Continuous functions on a compact Hausdorff space

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 ${\displaystyle f(x)\ne f(y).}$
• The space ${\displaystyle \mathcal{C}(X)}$ is infinite-dimensional whenever ${\displaystyle X}$ 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 ${\displaystyle X}$ (regular Borel measures), denoted by ${\displaystyle \mathrm{rca}(X).}$ 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 ${\displaystyle X}$ is infinite, then ${\displaystyle \mathcal{C}(X)}$ is not reflexive, nor is it weakly complete.
• The Arzelà–Ascoli theorem holds: A subset ${\displaystyle K}$ 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 ${\displaystyle A}$ is a subring of ${\displaystyle \mathcal{C}(X)}$ that contains all constants and separates points, then the closure of ${\displaystyle A}$ is ${\displaystyle \mathcal{C}(X).}$ In the case of complex functions, the statement holds with the additional hypothesis that ${\displaystyle A}$ is closed under complex conjugation.
• If ${\displaystyle X}$ and ${\displaystyle Y}$ 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 ${\displaystyle F}$ is continuous. Furthermore, ${\displaystyle F}$ has the form ${\displaystyle F(h)(y)=h(f(y))}$ for some continuous function ${\displaystyle f:Y\to X.}$ In particular, if ${\displaystyle C(X)}$ and ${\displaystyle C(Y)}$ are isomorphic as algebras, then ${\displaystyle X}$ and ${\displaystyle Y}$ are homeomorphic topological spaces.
• Let ${\displaystyle \mathrm{\Delta }}$ 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, ${\displaystyle \mathrm{\Delta }}$ can be identified with the collection of all complex homomorphisms ${\displaystyle \mathcal{C}(X)\to \mathbb{C}.}$ Equip ${\displaystyle \mathrm{\Delta }}$with the initial topology with respect to this pairing with ${\displaystyle \mathcal{C}(X)}$ (that is, 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 ${\displaystyle C(X)}$ for some ${\displaystyle X.}$