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Completeness and sequential completeness in certain spaces of measures. (English) Zbl 0832.46016
Summary: Let \(X\) be a completely regular Hausdorff space, \(E\) a Banach space over \(K\), the field of real or complex numbers, \(C(X, E)\) (\(C(X)\) if \(E= K\)) the space of all \(E\)-valued continuous functions on \(X\), and \(C_b(X, E)\) (\(C_b(X)\) if \(E= K\)) the space of all \(E\)-valued bounded continuous functions on \(X\). Put \(F_z= (C_b(X, E), \beta_z)\) (\(\beta_z\) the so called strict topologies), and \(F= (C(X, E), \beta_{\infty C})\).
It is proved that \((F_z', \sigma(F_z', F_z))\) is sequentially complete for \(z= \sigma, \infty, g\); if, in addition \(X\) is meta-compact and normal, then the result is also true for \(z= \tau\). Also it is proved that \((F', \sigma(F', F))\) is sequentially complete. For the Mackey topology it is proved that \((F_z', \tau(F_z', F_z))\) is complete for \(z= \sigma, \infty, g\) and for \(z= \tau(t)\) it is complete if and only if \(M_g(X)= M_\tau(x)\) \((M_t(X))\). Further, it is proved that \((F', \tau(F', F))\) is complete. Some additional results are proved for sequential convergence.

MSC:
46E10 Topological linear spaces of continuous, differentiable or analytic functions
47A70 (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces
46E40 Spaces of vector- and operator-valued functions
46G10 Vector-valued measures and integration
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