Locally convex spaces with hemicompact weak* dual

Abstract

For a locally convex space $L$ let $L'_σ$ denote its dual endowed with the weak* topology $σ(L',L)$. We prove that $L'_σ$ is hemicompact if and only if $L$ with the Mackey topology $τ(L,L')$ is metrizable and barrelled, and if and only if $L'_σ$ is $σ$-compact and quasi-complete. The key step is that a hemicompact $L'_σ$ is quasi-complete. The Mackey topology $τ(L,L')$ is metrizable if and only if $L'$ is a countable union of absolutely convex weak*-compact sets; spaces with normable barrelled Mackey topology are characterized similarly. As a consequence, a metrizable (in particular, normed) space $L$ is barrelled if and only if $L'_σ$ is hemicompact. For function spaces we obtain: $(C_k(X))'_σ$ is hemicompact if and only if $X$ is hemicompact, and $(C_p(X))'_σ$ is hemicompact if and only if $X$ is countable and all compact subsets of $X$ are finite.