# 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 space with values in the real or complex numbers. This space, denoted by C(X), is a vector space with respect to the pointwise addition of functions and scalar multiplication by constants. It is, moreover, a normed space with norm defined by

${\displaystyle \|f\|=\sup _{x\in X}|f(x)|,}$

the uniform norm. The uniform norm defines the topology of uniform convergence of functions on X. The space C(X) is a Banach algebra with respect to this norm. (Rudin 1973, §11.3)

## 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 CB(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 CB(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 KX 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.

## References

• Dunford, N.; Schwartz, J.T. (1958), Linear operators, Part I, Wiley-Interscience.
• Hewitt, Edwin; Stromberg, Karl (1965), Real and abstract analysis, Springer-Verlag.
• Rudin, Walter (1973), Functional analysis, McGraw-Hill, ISBN 0-07-054236-8.
• Rudin, Walter (1966), Real and complex analysis, McGraw-Hill, ISBN 0-07-054234-1.