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On Schwartz groups. (English) Zbl 1126.22001
The notion of a Schwartz topological group is introduced: a Hausdorff Abelian group $$G$$ is a Schwartz group if for every neighborhood $$U$$ of zero in $$G$$ there are another neighborhood $$V$$ of zero and a sequence $$(F_n)$$ of finite subsets of $$G$$ such that for each $$n$$, $$V\subset F_n+ \{x\in G:x\in U, 2x\in U, \cdots, nx\in U\}$$. If $$E$$ is a topological vector space, then the underlying additive group of $$E$$ is a Schwartz group if and only if $$E$$ is a Schwartz vector space in the sense of A. Grothendieck [Mem. Am. Math. Soc. 16, 140 p. (1955; Zbl 0064.35501)]. All hemicompact $$k$$-space topological groups, the free Abelian group $$A(X)$$ over a hemicompact $$k$$-space $$X$$ and the Pontryagin dual group of a metrizable group are Schwartz groups. Several other basic properties of Schwartz groups are established.

##### MSC:
 22A05 Structure of general topological groups 46A11 Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)
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