Abstract
Publisher Summary This chapter focuses on spaces of continuous functions. Assuming X be a completely regular and Hausdorff space and E be a locally convex topological Hausdorff vector space, it is possible to denote C (X) the space of the continuous functions on X and by C (X;E) the space of the continuous functions on X with values in E. A natural way to introduce new locally convex topologies on C(X) is the use of semi-norms of uniform convergence on suitable subsets of X. However, this is not the best way. A much better technique needs a large roundabout, passing through the characters of C (X) and the repletion of X. The linear space of the continuous functions on X with values in E is denoted by C (X;E) and its elements are represented by φ.
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