )
21
119
coo ( S ) csc"
23
119
120,
158,
172,
22,
161,
27,
298
275
161
268
198
22
198
77
204
77
78
113
119
91
5,
120
1 7 2 , 179,
234
6
123
13 41
319
320
INDEX OF NOTATION
6 6
22,
77,
273 23
13 6
81
13
197 157
272
177
272
157
19
171
55 120 4
EB
120 120
Eitr
15
E'
Y
15
21
Eh
15
21
E 6 E G
Y
Y
F
35
F
35
F
77 285
GRCPTOL
204 204 282 283
22,
226 230
37 174 187
236 175 95 98 96
175 251 251
INDEX OF NOTATION
321
251
293
251
268
112 125
RCPTOL
274
102
33
125 102
U
158
S
77
125 269 106
P
SC*COG
288
SC”COGR
288
126
SGRCPTOL
285
126
S&m{
100
S
Ea 1
TI
126
CrEA Spec ( A )
126
ss
31
{Ea)
31 297 268
79
24 22
22 204
88,
156,
226
21,
197
204
204
21 4
197
21 4 82
22,
77 113 35
PMW
269
172
INDEX OF NOTATION
322
160 12
U
6
us
21 4
usJt
21 4
uu
21 4
270
UX
98
X"
100
XK
127
96 96
4 113 178 21 21,
22,
77
INDEX OF TERMS
Approximation property, 53 associated topology, 29 Ball, 4 Banach ball, 4 BanachSteinhaus property, 45 Blaschke product, 243 Bohr compactification, 291 bounded, 5 , 7 7 bornology, 5 basis, 5 convex bornology, 5 complete, 5 of countable type, 5 von Neumann bornology, 5 Compactology, 273 regular, 273 of countable type, 273 compactological semigroup, group, 285 condition C,, 46 contraction (completelynonunitary), 2 5 0 CoSaks space, 2 7 0 Dual, 15 ydual, 33 DunfordPettis property, 168 Generalised Dirac transform, 9 0 generalised GelfandNaimark transform, 9 2 generalised zero set, 256 Gspace, 6 2
323
324
INDEX OF TERMS
Hardy space, 230 hemicompact (topological space), 88 Inner function, 243 Lifting, 184 Lindelof space, 132 Mackey space, 4 8 measure, Baire measure, 125 Bore1 measure, 125 Choquet measure, 104 compact regular measure, 104 vcontinuous, 175 vregular, 175 of compact support, 102 Radon measure, 104 regular measure, 170 aadditive measure, 125 radditive measure, 125 tight measure, 101 , 104 measurable function, 177, 178 Lusin measurable function, 177 measure compact, 131 metric approximation property, 53 minimal function, 256 pspace, 100 Normal (operator between W"algebras) , 206 nuclear operator, 200 Outer function, 244 Partition of unity, 52 pequiintegrable, 158 PMWmorphism, 269 premeasure , 101
INDEX OF TERMS Prohorov space, 1 2 9 pseudocompact, 1 2 2 Radon Nikodym property, 1 8 8 real compact, 9 8 real compactification, 9 8 Saks algebra, 3 6 perfect, 9 5 spectrum, 3 7 Saks Catalgebra,3 6 Saks space, 2 8 complete, 2 8 completion, 3 0 products of, 3 1 projective limits of, 3 1 quotient, 2 9 subspace, 2 9 tensor products, 3 5 saturated family, 7 7 of countable type, 7 7 Scomplete, 8 0 semivariation (of a measure), 1 7 3 SS1 normal, 7 9 strict Dunford Pettis property, 1 6 8 strongly measure compact, 1 3 4 trace (normal, faithful, semifinite)I 2 1 8 trace class (operator of) t 2 0 1 Tspace, 1 2 9 Ultrastrong topology, 2 1 4 ultraweak topology, 2 1 4 uniformly tight , 1 0 5 von Neumann algebra, 2 0 3 W"algebra, 2 1 6 enveloping W"algebra, 2 2 1
325