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
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)|,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8af16f0b74844bf05d9d112fe964e6e6e2618d86)
the uniform norm. The uniform norm defines the topology of uniform convergence of functions on
The space
is a Banach algebra with respect to this norm.(Rudin 1973, §11.3)
Generalizations
The space
of real or complex-valued continuous functions can be defined on any topological space
In the non-compact case, however,
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
: (Hewitt & Stromberg 1965, §II.7)
the subset of
consisting of functions with compact support. This is called the space of functions vanishing in a neighborhood of infinity.
the subset of
consisting of functions such that for every
there is a compact set
such that
for all
This is called the space of functions vanishing at infinity.
The closure of
is precisely
In particular, the latter is a Banach space.