## 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.

Original language | English |
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Article number | 236 |

Journal | Results in Mathematics |

Volume | 77 |

Issue number | 6 |

DOIs | |

State | Published - 1 Dec 2022 |

## Keywords

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

## ASJC Scopus subject areas

- Mathematics (miscellaneous)
- Applied Mathematics