Continuous Functions On A Compact Hausdorff Space - Generalizations

Generalizations

The space C(X) of real or complex-valued continuous functions can be defined on any topological space X. In the non-compact case, however, C(X) is not in general a Banach space with respect to the uniform norm since it may contain unbounded functions. Hence it is more typical to consider the space, denoted here C_B(X) of bounded continuous functions on X. 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 when X is a locally compact Hausdorff space. In this case, it is possible to identify a pair of distinguished subsets of C_B(X): (Hewitt & Stromberg 1965, §II.7)

• C₀₀(X), the subset of C(X) consisting of functions with compact support. This is called the space of functions vanishing in a neighborhood of infinity.

• C₀(X), the subset of C(X) consisting of functions such that for every ε > 0, there is a compact set K⊂X such that |f(x)| < ε for all x ∈ X\K. This is called the space of functions vanishing at infinity.

The closure of C₀₀(X) is precisely C₀(X). In particular, the latter is a Banach space.