Bounded sets structure of CpX and quasi-(DF)-spaces

0517562 - MÚ 2020 RIV DE eng J - Journal Article
Ferrando, J.C. - Gabriyelyan, S. - Kąkol, Jerzy
Bounded sets structure of CpX and quasi-(DF)-spaces.
Mathematische Nachrichten. Roč. 292, č. 12 (2019), s. 2602-2618. ISSN 0025-584X. E-ISSN 1522-2616
R&D Projects: GA ČR GF16-34860L
Institutional support: RVO:67985840
Keywords : (DF)-space * bounded resolution * free locally convex space
OECD category: Pure mathematics
Impact factor: 0.910, year: 2019
Method of publishing: Limited access
http://dx.doi.org/10.1002/mana.201800085

For wide classes of locally convex spaces, in particular, for the space Cp(X) of continuous real-valued functions on a Tychonoff space X equipped with the pointwise topology, we characterize the existence of a fundamental bounded resolution (i.e., an increasing family of bounded sets indexed by the irrationals which swallows the bounded sets). These facts together with some results from Grothendieck's theory of (DF)-spaces have led us to introduce quasi-(DF)-spaces, a class of locally convex spaces containing (DF)-spaces that preserves subspaces, countable direct sums and countable products. Regular (LM)-spaces as well as their strong duals are quasi-(DF)-spaces. Hence the space of distributions D '(omega) provides a concrete example of a quasi-(DF)-space not being a (DF)-space. We show that Cp(X) has a fundamental bounded resolution if and only if Cp(X) is a quasi-(DF)-space iff the strong dual of Cp(X) is a quasi-(DF)-space if and only if X is countable.
Permanent Link: http://hdl.handle.net/11104/0302892