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Weak sequential convergence and weak compactness in spaces of vector-valued continuous functions. (English) Zbl 0854.46032
Let \(X\) be a completely regular Hausdorff space, \(E\) be a Hausdorff locally convex space and \(C_b (X, E)\) be the space of all bounded continuous \(E\)-valued functions on \(X\). The necessary and sufficient conditions that a sequence \((f_n)\) of elements of \(C_b (X, E)\) would be weakly convergent to zero in \((C_b (X, E), \tau)\) and a subset \(H\subset C_b (X, E)\) would be weakly relatively (countably) compact in \((C_b (X, E), \tau)\) are given in the case, when \(\tau\) is one of the strict topologies defined on \(C_b (X, E)\).
Reviewer: M.Abel (Tartu)

MSC:
46E40 Spaces of vector- and operator-valued functions
46E10 Topological linear spaces of continuous, differentiable or analytic functions
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