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Duality theory of spaces of vector-valued continuous func- tions

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Marian Nowak, Aleksandra Rzepka

Duality theory of spaces of vector-valued continuous func- tions

Comment.Math.Univ.Carolinae 46,1 (2005) 55-73.

Abstract: LetX be a completely regular Hausdorff space,Ea real normed space, and letCb(X, E) be the space of all bounded continuousE-valued functions onX. We develop the general duality theory of the spaceCb(X, E) endowed with locally solid topologies; in particular with the strict topologies βz(X, E) for z = σ, τ, t.

As an application, we consider criteria for relative weak-star compactness in the spaces of vector measuresMz(X, E0) forz=σ, τ, t. It is shown that if a subsetH of Mz(X, E0) is relativelyσ(Mz(X, E0), Cb(X, E))-compact, then the set conv(S(H)) is still relatively σ(Mz(X, E0), Cb(X, E))-compact (S(H) = the solid hull ofH in Mz(X, E0)). A Mackey-Arens type theorem for locally convex-solid topologies on Cb(X, E) is obtained.

Keywords: vector-valued continuous functions, strict topologies, locally solid topologies, weak-star compactness, vector measures

AMS Subject Classification: 46E10, 46E15, 46E40, 46G10

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