Weakly Compact Sets Lectures Held at S.U.N.Y., Buffalo, in Spring 19
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801 Klaus Floret
Weakly Compact Sets Lectures Held at S.U.N.Y., Buffalo, in Spring 1978
Springer-Verlag Berlin Heidelberg New York 1980
Author Klaus Floret Mathematisches Seminar der Universit~.t Kiel OIshausenstr. 40-60 2300 Kiel Federal Republic of Germany
AMS Subject Classifications (1980): Primary: 46A05, 46A50 Secondary: 41A65, 46A25, 46B10, 46E15, 46E30, 54C35, 54D30, 54 D 60 ISBN 3-540-09991-3 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-09991-3 Springer-Verlag NewYork 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 the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1980 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
Weakly compact subsets~
i.e. sets which are compact with respect to the
weak topology of a Banach-space
or more generally:of a locally convex
space play an important r~le in many questions are characterizations
of reflexivity~
of analysis.
characterizations
Among them
of subsets with
elements of least distance in linear and convex approximation theory~ ranges of vector measures and existence theorems in optimal control theory~ pointwise convergence of sequences of functions~ minimax-theorems~
separation
properties of convex sets. The intention of these lecture notes is to prove the main results on weak compactness due to W.F. Eberlein~ and R.C.
V.L. Smulian, M. Krein~ A. Grothendieck~
James as well as to go into some of the questions mentioned above.
There are three loci:
the theorems on countable eompactness~
compaetness~ and the supremum of linear funetionals. is A. Grothendieck's
on sequential
The linking element
interchangeable double-limit property.
The results on
countable and sequential compactness are~ as usual~ first proved in spaces of continuous
functions~
equipped with the topology of pointwise convergence.
The approach to R.C. James' theorem and its various applications original one in the form which was given by J.D. Pryce:
is the
His proof is just
checked carefully and the result stated as a double-limit-theorem which implies many of the applications and Mo DeWilde.
of other versions due to S. Simons
A short look into the contents shows that emphasis is put
on R.C. James' theorem .
A reader who is just interested in this~ may start
with §5 provided she or he accepts the W.F. Eberlein-A. (1.6.)~ the W.F. Eberlein-V.L. Smulian-theorem
Grothendieck-theorem
(3.10.) and a consequence of
JV it~ A. Grothendieck's theorem on weak compactness in
C(K)
(4.2. and 4.4.).
The typical reader whom I have in mind knows the basic facts on locally convex spaces and bec
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