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Strict topologies on spaces of continuous functions and $$u$$-additive measure spaces. (English) Zbl 0914.46031
Let $$X$$ be a completely regular Hausdorff space, $$E$$ a locally convex Hausdorff space, and $$C_b(X, E)$$ the space of all bounded continuous mappings from $$X$$ into $$E$$. We put $$C_b(X, E)= C_b(X)$$ in case $$E$$ is the real line. The author studies topologies for $$C_b(X, E)$$ defined by inductive limits of seminorms determined by certain compact subsets of $$\beta X\smallsetminus X$$. Two of this main results are that
(i) $$C_b(X)\otimes E$$ is dense in $$C_b(X, E)$$ and
(ii) the dual space of $$C_b(X, E)$$ is isomorphic to a certain family of $$E'$$-valued additive measures on the zero sets of $$X$$, where $$E'$$ is the dual of $$E$$.
He first proves these results by assuming that $$E$$ is a normed space, then drops this assumptions and derives them again by somewhat different methods. A reference to the source of Proposition 3.7 would help the reader; the author says that it is known but gives no source.

##### MSC:
 46E40 Spaces of vector- and operator-valued functions 46E10 Topological linear spaces of continuous, differentiable or analytic functions 54C35 Function spaces in general topology 46E27 Spaces of measures
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##### References:
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