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Banach-Dieudonné theorem revisited. (English) Zbl 1034.22001

This paper deals with a variant of the Banach-Dieudonné theorem for topological Abelian groups. Let \(G\) be a topological Abelian group and let \(\Gamma\) be its dual group (i.e., the group of all continuous group homomorphisms of \(G\) into the unity circle of the complex plane with the standard product). The symbol \(\sigma(\Gamma, G)\) denotes the topology on \(\Gamma\) of pointwise convergence on \(G\). The authors consider the continuous convergence structure \(\Lambda\) on the dual group \(\Gamma\) which is defined by the property: the net \(\{\phi_i \}_{i\in I}\) \(\Lambda\)-converges to \(\phi\) if for every net \(\{g_j \}_{j\in J}\) in \(G\) convergent to \(g\), the combined net \(\{\phi_i(g_j) \}_{(i,j)\in I\times J}\) converges to \(\phi(g)\). Here the set \(I\times J\) is directed by the product direction, that is to say, \((i,j)<(i',j')\) if \(i<i'\) and \(j<j'\). This yields the coarsest convergence structure for which the evaluation mapping is continuous. The topology \(t_{\Lambda}\) canonically associated to the continuous convergence \(\Lambda\) is defined as follows: \(L\subset \Gamma\) is closed in \(t_{\Lambda}\) if and only if for every net \(\{\phi_i \}\subset L\) that \(\Lambda\)-converges to \(\phi\) it follows that \(\phi\in L\).
The main results of the authors are: (1) Let \(G\) be a topological Abelian group. The topology associated to the continuous convergence structure \(t_{\Lambda}\) is the finest of all topologies on \(\Gamma\) which induce \(\sigma(\Gamma, G)\) on all equicontinuous subsets of \(\Gamma\). (2) If \(G\) is an almost metrizable topological Abelian group, then the finest of all topologies which induce \(\sigma(\Gamma, G)\) on the equicontinuous subsets of \(\Gamma\) coincides with the compact-open topology in \(\Gamma\) and therefore is a group topology. (3) A locally quasi-convex metrizable Abelian group \(G\) is complete if and only if the quasi-convex hull of every compact subset \(K\subset G\) is again compact. In addition to these, the paper contains further interesting results.
Reviewer’s remark: A different proof of (3) (Theorem 3.6 of the paper) can be found in the reviewer’s paper [Math. Z. 238, 493–453 (2001; Zbl 1014.22001)].

MSC:

22A05 Structure of general topological groups
46A04 Locally convex Fréchet spaces and (DF)-spaces
46A19 Other “topological” linear spaces (convergence spaces, ranked spaces, spaces with a metric taking values in an ordered structure more general than \(\mathbb{R}\), etc.)

Citations:

Zbl 1014.22001
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References:

[1] Wheeler, Studia Math. 41 pp 243– (1972)
[2] DOI: 10.1007/BF01361183 · Zbl 0149.33604 · doi:10.1007/BF01361183
[3] Schaefer, Topological vector spaces (1970) · Zbl 0212.14001
[4] DOI: 10.1007/s000130050160 · Zbl 0899.22001 · doi:10.1007/s000130050160
[5] DOI: 10.1007/BF01223977 · Zbl 0356.46006 · doi:10.1007/BF01223977
[6] DOI: 10.1016/S0166-8641(99)00187-X · Zbl 0983.22003 · doi:10.1016/S0166-8641(99)00187-X
[7] DOI: 10.1023/A:1013730527108 · Zbl 1079.22500 · doi:10.1023/A:1013730527108
[8] Binz, Continuous Convergence in C(X) (1975) · doi:10.1007/BFb0088826
[9] DOI: 10.1016/S0022-4049(98)00034-6 · Zbl 0935.22004 · doi:10.1016/S0022-4049(98)00034-6
[10] Au{\(\beta\)}enhofer, Dissertations Math. 384 (1999)
[11] DOI: 10.1017/S0305004100048222 · doi:10.1017/S0305004100048222
[12] Martín-Peinador, Rev. Mat. Hispano-Americana, 4 34 pp 221– (1974)
[13] Köthe, Topological vector spaces I (1969)
[14] Banaszczyk, Additive subgroups of topological vector spaces (1991) · Zbl 0743.46002 · doi:10.1007/BFb0089147
[15] Jarchow, Locally convex spaces (1981) · doi:10.1007/978-3-322-90559-8
[16] DOI: 10.1007/s002090100263 · Zbl 1014.22001 · doi:10.1007/s002090100263
[17] García, Topología I (1975)
[18] DOI: 10.1007/BF01360965 · Zbl 0086.08803 · doi:10.1007/BF01360965
[19] DOI: 10.2307/1992847 · Zbl 0064.35502 · doi:10.2307/1992847
[20] Chasco, Studia Math. 132 pp 257– (1999)
[21] DOI: 10.1007/BF01171410 · Zbl 0293.46004 · doi:10.1007/BF01171410
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