omega(omega)-base and infinite-dimensional compact sets in locally convex spaces

0557877 - MÚ 2023 RIV ES eng J - Journal Article
Banakh, T. - Kąkol, Jerzy - Schürz, J. P.
omega(omega)-base and infinite-dimensional compact sets in locally convex spaces.
Revista Mathématica Complutense. Roč. 35, č. 2 (2022), s. 599-614. ISSN 1139-1138. E-ISSN 1988-2807
R&D Projects: GA ČR(CZ) GF20-22230L
Institutional support: RVO:67985840
Keywords : free space * locally convex space * networks * ωω-base
OECD category: Pure mathematics
Impact factor: 0.8, year: 2022
Method of publishing: Open access
https://doi.org/10.1007/s13163-021-00397-9

A locally convex space (lcs) E is said to have an ωω-base if E has a neighborhood base { Uα: α∈ ωω} at zero such that Uβ⊆ Uα for all α≤ β. The class of lcs with an ωω-base is large, among others contains all (LM)-spaces (hence (LF)-spaces), strong duals of distinguished Fréchet lcs (hence spaces of distributions D′(Ω)). A remarkable result of Cascales-Orihuela states that every compact set in an lcs with an ωω-base is metrizable. Our main result shows that every uncountable-dimensional lcs with an ωω-base contains an infinite-dimensional metrizable compact subset. On the other hand, the countable-dimensional vector space φ endowed with the finest locally convex topology has an ωω-base but contains no infinite-dimensional compact subsets. It turns out that φ is a unique infinite-dimensional locally convex space which is a kR-space containing no infinite-dimensional compact subsets. Applications to spaces Cp(X) are provided.
Permanent Link: http://hdl.handle.net/11104/0331725