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When is a Locally Convex Space Eberlein–Grothendieck?
- Source :
- Results in Mathematics / Resultate der Mathematik; Dec2022, Vol. 77 Issue 6, p1-16, 16p
- Publication Year :
- 2022
-
Abstract
- 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 C k (X) . We prove that (C k (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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 14226383
- Volume :
- 77
- Issue :
- 6
- Database :
- Complementary Index
- Journal :
- Results in Mathematics / Resultate der Mathematik
- Publication Type :
- Academic Journal
- Accession number :
- 159896042
- Full Text :
- https://doi.org/10.1007/s00025-022-01770-w