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 ${\displaystyle X}$ with values in the real or complex numbers. This space, denoted by ${\displaystyle {\mathcal {C}}(X)}$, 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

${\displaystyle \|f\|=\sup _{x\in X}|f(x)|,}$

the uniform norm. The uniform norm defines the topology of uniform convergence of functions on ${\displaystyle X}$. The space ${\displaystyle {\mathcal {C}}(X)}$ is a Banach algebra with respect to this norm. (Rudin 1973, §11.3)

Properties

• By Urysohn's lemma, ${\displaystyle {\mathcal {C}}(X)}$ separates points of ${\displaystyle X}$: If ${\displaystyle x,y\in X}$ are distinct points, then there is an ${\displaystyle f\in {\mathcal {C}}(X)}$ such that ${\displaystyle f(x)\neq f(y)}$.
• The space ${\displaystyle {\mathcal {C}}(X)}$ 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 ${\displaystyle {\mathcal {C}}(X)}$. 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. (Dunford & Schwartz 1958, §IV.6.3)
• Positive linear functionals on ${\displaystyle {\mathcal {C}}(X)}$ correspond to (positive) regular Borel measures on ${\displaystyle X}$, by a different form of the Riesz representation theorem. (Rudin 1966, Chapter 2)
• If X is infinite, then ${\displaystyle {\mathcal {C}}(X)}$ is not reflexive, nor is it weakly complete.
• The Arzelà-Ascoli theorem holds: A subset K of ${\displaystyle {\mathcal {C}}(X)}$ is relatively compact if and only if it is bounded in the norm of ${\displaystyle {\mathcal {C}}(X)}$, and equicontinuous.
• The Stone-Weierstrass theorem holds for ${\displaystyle {\mathcal {C}}(X)}$. In the case of real functions, if A is a subring of ${\displaystyle {\mathcal {C}}(X)}$ that contains all constants and separates points, then the closure of A is ${\displaystyle {\mathcal {C}}(X)}$. 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 ${\displaystyle F:{\mathcal {C}}(X)\to {\mathcal {C}}(Y)}$ 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 ${\displaystyle {\mathcal {C}}(X)}$. Then there is a one-to-one correspondence between Δ and the points of ${\displaystyle X}$. Furthermore, Δ can be identified with the collection of all complex homomorphisms ${\displaystyle {\mathcal {C}}(X)}$ → C. Equip Δ with the initial topology with respect to this pairing with ${\displaystyle {\mathcal {C}}(X)}$ (i.e., the Gelfand transform). Then ${\displaystyle X}$ is homeomorphic to Δ equipped with this topology. (Rudin 1973, §11.13)
• A sequence in ${\displaystyle {\mathcal {C}}(X)}$ is weakly Cauchy if and only if it is (uniformly) bounded in ${\displaystyle {\mathcal {C}}(X)}$ and pointwise convergent. In particular, ${\displaystyle {\mathcal {C}}(X)}$ is only weakly complete for ${\displaystyle X}$ a finite set.
• The vague topology is the weak* topology on the dual of ${\displaystyle {\mathcal {C}}(X)}$.
• 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. (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)

• C₀₀(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.
• C₀(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 C₀₀(X) is precisely C₀(X). 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 (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.
• Rudin, Walter (1966), Real and complex analysis, McGraw-Hill, ISBN 0-07-054234-1.