A Course on Topological Vector Spaces

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Sprache: Englisch

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Beschreibung

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.

Über den Autor

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

Zusammenfassung

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

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.

Details

Erscheinungsjahr:	2020
Fachbereich:	Analysis
Genre:	Mathematik , Medizin , Naturwissenschaften , Technik
Rubrik:	Naturwissenschaften & Technik
Medium:	Taschenbuch
Reihe:	Compact Textbooks in Mathematics
Inhalt:	viii 155 S. 1 farbige Illustr. 155 p. 1 illus. in color.
ISBN-13:	9783030329440
ISBN-10:	3030329445
Sprache:	Englisch
Ausstattung / Beilage:	Paperback
Einband:	Kartoniert / Broschiert
Autor:	Voigt, Jürgen
Auflage:	1st ed. 2020
Hersteller:	Springer International Publishing Springer International Publishing AG Compact Textbooks in Mathematics
Maße:	235 x 155 x 10 mm
Von/Mit:	Jürgen Voigt
Erscheinungsdatum:	07.03.2020
Gewicht:	0,26 kg

Artikel-ID: 117393267