Normed and Banach groups. (English) Zbl 1537.22004
In [Isr. J. Math. 8, 230–252 (1970; Zbl 0214.28402)], P. Enflo introduced UFSS groups: An abelian topological group \((G,\tau)\) is called a UFSS (uniformly free from small subgroups) group, if there is a symmetric set \(U\) such that \((U_{(n)})=(\{x\in G:\ x,2x,\ldots,nx\in U\})\) forms a neighborhood base at \(0\).
In the paper under review, a locally quasi-convex UFSS group is called a normed group and in case the topological group is additionally complete, it is called a Banach group. The names are justified, as a locally convex space is a normed, respectively Banach group if and only if it is a normed space, respectively Banach space. Then the author shows permanence properties of normed and Banach groups. Most of them are consequences of the corresponding permanence properties of locally quasi-convex or UFSS groups proved in [W. Banaszczyk, Additive subgroups of topological vector spaces. Berlin etc.: Springer-Verlag (1991; Zbl 0743.46002)] or [the reviewer et al., J. Math. Anal. Appl. 380, No. 2, 552–570 (2011; Zbl 1216.22002)], respectively.
Afterwards, normed LCA groups are characterized to be exactly the duals of compactly generated LCA groups and it is shown that every normed Schwartz group is locally precompact.
Then the author gives examples for Banach groups. If \(G\) is a Banach group and \(K\) a compact space then the group \(\mathcal{C}(K,G)\) of all continuous functions \(K\to G\) endowed with the topology of uniform convergence is a Banach group. As a consequence, the space \(\ell_\infty(\kappa,\mathbb T)\) of all bounded families with index set \(\kappa\) and values in the complex torus \(\mathbb T\) is a Banach group. As well the subgroup \(c_0(\kappa,\mathbb T)\), which generalizes the group of \(\mathbb T\)-valued null-sequences, is a Banach group, even more, the author shows that this group is reflexive.
Then he proceeds in proving that the group \(\ell_1(\kappa;\mathbb T)\) of all absolutely summable families (with index set \(\kappa\)) is a reflexive Banach group which has the Schur property. (In [the reviewer, Axioms 11, No. 5, Paper No. 218, 22 p. (2022; doi:10.3390/axioms11050218)] the group \(\ell_1(\mathbb N;G)\) was studied (\(G\) an abelian topological group) and it was shown there that \(\ell_1(\mathbb N,\mathbb T)\) has the properties stated above.)
Finally, he shows that the dual group of \((G,T_{\mathbf{u}})\), where \(\mathbf{u}\) is a \(T\)-sequence which algebraically generates the abelian group \(G\), is a Banach group.
Next to that, the author gives representation theorems for normed groups (modifying (9.6) in [Banaszczyk, loc. cit.] and (10.1) (10.6) in [the reviewer, 2022, loc. cit.]) and shows that every metrizable locally quasi-convex group embeds into a product of reflexive Banach groups.
Finally, the author studies almost normed (almost Banach groups) which are abelian groups which have a compact subgroup such that the quotient group is a normed group (Banach group).
In the paper under review, a locally quasi-convex UFSS group is called a normed group and in case the topological group is additionally complete, it is called a Banach group. The names are justified, as a locally convex space is a normed, respectively Banach group if and only if it is a normed space, respectively Banach space. Then the author shows permanence properties of normed and Banach groups. Most of them are consequences of the corresponding permanence properties of locally quasi-convex or UFSS groups proved in [W. Banaszczyk, Additive subgroups of topological vector spaces. Berlin etc.: Springer-Verlag (1991; Zbl 0743.46002)] or [the reviewer et al., J. Math. Anal. Appl. 380, No. 2, 552–570 (2011; Zbl 1216.22002)], respectively.
Afterwards, normed LCA groups are characterized to be exactly the duals of compactly generated LCA groups and it is shown that every normed Schwartz group is locally precompact.
Then the author gives examples for Banach groups. If \(G\) is a Banach group and \(K\) a compact space then the group \(\mathcal{C}(K,G)\) of all continuous functions \(K\to G\) endowed with the topology of uniform convergence is a Banach group. As a consequence, the space \(\ell_\infty(\kappa,\mathbb T)\) of all bounded families with index set \(\kappa\) and values in the complex torus \(\mathbb T\) is a Banach group. As well the subgroup \(c_0(\kappa,\mathbb T)\), which generalizes the group of \(\mathbb T\)-valued null-sequences, is a Banach group, even more, the author shows that this group is reflexive.
Then he proceeds in proving that the group \(\ell_1(\kappa;\mathbb T)\) of all absolutely summable families (with index set \(\kappa\)) is a reflexive Banach group which has the Schur property. (In [the reviewer, Axioms 11, No. 5, Paper No. 218, 22 p. (2022; doi:10.3390/axioms11050218)] the group \(\ell_1(\mathbb N;G)\) was studied (\(G\) an abelian topological group) and it was shown there that \(\ell_1(\mathbb N,\mathbb T)\) has the properties stated above.)
Finally, he shows that the dual group of \((G,T_{\mathbf{u}})\), where \(\mathbf{u}\) is a \(T\)-sequence which algebraically generates the abelian group \(G\), is a Banach group.
Next to that, the author gives representation theorems for normed groups (modifying (9.6) in [Banaszczyk, loc. cit.] and (10.1) (10.6) in [the reviewer, 2022, loc. cit.]) and shows that every metrizable locally quasi-convex group embeds into a product of reflexive Banach groups.
Finally, the author studies almost normed (almost Banach groups) which are abelian groups which have a compact subgroup such that the quotient group is a normed group (Banach group).
Reviewer: Lydia Außenhofer (Passau)
MSC:
| 22A10 | Analysis on general topological groups |
| 22A20 | Analysis on topological semigroups |
| 46A30 | Open mapping and closed graph theorems; completeness (including \(B\)-, \(B_r\)-completeness) |
| 22A25 | Representations of general topological groups and semigroups |
Keywords:
normed group; Banach group; almost normed group; almost Banach group; locally quasi-convex group; Schwartz group; Pontryagin reflexive groupReferences:
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