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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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