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Additive subgroups of topological vector spaces. (English) Zbl 0743.46002

Lecture Notes in Mathematics. 1466. Berlin etc.: Springer-Verlag. vi, 178 p. (1991).
The monograph is a habilitation work of the author. It contains almost all of his results which belong to the area of topological groups. The book is self-contained and is written in a clear and strict manner.
The central subject of the work is the class of abelian nuclear groups – some natural class which contains all locally compact groups and nuclear locally convex vector-spaces.
Chapter 1 (Preliminaries). The first two sections are devoted to standard definitions and facts about topological groups and vector spaces. The third one contains some information on finite-dimensional geometry ( lattices, convex sets, ellipsoids, Kolmogorov diameters) and is the base of further valuable geometrical considerations.
Chapter 2 ( Exotic groups). The sense of this chapter is to demonstrate some pathological effects taking place outside the class of nuclear groups. For example:
Theorem 6.1. Let \(E\) be a metrizable locally convex space. If it is not nuclear, then it contains a discrete subgroup \(K\) such that the quotient group \((\hbox{span} K)/K\) is exotic (i.e. without nontrivial continuous unitary representations).
Chapter 3 (Nuclear groups) is the main one. It is seen from the title that the foundations of the nuclear groups theory are constructed here. There are considered different definitions and relations between them, properties of the class and so on. The last section is devoted to the most impressing author’s result:
Theorem 10.3. Let \(\sum g_ i\) be a convergent series in a metrizable nuclear group \(G\). Then the set of sums of all convergent permutations of the series is a translate of some (not necessarily closed) subgroup. If \(G\) is complete then the set is a translate of a continuous homomorphic image of a nuclear Fréchet space.
This theorem is a very strong generalization of the famous Levi-Steinitz theorem. Before the Banaszczyk’s theorem the only known earlier result in this direction was the case \(G=\mathbb{R}^ \omega\).
Chapter 4 (The Bochner theorem) is devoted to generalization for nuclear groups of the classical Bochner theorem on positive-definite functions. This generalization contains in it the three most known previous ones: the Weil-Raikov theorem for locally compact abelian groups, the Minlos theorem for nuclear locally convex spaces and the Madrecki theorem for locally convex spaces over \(p\)-adic fields.
Chapter 5 ( Pontryagin duality). The main result: Theorem 16.1. Let \(G\) be a metrizable nuclear group. Then the dual group is nuclear in the topology of precompact convergence.

MSC:

46A11 Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)
43A35 Positive definite functions on groups, semigroups, etc.
22B05 General properties and structure of LCA groups
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
22-02 Research exposition (monographs, survey articles) pertaining to topological groups
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