Metrizable bounded sets in C(X) spaces and distinguished Cp(X) spaces

0510255 - MÚ 2020 RIV DE eng J - Journal Article
Ferrando, J.C. - Kąkol, Jerzy
Metrizable bounded sets in C(X) spaces and distinguished Cp(X) spaces.
Journal of Convex Analysis. Roč. 26, č. 4 (2019), s. 1337-1346. ISSN 0944-6532. E-ISSN 0944-6532
R&D Projects: GA ČR GF16-34860L
Institutional support: RVO:67985840
Keywords : countable tightness * Frechet-Urysohn space * strong dual
OECD category: Pure mathematics
Impact factor: 0.527, year: 2019
Method of publishing: Limited access
http://www.heldermann.de/JCA/JCA26/JCA264/jca26070.htm

Quite recently W. Ruess [17] has shown that a wide class of locally convex spaces for which all bounded sets are metrizable enjoy Rosenthal's ∂1-dichotomy. Being motivated by this fact we show that for a Tychonoff space X the bounded sets of Cp (X) are metrizable (respectively, the bounded sets of Ck (X) are weakly metrizable) if and only if X is countable. If X is a P-space we show that every bounded set in Cp (X) is metrizable if and only if X is countable and discrete. The second part of the paper deals with distinguished Cp (X) spaces. Among other things we show that Cp (X) is distinguished if and only if the strong topology of the dual coincides with its strongest locally convex topology, and that Cp (X) is always distinguished whenever X is countable.
Permanent Link: http://hdl.handle.net/11104/0300770