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.
Author(s): J. Voigt
Series: Compact Textbooks in Mathematics
Edition: 1
Publisher: Birkhäuser
Year: 2020
Language: English
Pages: 152
Tags: topology, vector spaces
Preface
Contents
1 Initial Topology, Topological Vector Spaces, Weak Topology
2 Convexity, Separation Theorems, Locally Convex Spaces
3 Polars, Bipolar Theorem, Polar Topologies
4 The Tikhonov and Alaoglu–Bourbaki Theorems
5 The Mackey–Arens Theorem
6 Topologies on E'', Quasi-barrelled and Barrelled Spaces
7 Fréchet Spaces and DF-Spaces
8 Reflexivity
9 Completeness
10 Locally Convex Final Topology, Topology of D(Ω)
11 Precompact – Compact – Complete
12 The Banach–Dieudonné and Krein–Šmulian Theorems
13 The Eberlein–Šmulian and Eberlein–Grothendieck Theorems
14 Krein's Theorem
15 Weakly Compact Sets in L1(μ)
16 B0=B
17 The Krein–Milman Theorem
A The Hahn–Banach Theorem
B Baire's Theorem and the Uniform Boundedness Theorem
References
Index of Notation
Index
