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 ${\mathcal {C}}(X),$ 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

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

the uniform norm. The uniform norm defines the topology of uniform convergence of functions on

X.

The space

{\mathcal {C}}(X)

is a Banach algebra with respect to this norm.(Rudin 1973, §11.3)

Properties[edit]

By Urysohn's lemma, ${\mathcal {C}}(X)$ separates points of $X$ : If $x,y\in X$ are distinct points, then there is an $f\in {\mathcal {C}}(X)$ such that $f(x)\neq f(y).$
The space ${\mathcal {C}}(X)$ 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 ${\mathcal {C}}(X).$ Specifically, this dual space is the space of Radon measures on $X$ (regular Borel measures), denoted by $\operatorname {rca} (X).$ 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 ${\mathcal {C}}(X)$ correspond to (positive) regular Borel measures on $X,$ by a different form of the Riesz representation theorem. (Rudin 1966, Chapter 2)
If $X$ is infinite, then ${\mathcal {C}}(X)$ is not reflexive, nor is it weakly complete.
The Arzelà–Ascoli theorem holds: A subset $K$ of ${\mathcal {C}}(X)$ is relatively compact if and only if it is bounded in the norm of ${\mathcal {C}}(X),$ and equicontinuous.
The Stone–Weierstrass theorem holds for ${\mathcal {C}}(X).$ In the case of real functions, if $A$ is a subring of ${\mathcal {C}}(X)$ that contains all constants and separates points, then the closure of $A$ is ${\mathcal {C}}(X).$ 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 $F:{\mathcal {C}}(X)\to {\mathcal {C}}(Y)$ 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 $f:Y\to X.$ In particular, if $C(X)$ and $C(Y)$ are isomorphic as algebras, then $X$ and $Y$ are homeomorphic topological spaces.
Let $\Delta$ be the space of maximal ideals in ${\mathcal {C}}(X).$ Then there is a one-to-one correspondence between Δ and the points of $X.$ Furthermore, $\Delta$ can be identified with the collection of all complex homomorphisms ${\mathcal {C}}(X)\to \mathbb {C} .$ Equip $\Delta$ with the initial topology with respect to this pairing with ${\mathcal {C}}(X)$ (that is, the Gelfand transform). Then $X$ is homeomorphic to Δ equipped with this topology. (Rudin 1973, §11.13)
A sequence in ${\mathcal {C}}(X)$ is weakly Cauchy if and only if it is (uniformly) bounded in ${\mathcal {C}}(X)$ and pointwise convergent. In particular, ${\mathcal {C}}(X)$ is only weakly complete for $X$ a finite set.
The vague topology is the weak* topology on the dual of ${\mathcal {C}}(X).$
The Banach–Alaoglu theorem implies that any normed space is isometrically isomorphic to a subspace of $C(X)$ for some $X.$

Generalizations[edit]

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)

$C_{00}(X),$ 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 $x\in X\backslash K.$ 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[edit]

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. Vol. 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.

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