Locally convex spaces with hemicompact weak* dual

For a locally convex space $L$ let $L_{\sigma }^{\mathrm{\prime }}$ denote its dual endowed with the weak* topology $\sigma (L^{\mathrm{\prime }},L)$. We prove that $L_{\sigma }^{\mathrm{\prime }}$ is hemicompact if and only if $L$ with the Mackey topology $\tau (L,L^{\mathrm{\prime }})$ is metrizable and barrelled, and if and only if $L_{\sigma }^{\mathrm{\prime }}$ is $\sigma$-compact and quasi-complete. The key step is that a hemicompact $L_{\sigma }^{\mathrm{\prime }}$ is quasi-complete. The Mackey topology $\tau (L,L^{\mathrm{\prime }})$ is metrizable if and only if $L^{\mathrm{\prime }}$ 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 $L_{\sigma }^{\mathrm{\prime }}$ is hemicompact. For function spaces we obtain: $(C_k(X){)}_{\sigma }^{\mathrm{\prime }}$ is hemicompact if and only if $X$ is hemicompact, and $(C_p(X){)}_{\sigma }^{\mathrm{\prime }}$ is hemicompact if and only if $X$ is countable and all compact subsets of $X$ are finite.