UID:
almahu_9948276330702882
Format:
VIII, 155 p. 1 illus. in color.
,
online resource.
Edition:
1st ed. 2020.
ISBN:
9783030329457
Series Statement:
Compact Textbooks in Mathematics,
Content:
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. .
Note:
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.
In:
Springer eBooks
Additional Edition:
Printed edition: ISBN 9783030329440
Additional Edition:
Printed edition: ISBN 9783030329464
Language:
English
DOI:
10.1007/978-3-030-32945-7
URL:
https://doi.org/10.1007/978-3-030-32945-7
