When is a locally convex space Eberlein-Grothendieck?

0563646 - MÚ 2023 RIV CH eng J - Journal Article
Kąkol, Jerzy - Leiderman, A. G.
When is a locally convex space Eberlein-Grothendieck?
Results in Mathematics. Roč. 77, č. 6 (2022), č. článku 236. ISSN 1422-6383. E-ISSN 1420-9012
R&D Projects: GA ČR(CZ) GF20-22230L
Institutional support: RVO:67985840
Keywords : Barrelled space * compact space * locally convex space * weak topology
OECD category: Pure mathematics
Impact factor: 2.2, year: 2022
Method of publishing: Limited access
https://doi.org/10.1007/s00025-022-01770-w

The weak topology of a locally convex space (lcs) E is denoted by w. In this paper we undertake a systematic study of those lcs E such that (E, w) is (linearly) Eberlein-Grothendieck (see Definitions 1.2 and 3.1). The following results obtained in our paper play a key role: for every barrelled lcs E, the space (E, w) is Eberlein-Grothendieck (linearly Eberlein-Grothendieck) if and only if E is metrizable (E is normable, respectively). The main applications concern to the space of continuous real-valued functions on a Tychonoff space X endowed with the compact-open topology Ck(X). We prove that (Ck(X) , w) is Eberlein-Grothendieck (linearly Eberlein-Grothen-dieck) if and only if X is hemicompact (X is compact, respectively). Besides this, we show that the class of E for which (E, w) is linearly Eberlein-Grothendieck preserves linear continuous quotients. Various illustrating examples are provided.
Permanent Link: https://hdl.handle.net/11104/0335546