Continuous functions on a compact Hausdorff space

Properties

• By Urysohn's lemma, C separates points of X: If x, y ∈ X and x ≠ y, then there is an f ∈ C such that f ≠ f.
• The space C is infinite-dimensional whenever X is an infinite space. 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. Specifically, this dual space is the space of Radon measures on X, denoted by rca. 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.
• Positive linear functionals on C correspond to regular Borel measures on X, by a different form of the Riesz representation theorem.
• If X is infinite, then C is not reflexive, nor is it weakly complete.
• The Arzelà-Ascoli theorem holds: A subset K of C is relatively compact if and only if it is bounded in the norm of C, and equicontinuous.
• The Stone-Weierstrass theorem holds for C. In the case of real functions, if A is a subring of C that contains all constants and separates points, then the closure of A is C. 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 → C is a homomorphism of algebras which commutes with complex conjugation, then F is continuous. Furthermore, F has the form F = h for some continuous function ƒ : Y → X. In particular, if C and C are isomorphic as algebras, then X and Y are homeomorphic topological spaces.
• Let Δ be the space of maximal ideals in C. 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 → C. Equip Δ with the initial topology with respect to this pairing with C. Then X is homeomorphic to Δ equipped with this topology.
• A sequence in C is weakly Cauchy if and only if it is bounded in C and pointwise convergent. In particular, C is only weakly complete for X a finite set.
• The vague topology is the weak* topology on the dual of C.
• The Banach–Alaoglu theorem implies that any normed space is isometrically isomorphic to a subspace of C for some X.

  Generalizations

• C₀₀, the subset of C consisting of functions with compact support. This is called the space of functions vanishing in a neighborhood of infinity.
• C₀, the subset of C consisting of functions such that for every ε > 0, there is a compact set K⊂X such that |f| < ε for all x ∈ X\K. This is called the space of functions vanishing at infinity.