Mackey Topologies on Vector-Valued Function Spaces

Abstract

Let $E$ be an ideal of $L^0$ over a $\sigma$-finite measure space $(\Omega ,\Sigma ,\mu )$, and let $(X,\Vert \cdot {\Vert }_X)$ be a real Banach space. Let $E(X)$ be a subspace of the space $L^0(X)$ of $\mu$-equivalence classes of all strongly $\Sigma$-measurable functions $f:\Omega \to X$ and consisting of all those $f\in L^0(X)$ for which the scalar function $\Vert f(\cdot ){\Vert }_X$ belongs to $E$. Let $E(X{)}_n^{\sim }$ stand for the order continuous dual of $E(X)$. We examine the Mackey topology $\tau (E(X),E(X{)}_n^{\sim })$ in case when it is locally solid. It is shown that $\tau (E(X),E(X{)}_n^{\sim })$ is the finest Hausdorff locally convex-solid topology on $E(X)$ with the Lebesgue property. We obtain that the space $(E(X),\tau (E(X),E(X{)}_n^{\sim }))$ is complete and sequentially barreled whenever $E$ is perfect. As an application, we obtain the Hahn-Vitali-Saks type theorem for sequences in $E(X{)}_n^{\sim }$. In particular, we consider the Mackey topology $\tau (L^{\Phi }(X),L^{\Phi }(X{)}_n^{\sim })$ on Orlicz-Bochner spaces $L^{\Phi }(X)$. We show that the space $(L^{\Phi }(X),\tau (L^{\Phi }(X),L^{\Phi }(X{)}_n^{\sim }))$ is complete iff $L^{\Phi }$ is perfect. Moreover, it is shown that the Mackey topology $\tau (L^{\infty }(X),L^{\infty }(X{)}_n^{\sim })$ is a mixed topology.