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If not distinguished, is \(C_p( X)\) even close? (English) Zbl 1471.46002

If \(X\) is a topological space, \(\mathcal{C}_p(X)\) denotes the linear space of all continuous real-valued functions, endowed with the topology of pointwise convergence. As the authors state, this paper “consolidates and complements initial studies of (non)-distinguished \(\mathcal{C}_p(X)\)-spaces” – a locally convex topological vector space is called distinguished is its strong dual is barrelled. Within the new results there is an internal characterization of spaces \(X\) with the property that \(\mathcal{C}_p(X)\) is distinguished, the fact that \(\mathcal{C}_p(X)\) is distinguished if its strong dual is primitive (a weak barrelledness property usually much weaker than barrelledness) and the statement that the strong dual of \(\mathcal{C}_p(X)\) is always distinguished. Various examples and problems are included in this paper.

MSC:

46A08 Barrelled spaces, bornological spaces
54C35 Function spaces in general topology
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