Continuous functions on a compact Hausdorff space

In mathematical analysis, and especially functional analysis, a fundamental role is played by the space of continuous functions on a compact Hausdorff spacewith values in the real or complex numbers.is a vector space with respect to the pointwise addition of functions and scalar multiplication by constants.of real or complex-valued continuous functions can be defined on any topological spaceis not in general a Banach space with respect to the uniform norm since it may contain unbounded functions.This is a Banach space (in fact a commutative Banach algebra with identity) with respect to the uniform norm.(Hewitt & Stromberg 1965, Theorem 7.9) It is sometimes desirable, particularly in measure theory, to further refine this general definition by considering the special case whenis a locally compact Hausdorff space.In this case, it is possible to identify a pair of distinguished subsets of

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