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Metrizability of precompact sets; an elementary proof. (English) Zbl 1119.46007

According to [B. Cascales and J. Orihuela, Math. Z. 195, No. 3, 365–381 (1987; Zbl 0604.46011)], a locally convex space \(E\) is said to be in the class \({\mathfrak G}\) if there is a family \(\{A_\alpha: \alpha\in \mathbb{N}^{\mathbb{N}}\}\) of subsets of its topological dual \(E'\) such that
(i) \(E'=\bigcup\{A_\alpha: \alpha\in\mathbb{N}^{\mathbb{N}}\}\), (ii) \(A_\alpha\subseteq A_\beta\) if \(a\leq\beta\) (i.e., \(\alpha(i)\leq\beta(i)\) for all \(i\in\mathbb{N}\)), and (iii) in each set \(A_\alpha\), sequences are equicontinuous.
It was shown there that (A) every precompact subset of a locally convex space in class \({\mathfrak G}\) is metrizable. In [N. Robertson, Bull. Aust. Math. Soc. 43, No. 1, 131–135 (1991; Zbl 0703.54016)], it was proved that (B) every uniform space \((X,{\mathcal N})\) covered by a family \(\{A_\alpha: \alpha\in\mathbb{N}^{\mathbb{N}}\}\) of precompact subsets is trans-separable. Let us recall that a locally convex space \(E\) is trans-separable if for every absolutely convex neighborhood of the origin \(U\) in \(E\) there exists a countable set \(N\) in \(E\) such that \(E= N+ U\). Among other permanence properties, trans-separable locally convex spaces are stable under transition to linear subspaces, topological products and continuous linear images.
In the present paper, the authors essentially adapt Robertson’s argument of the proof of result (B) to provide a short proof of (C) if \(E\) is a locally convex space in the class \({\mathfrak G}\) and \(P\) is a precompact subset of \(E\) there is a countable set \(N\) in \(E'\) such that \(E'= N+ P^{\text o}\). Result (C) shows that if \(E\in{\mathfrak G}\), then \(E'\) is trans-separable under the topology of uniform convergence on the precompact subsets of \(E\). According to [H. H. Pfister, Arch. Math. 27, 86–92 (1976; Zbl 0318.46005)], this fact assures that the precompact subsets of \(E\) are metrizable; providing an elementary proof of result (A).

MSC:

46A50 Compactness in topological linear spaces; angelic spaces, etc.
46A03 General theory of locally convex spaces
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