Topological properties of some spaces of continuous operators

Nowak, Marian

Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 36 (2016) no. 1, pp. 79-86.

Résumé

Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let C_b(X,E) be the space of all E-valued bounded continuous functions on X, equipped with the strict topology β. We study topological properties of the space L_β(C_b(X,E),F) of all (β,||·||_F)-continuous linear operators from C_b(X,E) to F, equipped with the topology τ_s of simple convergence. If X is a locally compact paracompact space (resp. a P-space), we characterize τ_s-compact subsets of L_β(C_b(X,E),F) in terms of properties of the corresponding sets of the representing operator-valued Borel measures. It is shown that the space (L_β(C_b(X,E),F),τ_s) is sequentially complete if X is a locally compact paracompact space.

Détail

Keywords: spaces of vector-valued continuous functions, strict topologies, operator measures, topology of simple convergence, continuous operators

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