Voigt A Course on Topological Vector Spaces

Erscheinungsjahr 2020
ISBN: 978-3-030-32945-7
Verlag: Springer International Publishing
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 155 Seiten

Reihe: Compact Textbooks in Mathematics

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 L-spaces. Lastly, it addresses topics such as the locally convex final topology, with the application to test functions D(O) 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. 

Zielgruppe

Graduate

Autoren/Hrsg.

Voigt, Jürgen

Inhaltsverzeichnis

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.

Über Autor(innen)

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