When is a Locally Convex Space Eberlein–Grothendieck?

Jerzy Ka̧kol, Arkady Leiderman

Research output: Contribution to journalArticlepeer-review

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 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.

Original languageEnglish
Article number236
JournalResults in Mathematics
Volume77
Issue number6
DOIs
StatePublished - 1 Dec 2022

Keywords

  • Barrelled space
  • C(X) space
  • Compact space
  • Locally convex space
  • Weak topology

ASJC Scopus subject areas

  • Mathematics (miscellaneous)
  • Applied Mathematics

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