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
the uniform norm. The uniform norm defines the topology of uniform convergence of functions on X. The space C(X) is a Banach space with respect to this norm.
The space C(X) of real or complexvalued continuous functions can be defined on any topological space X. In the noncompact 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)
The closure of C_{00}(X) is precisely C_{0}(X). In particular, the latter is a Banach 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
the uniform norm. The uniform norm defines the topology of uniform convergence of functions on X. The space C(X) is a Banach space with respect to this norm.
The space C(X) of real or complexvalued continuous functions can be defined on any topological space X. In the noncompact 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)
The closure of C_{00}(X) is precisely C_{0}(X). In particular, the latter is a Banach space.
