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 [math]\displaystyle{ X }[/math] with values in the real or complex numbers. This space, denoted by [math]\displaystyle{ \mathcal{C}(X), }[/math] 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 [math]\displaystyle{ \|f\| = \sup_{x\in X} |f(x)|, }[/math] the uniform norm. The uniform norm defines the topology of uniform convergence of functions on [math]\displaystyle{ X. }[/math] The space [math]\displaystyle{ \mathcal{C}(X) }[/math] is a Banach algebra with respect to this norm.(Rudin 1973)

Properties

• By Urysohn's lemma, [math]\displaystyle{ \mathcal{C}(X) }[/math] separates points of [math]\displaystyle{ X }[/math]: If [math]\displaystyle{ x, y \in X }[/math] are distinct points, then there is an [math]\displaystyle{ f \in \mathcal{C}(X) }[/math] such that [math]\displaystyle{ f(x) \neq f(y). }[/math]
• The space [math]\displaystyle{ \mathcal{C}(X) }[/math] is infinite-dimensional whenever [math]\displaystyle{ X }[/math] 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 [math]\displaystyle{ \mathcal{C}(X). }[/math] Specifically, this dual space is the space of Radon measures on [math]\displaystyle{ X }[/math] (regular Borel measures), denoted by [math]\displaystyle{ \operatorname{rca}(X). }[/math] 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. (Dunford Schwartz)
• Positive linear functionals on [math]\displaystyle{ \mathcal{C}(X) }[/math] correspond to (positive) regular Borel measures on [math]\displaystyle{ X, }[/math] by a different form of the Riesz representation theorem. (Rudin 1966)
• If [math]\displaystyle{ X }[/math] is infinite, then [math]\displaystyle{ \mathcal{C}(X) }[/math] is not reflexive, nor is it weakly complete.
• The Arzelà–Ascoli theorem holds: A subset [math]\displaystyle{ K }[/math] of [math]\displaystyle{ \mathcal{C}(X) }[/math] is relatively compact if and only if it is bounded in the norm of [math]\displaystyle{ \mathcal{C}(X), }[/math] and equicontinuous.
• The Stone–Weierstrass theorem holds for [math]\displaystyle{ \mathcal{C}(X). }[/math] In the case of real functions, if [math]\displaystyle{ A }[/math] is a subring of [math]\displaystyle{ \mathcal{C}(X) }[/math] that contains all constants and separates points, then the closure of [math]\displaystyle{ A }[/math] is [math]\displaystyle{ \mathcal{C}(X). }[/math] In the case of complex functions, the statement holds with the additional hypothesis that [math]\displaystyle{ A }[/math] is closed under complex conjugation.
• If [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are two compact Hausdorff spaces, and [math]\displaystyle{ F : \mathcal{C}(X) \to \mathcal{C}(Y) }[/math] is a homomorphism of algebras which commutes with complex conjugation, then [math]\displaystyle{ F }[/math] is continuous. Furthermore, [math]\displaystyle{ F }[/math] has the form [math]\displaystyle{ F(h)(y) = h(f(y)) }[/math] for some continuous function [math]\displaystyle{ f : Y \to X. }[/math] In particular, if [math]\displaystyle{ C(X) }[/math] and [math]\displaystyle{ C(Y) }[/math] are isomorphic as algebras, then [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are homeomorphic topological spaces.
• Let [math]\displaystyle{ \Delta }[/math] be the space of maximal ideals in [math]\displaystyle{ \mathcal{C}(X). }[/math] Then there is a one-to-one correspondence between Δ and the points of [math]\displaystyle{ X. }[/math] Furthermore, [math]\displaystyle{ \Delta }[/math] can be identified with the collection of all complex homomorphisms [math]\displaystyle{ \mathcal{C}(X) \to \Complex. }[/math] Equip [math]\displaystyle{ \Delta }[/math]with the initial topology with respect to this pairing with [math]\displaystyle{ \mathcal{C}(X) }[/math] (that is, the Gelfand transform). Then [math]\displaystyle{ X }[/math] is homeomorphic to Δ equipped with this topology. (Rudin 1973)
• A sequence in [math]\displaystyle{ \mathcal{C}(X) }[/math] is weakly Cauchy if and only if it is (uniformly) bounded in [math]\displaystyle{ \mathcal{C}(X) }[/math] and pointwise convergent. In particular, [math]\displaystyle{ \mathcal{C}(X) }[/math] is only weakly complete for [math]\displaystyle{ X }[/math] a finite set.
• The vague topology is the weak* topology on the dual of [math]\displaystyle{ \mathcal{C}(X). }[/math]
• The Banach–Alaoglu theorem implies that any normed space is isometrically isomorphic to a subspace of [math]\displaystyle{ C(X) }[/math] for some [math]\displaystyle{ X. }[/math]

Generalizations

The space [math]\displaystyle{ C(X) }[/math] of real or complex-valued continuous functions can be defined on any topological space [math]\displaystyle{ X. }[/math] In the non-compact case, however, [math]\displaystyle{ C(X) }[/math] 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 [math]\displaystyle{ C_B(X) }[/math] of bounded continuous functions on [math]\displaystyle{ X. }[/math] This is a Banach space (in fact a commutative Banach algebra with identity) with respect to the uniform norm. (Hewitt Stromberg)

It is sometimes desirable, particularly in measure theory, to further refine this general definition by considering the special case when [math]\displaystyle{ X }[/math] is a locally compact Hausdorff space. In this case, it is possible to identify a pair of distinguished subsets of [math]\displaystyle{ C_B(X) }[/math]: (Hewitt Stromberg)

• [math]\displaystyle{ C_{00}(X), }[/math] the subset of [math]\displaystyle{ C(X) }[/math] consisting of functions with compact support. This is called the space of functions vanishing in a neighborhood of infinity.
• [math]\displaystyle{ C_0(X), }[/math] the subset of [math]\displaystyle{ C(X) }[/math] consisting of functions such that for every [math]\displaystyle{ r \gt 0, }[/math] there is a compact set [math]\displaystyle{ K \subseteq X }[/math] such that [math]\displaystyle{ |f(x)| \lt r }[/math] for all [math]\displaystyle{ x \in X \backslash K. }[/math] This is called the space of functions vanishing at infinity.

The closure of [math]\displaystyle{ C_{00}(X) }[/math] is precisely [math]\displaystyle{ C_0(X). }[/math] 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 (January 1, 1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. https://archive.org/details/functionalanalys00rudi. 
• Rudin, Walter (1966), Real and complex analysis, McGraw-Hill, ISBN 0-07-054234-1 .