Locally convex properties of free locally convex spaces

Abstract

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∞-barrelled, (ii) L(X) is ℓ∞-quasibarrelled, (iii) L(X) is c0-barrelled, (iv) L(X) is ℵ0-quasibarrelled, and (v) X is a P-space. If X is a non-discrete metrizable space, then L(X) is c0-quasibarrelled but it is neither c0-barrelled nor ℓ∞-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 c0-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 μ-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 μ-space (for example, metrizable), then L(X) has the sequential Dunford–Pettis property iff X is discrete.