JournalszaaVol. 24 , No. 2DOI 10.4171/zaa/1243

Mackey Topologies on Vector-Valued Function Spaces

  • Marian Nowak

    University of Zielona Góra, Poland
Mackey Topologies on Vector-Valued Function Spaces cover

Abstract

Let E\,E\, be an ideal of L0\,L^{0}\, over a σ\,\sigma-finite measure space (Ω,Σ,μ)\,(\Omega,\Sigma,\mu), and let (X,X)\,(X,\|\cdot\|_X)\, be a real Banach space. Let E(X)\,E(X)\, be a subspace of the space L0(X)\,L^{0}(X)\, of μ\,\mu-equivalence classes of all strongly Σ\,\Sigma-measurable functions f:ΩX\,f: \Omega\to X\, and consisting of all those fL0(X)\,f\in L^{0}(X)\, for which the scalar function f()X\,\|f(\cdot)\|_X\, belongs to E\,E. Let E(X)n\,E(X)_n^{\sim}\, stand for the order continuous dual of E(X)\,E(X). We examine the Mackey topology τ(E(X),E(X)n)\,\tau(E(X),E(X)_n^{\sim}) in case when it is locally solid. It is shown that τ(E(X),E(X)n)\,\tau(E(X),E(X)_n^{\sim})\, is the finest Hausdorff locally convex-solid topology on E(X)\,E(X)\, with the Lebesgue property. We obtain that the space (E(X),τ(E(X),E(X)n))\,(E(X),\tau(E(X), E(X)_n^{\sim}))\, is complete and sequentially barreled whenever E\,E\, is perfect. As an application, we obtain the Hahn-Vitali-Saks type theorem for sequences in E(X)n\,E(X)_n^{\sim}. In particular, we consider the Mackey topology τ(LΦ(X),LΦ(X)n)\,\tau(L^{\Phi}(X), L^{\Phi}(X)_n^{\sim})\, on Orlicz-Bochner spaces LΦ(X)\,L^{\Phi}(X). We show that the space (LΦ(X),τ(LΦ(X),LΦ(X)n))\,(L^{\Phi}(X), \tau(L^{\Phi}(X), L^{\Phi}(X)_n^{\sim}))\, is complete iff LΦ\,L^{\Phi}\, is perfect. Moreover, it is shown that the Mackey topology τ(L(X),L(X)n)\,\tau(L^{\infty}(X), L^{\infty}(X)_n^{\sim})\, is a mixed topology.