Bundles of Topological Vector Spaces and Their Duality
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955 Gerhard Gierz
Bundles of Topological Vector Spaces and Their Duality
Springer-Verlag Berlin Heidelberg New York 1982
Author
Gerhard Gierz Department of Mathematics, University of California Riverside, CA 92521, USA
AMS Subject Classifications (1980): 46ElO, 46E15, 46E40, 46H 25, 46B20, 55R25, 28C20 ISBN 3-540-11610-9 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-11610-9 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschatt Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1982 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
Contents
Introduction Notational remarks
7
1•
Basic definitions
8
2.
Full bundles and bundles with completely regular base spaces
22
3.
Bundles with locally paracompact base spaces
28
4.
Stone - WeierstraB theorems for bundles
39
5.
An alternative description of spaces of sections: Function modules
6.
Some algebraic aspects of
7.
A third description of spaces of sections: C(X)-
44
60
-convex modules
62
8.
C(X)-submodules of rIp)
80
9.
Quotients of bundles of C(X)-modules
86
10.
r·1orphisms between bundles
95
11 •
Bundles of operators
112
12
Excursion:
136
13.
M-structure and bundles .
144
14.
An adequate M-theory for n-spaces
154
15.
Duality
159
16.
The closure of the "unit ball" of a bundle and
Continuous lattices, and bundles
separation axioms A definition
183 200
17.
Locally trivial bundles:
18.
Local linear independence
202
19.
The space Mod(r(p) ,C(X))
209
2C'
Internal duality of C(X)-modules
232
IV
21.
The dual space rip)
Appendix:
I
of a space of sections
252
Integral representation of linear functionals on a space of sections and Klaus Keirnel)
(by Gerhard Gierz 260
References
284
Index
291
Introduction.
In the present notes we are dealing with topological vector spaces which vary continuously over a topological space. Among the first authors formulating this idea were Godement [Go 49J, Kaplansky [Ka 51J, Gelfand and Naimark. In these early papers, theyaxiomatized the idea of subdirect continuous representation of Banach spaces. To be precise, they considered spaces E of functions a defined on a topological space X with values in given Banach spaces Ex, x
E
X,
satisfying axioms like
(1)
The function x ....
[l o I x ) II : X
and bounded for every a
(2)
E
....
:rn. is (upper semi-) continuous
E.
The space E is complete in the norm {o Cx) : a
E} for every x
II 0 II
sup
II a (x) II .
XEX
X.
(3)
E
(4)
E is a Cb(X)-module relative to the multiplication (f,o) .... f-o:
x
E
Cb(X)XE .... E, where (f·o) (x)
E
:= fix) ·o(
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