# Mackey Topologies on Vector-Valued Function Spaces

### Marian Nowak

University of Zielona Góra, Poland

## Abstract

Let $E$ be an ideal of $L_{0}$ over a $σ$-finite measure space $(Ω,Σ,μ)$, and let $(X,∥⋅∥_{X})$ be a real Banach space. Let $E(X)$ be a subspace of the space $L_{0}(X)$ of $μ$-equivalence classes of all strongly $Σ$-measurable functions $f:Ω→X$ and consisting of all those $f∈L_{0}(X)$ for which the scalar function $∥f(⋅)∥_{X}$ belongs to $E$. Let $E(X)_{n}$ stand for the order continuous dual of $E(X)$. We examine the Mackey topology $τ(E(X),E(X)_{n})$ in case when it is locally solid. It is shown that $τ(E(X),E(X)_{n})$ is the finest Hausdorff locally convex-solid topology on $E(X)$ with the Lebesgue property. We obtain that the space $(E(X),τ(E(X),E(X)_{n}))$ is complete and sequentially barreled whenever $E$ is perfect. As an application, we obtain the Hahn-Vitali-Saks type theorem for sequences in $E(X)_{n}$. In particular, we consider the Mackey topology $τ(L_{Φ}(X),L_{Φ}(X)_{n})$ on Orlicz-Bochner spaces $L_{Φ}(X)$. We show that the space $(L_{Φ}(X),τ(L_{Φ}(X),L_{Φ}(X)_{n}))$ is complete iff $L_{Φ}$ is perfect. Moreover, it is shown that the Mackey topology $τ(L_{∞}(X),L_{∞}(X)_{n})$ is a mixed topology.

## Cite this article

Marian Nowak, Mackey Topologies on Vector-Valued Function Spaces. Z. Anal. Anwend. 24 (2005), no. 2, pp. 327–340

DOI 10.4171/ZAA/1243