Spaces of Vector-Valued Continuous Functions
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1003
Jean Schmets
Spaces of Vector-Valued Continuous Functions
Springer-Verlag Berlin Heidelberg New York Tokyo 1983
Author
Jean Schmets Institut de Mathematique, Universite de Liege 15, avenue des Tilleuls, 4000 Liege, Belgique
AMS Subject Classifications (1980): 46A05, 46 ElO ISBN 3-540-12327-X Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-12327-X Springer-Verlag New York Heidelberg Berlin Tokyo 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 "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1983 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
INTRODUCTION
Let X be a Hausdorff completely regular space and E be a Hausdorff locally convex topological vector space. Then C(X;E) denotes the linear space of the continuous functions on X with values in E; in the scalar case (i.e. if E is ill or
we simply write C(X).
The purpose of these notes is to characterize locally convex properties of C(X;E) by means of topological properties of X and of ly convex properties of E. The scalar case has already been developed quite convenient to recall briefly its contents.
in [201
and it is
Its chapter I deals
with the ultrabornological (resp. bornological; barrelled; quasibarrelled) space associated to E. One finds in its chapter II a description of the realcompactification uX of X as well as the definition of the space Cp(X), i.e.
C(X) endowed with the most general locally con-
vex topology of uniform convergence on subsets of uX.
In its chapter
III, the ultrabornological (resp. bornological; barrelled; quasibarrelled)
space associated to Cp(X) is characterized. This gives of
course a necessary and sufficient condition for Cp(X) to have that property. Its chapter IV concerns conditions of separability or of weakcompactness in Cp(X). Finally its chapter V gives an attempt to study the vectorvalued case: it deals with the space Cs(X;E), i.e. C(X;E) endowed with the simple or pointwise topology. Since the pu b Li c a t i o n of [201, its chapters I to IV have got few complementary results. It is not the case of its chapter V to which significant results have been added. This is due mostly to A. Defant, W. Govaerts, R.
Hollstein, J. Mendoza Casas, J. Mujica, myself,
Now it is possible to say that large parts of the study of the Cp(X;E) spaces are settled. The aim of these notes is to present these new results as a complement to the chapters I to IV of [201 . The chapter I contains a description of the spaces Cp(X;E), i.e. C(X;E) endowed with the most general locally convex topology of uni
IV
form convergence on subsets of uX. The notion of the support of an absolutely convex s
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