Locally convex properties of free locally convex spaces

[摘要] Let L(X) be the free locally convex space over a Tychonoff space X. We show that the following assertions are equivalent: (i) L(X) is l(infinity)-barrelled, (ii) L(X) is l(infinity)-quasibarrelled, (iii) L(X) is c(0)-barrelled, (iv) L(X) is N-0-quasibarrelled, and (v) X is a P-space. If X is a non-discrete metrizable space, then L(X) is c(0)-quasibarrelled but it is neither c(0)-barrelled nor l(infinity)-quasibarrelled. We prove that L(X) is a (DF)-space iff X is a countable discrete space. We show that there is a countable Tychonoff space X such that L(X) is a quasi-(DF)-space but is not a c(0)-quasibarrelled space. For each non-metrizable compact space K, the space L(K) is a (df)-space but is not a quasi-(DF)-space. If X is a mu-space, then L(X) has the Grothendieck property iff every compact subset of X is finite. We show that L(X) has the Dunford-Pettis property for every Tychonoff space X. If X is a sequential space and a mu-space (for example, metrizable), then L(X) has the sequential Dunford-Pettis property iff X is discrete. (C) 2019 Elsevier Inc. All rights reserved.

[发布日期] 2019-12-15

[关键词] Free locally convex space;Weak barrelledness;(DF)-, (df)- and quasi-(DF)-spaces;Grothendieck property;Dunford-Pettis property;Sequential Dunford-Pettis property