Abstract
Let C(X; E) be the space of the continuous functions on the completely regular and Hausdorff space X, with values in the locally convex topological vector space E. We introduce locally convex topologies on C(X; E) by means of uniform convergence on subsets of X or of the repletion of X. A generalization of a result of Singer gives a representation of the continuous linear functionals on these spaces by means of vector-valued measures which admit a kind of support. This allows a study of the bounded subsets of the dual, comparable to the one of the scalar case. We also give some results stating when these spaces are ultrabornological or bornological.
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