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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.

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
Full Text: DOI
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