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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.

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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