Additive Subgroups of Topological Vector Spaces

1. Front Matter

   Pages I-VII

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2. Preliminaries

   • Wojciech Banaszczyk

   Pages 1-44

3. Exotic groups

   • Wojciech Banaszczyk

   Pages 45-71

4. Nuclear groups

   • Wojciech Banaszczyk

   Pages 72-109

5. The bochner theorem

   • Wojciech Banaszczyk

   Pages 110-131

6. Pontryagin duality

   • Wojciech Banaszczyk

   Pages 132-167

7. Back Matter

   Pages 168-178

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The Pontryagin-van Kampen duality theorem and the Bochner theorem on positive-definite functions are known to be true for certain abelian topological groups that are not locally compact. The book sets out to present in a systematic way the existing material. It is based on the original notion of a nuclear group, which includes LCA groups and nuclear locally convex spaces together with their additive subgroups, quotient groups and products. For (metrizable, complete) nuclear groups one obtains analogues of the Pontryagin duality theorem, of the Bochner theorem and of the Lévy-Steinitz theorem on rearrangement of series (an answer to an old question of S. Ulam). The book is written in the language of functional analysis. The methods used are taken mainly from geometry of numbers, geometry of Banach spaces and topological algebra. The reader is expected only to know the basics of functional analysis and abstract harmonic analysis.

• Abstract harmonic analysis
• Banasche Algebra
• Kombinatorik
• Vector space
• combinatorics
• functional analysis
• geometry of Banach spaces
• geometry of numbers
• topological groups

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