Let (Ω, Σ, μ) be a σ-finite measure space and let % and <%o denote the usual metrizable topologies on L° and L°°, respectively. In this paper the space L°° with the mixed topology 7(^,^|L°°) is examined. It is proved that γ(<%o,<%\L°°) is the finest Lebesgue topology on L°°, and that it coincides with the Mackey topology τ(L°°, L 1 ).
LetXbe a completely regular Hausdorff space, and let(E,‖·‖E)and(F,‖·‖F)be Banach spaces. LetCb(X,E)be the space of allE-valued bounded, continuous functions defined onX, equipped with the strict topologiesβz, where z=σ,∞,p,τ,t. General integral representation theorems of(βz,‖·‖F)-continuous linear operators T:Cb(X,E)→F with respect to the corresponding operator-valued measures are established. Strongly bounded and(βz,‖·‖F)-continuous operatorsT:Cb(X,E)→Fare studied. We extend to “the completely regular setting” some classical results concerning operators on the spacesC(X,E)andCo(X,E), whereX is a compact or a locally compact space.
T is (γ ϕ , · Y )-continuous, where γ ϕ stands for a natural mixed topology on L ϕ (µ, X). As an application, we derive Vitali-Hahn-Saks type theorems for families of operator measures.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.