Barrelledness in Topological and Ordered Vector Spaces
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692 T. Husain S. M. Khaleelulla
Barrelled ness in Topological and Ordered Vector Spaces
Springer-Verlag Berlin Heidelberg New York 1978
Authors T. Husain Mathematics Department McMaster University Hamilton, Ontario Canada, LBS 4K1 S. M. Khaleelulla Department of Mathematics Malnad College of Engineering Hassan - 573201/lndia
AMS Subject Classifications (1970): 46A07, 46A40
ISBN ISBN
3-540-09096-7 0-387-09096-7
Springer-Verlag Berlin Heidelberg New York 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 the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin Heidelberg 1978 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
PREFACE It is well known that in Functional Analysis the notion of barrelled space
plays an important role.
For example, the
open mapping and closed graph theorems and the Banach-Steinhaus theorem which are true for Banach spaces require an additional property on locally convex spaces for their validity and that is barrelledness.
A number of generalizations of barrelled
spaces have since been studied.
The aim of these authors in
writing this monograph is to give an account of certain classes of topological vector spaces and ordered topological vector spaces which generalize barrelled spaces and also to see if certain theorems true for Banach space can be carried over to these generalized classes of spaces.
To mention a few general-
izations of the above-mentioned notjon, they include, quasibarrelled, countably barrelled, o-barrelled spaces etc.
We
wish to give an up-to-date listing of such spaces and their usefulness in carrying over certain classical theorems in this monograph. This book consists of nine chapters, starting with Chapter 1 in which elementary results in topological and ordered topological vector spaces are given.
Their proofs are omitted
since they are supposed to be known to the reader and are available in any standard book on topological vector spaces or ordered
IV
topological vector spaces, e.g. Bourbaki [4J, Kothe [31], Schaefer [47], Horvath ~1], Peressini [41], Namioka [39] or Wong and Ng [63]. Chapter 2 also consists of known results on classical locally convex topological vector spaces and can easily be found in [4], [12],
[31],
[47] or[lOJ.
In Chapter 3, we study ultrabarrelled, ultrabornologic al and guasiultrabarre lled spaces.
They were first introduced
and studied by W. Robertson [44], and Iyahen [19]. Chapter 4 deals with order-guasibarr elled vector lattices which were first introduced by Wong [61J.
A kind of
closed graph and Banach-Steinhau s theorems f
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