A Course on Topological Vector Spaces, 1st ed. 2020 Compact Textbooks in Mathematics Series

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(?) 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. 

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.

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

Includes a streamlined introduction to the duality theory of locally convex spaces, culminating in the Mackey-Arens theorem

Treats various important topics concerning the weak topology of Banach spaces

Discusses examples of function spaces which occur in applications to differential operators and measure theory

Provides as a highlight the treatment of weak compactness in L_1-spaces