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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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