# 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

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. Template:Harv

## Properties

- By Urysohn's lemma,
*C*(*X*) separates points of*X*: If*x*,*y*∈*X*and*x*≠*y*, then there is an*f*∈*C*(*X*) such that*f*(*x*) ≠*f*(*y*).

- The space
*C*(*X*) is infinite-dimensional whenever*X*is an infinite space (since it separates points). Hence, in particular, it is generally not locally compact.

- The Riesz–Markov–Kakutani representation theorem gives a characterization of the continuous dual space of
*C*(*X*). Specifically, this dual space is the space of Radon measures on*X*(regular Borel measures), denoted by*rca*(*X*). This space, with the norm given by the total variation of a measure, is also a Banach space belonging to the class of ba spaces. Template:Harv

- Positive linear functionals on
*C*(*X*) correspond to (positive) regular Borel measures on*X*, by a different form of the Riesz representation theorem. Template:Harv

- The Arzelà-Ascoli theorem holds: A subset
*K*of*C*(*X*) is relatively compact if and only if it is bounded in the norm of*C*(*X*), and equicontinuous.

- The Stone-Weierstrass theorem holds for
*C*(*X*). In the case of real functions, if*A*is a subring of*C*(*X*) that contains all constants and separates points, then the closure of*A*is*C*(*X*). In the case of complex functions, the statement holds with the additional hypothesis that*A*is closed under complex conjugation.

- If
*X*and*Y*are two compact Hausdorff spaces, and*F*:*C*(*X*) →*C*(*Y*) is a homomorphism of algebras which commutes with complex conjugation, then*F*is continuous. Furthermore,*F*has the form*F*(*h*)(*y*) =*h*(*f*(*y*)) for some continuous function*ƒ*:*Y*→*X*. In particular, if*C*(*X*) and*C*(*Y*) are isomorphic as algebras, then*X*and*Y*are homeomorphic topological spaces.

- Let Δ be the space of maximal ideals in
*C*(*X*). Then there is a one-to-one correspondence between Δ and the points of*X*. Furthermore Δ can be identified with the collection of all complex homomorphisms*C*(*X*) →**C**. Equip Δ with the initial topology with respect to this pairing with*C*(*X*) (i.e., the Gelfand transform). Then*X*is homeomorphic to Δ equipped with this topology. Template:Harv

- A sequence in
*C*(*X*) is weakly Cauchy if and only if it is (uniformly) bounded in*C*(*X*) and pointwise convergent. In particular,*C*(*X*) is only weakly complete for*X*a finite set.

- The vague topology is the weak* topology on the dual of
*C*(*X*).

- The Banach–Alaoglu theorem implies that any normed space is isometrically isomorphic to a subspace of
*C*(*X*) for some*X*.

## 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. Template:Harv

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*): Template:Harv

*C*_{00}(*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**.

*C*_{0}(*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 *C*_{00}(*X*) is precisely *C*_{0}(*X*). In particular, the latter is a Banach space.

## References

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