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.
is a vector space with respect to the pointwise addition of functions and scalar multiplication by constants.
is a Banach algebra with respect to this norm.
of real or complex-valued continuous functions can be defined on any topological space
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
of bounded continuous functions on
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
is a locally compact Hausdorff space.
In this case, it is possible to identify a pair of distinguished subsets of