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 ${\displaystyle X}$ 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 ${\displaystyle \operatorname {rca} (X).}$ 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 ${\displaystyle {\mathcal {C}}(X)}$ 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 ${\displaystyle \Delta }$ 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, ${\displaystyle \Delta }$ can be identified with the collection of all complex homomorphisms ${\displaystyle {\mathcal {C}}(X)\to \mathbb {C} .}$ Equip ${\displaystyle \Delta }$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 ${\displaystyle X.}$

Generalizations

The space ${\displaystyle C(X)}$ of real or complex-valued continuous functions can be defined on any topological space ${\displaystyle X.}$ In the non-compact case, however, ${\displaystyle C(X)}$ 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 ${\displaystyle C_{B}(X)}$ of bounded continuous functions on ${\displaystyle X.}$ 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 ${\displaystyle X}$ is a locally compact Hausdorff space. In this case, it is possible to identify a pair of distinguished subsets of ${\displaystyle C_{B}(X)}$: ( Hewitt & Stromberg 1965 , §II.7)

• ${\displaystyle C_{00}(X),}$ the subset of ${\displaystyle C(X)}$ consisting of functions with compact support. This is called the space of functions vanishing in a neighborhood of infinity.
• ${\displaystyle C_{0}(X),}$ the subset of ${\displaystyle C(X)}$ consisting of functions such that for every ${\displaystyle r>0,}$ there is a compact set ${\displaystyle K\subseteq X}$ such that ${\displaystyle |f(x)|<r}$ for all ${\displaystyle x\in X\backslash K.}$ This is called the space of functions vanishing at infinity .

The closure of ${\displaystyle C_{00}(X)}$ is precisely ${\displaystyle C_{0}(X).}$ 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. 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 .