Studies in the space C(X) of real-valued continuous functions on a topological space X

Parsons, Wade William (1987) Studies in the space C(X) of real-valued continuous functions on a topological space X. Masters thesis, Memorial University of Newfoundland.

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Abstract

In this work, we study some aspects of the ring C(X) of real-valued continuous functions defined on a topological space X. Explicitly we consider C(X) as a ring, as a Banach space and as a Topological Vector Space. Our aim, throughout this work, is to relate the topological properties of X with appropriate properties of a suitable structure on C(X). -- In Chapter 0, we develop the necessary prerequisites, and show that completely regular spaces are the right kind of topological spaces for such a study. The ideals and filters in the ring C(X) are studied in Chapter 1, and a one-to-one correspondence is established between maximal ideals in C(X) and ultrafilters on X. These are then employed to construct the classical Stone-Cech Compactification βX and Hewitt's realcompactification ʊX. This leads to the consideration of how to "recover" the underlying space X from the ring C(X), generating some "Banach-Stone" type theorems. Alternative constructions of βX and ʊX are furnished. -- Taking X to be a compact Hausdorff space, we then study the Banach space C(X) under the uniform norm, and describe its dual space via the Riesz Representation Theorem. The necessary and sufficient conditions for an arbitrary Banach space B, to be C(X) for some X, are then obtained. A version of the Banach-Stone theorem is proved, namely, the Banach spaces C(X) and C(Y) are isometric if and only if X and Y are homeomorphic. The class of spaces C(X), for X compact Hausdorff and extremally disconnected, is precisely the class of " Hahn-Banach spaces", (i.e., they enjoy the Hahn-Banach extension property). -- The final chapter is devoted to the study of C(X) as a topological vector space, when equipped with the compact-open topology. The conditions on X, necessary and sufficient for C(X) to be metrizable, barreled, and bornological, respectively, are then given. It turns out that C(X) is bornological when and only when X is realcompact.

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