A Course on Topological Vector Spaces

by Jürgen Voigt

Overview

This book provides an introduction to the theory of topological vector spaces, with a focus on locally convex spaces. It discusses topologies in dual pairs, culminating in the Mackey-Arens theorem, and also examines the properties of the weak topology on Banach spaces, for instance Banach’s theorem on weak*-closed subspaces on the dual of a Banach space (alias the Krein-Smulian theorem), the Eberlein-Smulian theorem, Krein’s theorem on the closed convex hull of weakly compact sets in a Banach space, and the Dunford-Pettis theorem characterising weak compactness in L₁-spaces. Lastly, it addresses topics such as the locally convex final topology, with the application to test functions D(Ω) and the space of distributions, and the Krein-Milman theorem.

The book adopts an “economic” approach to interesting topics, and avoids exploring all the arising side topics. Written in a concise mathematical style, it is intended primarily for advanced graduate students with a background in elementary functional analysis, but is also useful as a reference text for established mathematicians.

Product Details

ISBN-13:	9783030329440
Publisher:	Springer International Publishing
Publication date:	03/08/2020
Series:	Compact Textbooks in Mathematics
Edition description:	1st ed. 2020
Pages:	155
Product dimensions:	6.10(w) x 9.25(h) x 0.00(d)

About the Author

Jürgen Voigt is Professor at the Institute of Analysis of the Technische Universität in Dresden, Germany.

Table of Contents

Initial topology, topological vector spaces, weak topology.- Convexity, separation theorems, locally convex spaces.- Polars, bipolar theorem, polar topologies.- The theorems of Tikhonov and Alaoglu-Bourbaki.- The theorem of Mackey-Arens.- Topologies on E'', quasi-barrelled and barrelled spaces.- Reflexivity.- Completeness.- Locally convex final topology, topology of D(\Omega).- Precompact — compact – complete.- The theorems of Banach—Dieudonne and Krein—Smulian.- The theorems of Eberlein—Grothendieck and Eberlein—Smulian.- The theorem of Krein.- Weakly compact sets in L_1(\mu).- \cB_0''=\cB.- The theorem of Krein—Milman.- A The theorem of Hahn-Banach.- B Baire's theorem and the uniform boundedness theorem.

Editorial Reviews

The material of the book is very carefully developed and even includes an introduction into the basics of topological and metric spaces. … At the beginning of each chapter, a brief outline of the subjects treated therein is given, while at the end, notes, comments and suggestions for further reading are included. … The book ends with an extensive reference list, an index and a very helpful index of notations.” (Wolfgang Lusky, Mathematical Reviews, December, 2021)

“The book may be highly recommended to all students and researchers with some knowledge of Banach or Hilbert space oriented functional analysis who want to learn its general abstract foundations.” (Jochen Wengenroth, zbMATH 1453.46001, 2021)

From the Publisher