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 $X$ 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 $$\Vert f\Vert =\underset{x\in X}{\sup }|f(x)|,$$ 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 $x,y\in X$ are distinct points, then there is an $f\in 𝒞(X)$ such that $f(x)\ne f(y).$
• 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 $\mathrm{rca}(X).$ 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. 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
• If $X$ is infinite, then $𝒞(X)$ is not reflexive, nor is it weakly complete.
• 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 $F:𝒞(X)\to 𝒞(Y)$ 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 $f:Y\to X.$ In particular, if $C(X)$ and $C(Y)$ are isomorphic as algebras, then $X$ and $Y$ are homeomorphic topological spaces.
• Let $\mathrm{\Delta }$ 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, $\mathrm{\Delta }$ can be identified with the collection of all complex homomorphisms $𝒞(X)\to ℂ.$ Equip $\mathrm{\Delta }$with the initial topology with respect to this pairing with $𝒞(X)$ (that is, 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 $r>0,$ there is a compact set $K\subseteq X$ such that $|f(x)|<r$ for all $x\in X\mathrm{\setminus }K.$ 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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