A Course on Topological Vector Spaces / / by Jürgen Voigt

• Record Nr.: UNINA9910483620103321
• Autore: Voigt Jürgen
• Pubbl/distr/stampa: Cham : , : Springer International Publishing : , : Imprint : Birkhäuser, , 2020
• ISBN: 3-030-32945-3
• Edizione: [1st ed. 2020.]
• Descrizione fisica: 1 online resource (VIII, 155 p. 1 illus. in color.)
• Collana: Compact Textbooks in Mathematics, , 2296-4568
• Disciplina: 515.73
• Soggetti: Functional analysis; Functional Analysis
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di bibliografia: Includes bibliographical references and indexes.
• Nota di contenuto: 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.
• Sommario/riassunto: 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. .