Английская Википедия: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 <math>X</math> with values in the real or complex numbers. This space, denoted by <math>\mathcal{C}(X),</math> 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 <math display=block>\|f\| = \sup_{x\in X} |f(x)|,</math> the uniform norm. The uniform norm defines the topology of uniform convergence of functions on <math>X.</math> The space <math>\mathcal{C}(X)</math> is a Banach algebra with respect to this norm.Шаблон:Harv
Properties
- By Urysohn's lemma, <math>\mathcal{C}(X)</math> separates points of <math>X</math>: If <math>x, y \in X</math> are distinct points, then there is an <math>f \in \mathcal{C}(X)</math> such that <math>f(x) \neq f(y).</math>
- The space <math>\mathcal{C}(X)</math> is infinite-dimensional whenever <math>X</math> 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 <math>\mathcal{C}(X).</math> Specifically, this dual space is the space of Radon measures on <math>X</math> (regular Borel measures), denoted by <math>\operatorname{rca}(X).</math> 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. Шаблон:Harv
- Positive linear functionals on <math>\mathcal{C}(X)</math> correspond to (positive) regular Borel measures on <math>X,</math> by a different form of the Riesz representation theorem. Шаблон:Harv
- If <math>X</math> is infinite, then <math>\mathcal{C}(X)</math> is not reflexive, nor is it weakly complete.
- The Arzelà–Ascoli theorem holds: A subset <math>K</math> of <math>\mathcal{C}(X)</math> is relatively compact if and only if it is bounded in the norm of <math>\mathcal{C}(X),</math> and equicontinuous.
- The Stone–Weierstrass theorem holds for <math>\mathcal{C}(X).</math> In the case of real functions, if <math>A</math> is a subring of <math>\mathcal{C}(X)</math> that contains all constants and separates points, then the closure of <math>A</math> is <math>\mathcal{C}(X).</math> In the case of complex functions, the statement holds with the additional hypothesis that <math>A</math> is closed under complex conjugation.
- If <math>X</math> and <math>Y</math> are two compact Hausdorff spaces, and <math>F : \mathcal{C}(X) \to \mathcal{C}(Y)</math> is a homomorphism of algebras which commutes with complex conjugation, then <math>F</math> is continuous. Furthermore, <math>F</math> has the form <math>F(h)(y) = h(f(y))</math> for some continuous function <math>f : Y \to X.</math> In particular, if <math>C(X)</math> and <math>C(Y)</math> are isomorphic as algebras, then <math>X</math> and <math>Y</math> are homeomorphic topological spaces.
- Let <math>\Delta</math> be the space of maximal ideals in <math>\mathcal{C}(X).</math> Then there is a one-to-one correspondence between Δ and the points of <math>X.</math> Furthermore, <math>\Delta</math> can be identified with the collection of all complex homomorphisms <math>\mathcal{C}(X) \to \Complex.</math> Equip <math>\Delta</math>with the initial topology with respect to this pairing with <math>\mathcal{C}(X)</math> (that is, the Gelfand transform). Then <math>X</math> is homeomorphic to Δ equipped with this topology. Шаблон:Harv
- A sequence in <math>\mathcal{C}(X)</math> is weakly Cauchy if and only if it is (uniformly) bounded in <math>\mathcal{C}(X)</math> and pointwise convergent. In particular, <math>\mathcal{C}(X)</math> is only weakly complete for <math>X</math> a finite set.
- The vague topology is the weak* topology on the dual of <math>\mathcal{C}(X).</math>
- The Banach–Alaoglu theorem implies that any normed space is isometrically isomorphic to a subspace of <math>C(X)</math> for some <math>X.</math>
Generalizations
The space <math>C(X)</math> of real or complex-valued continuous functions can be defined on any topological space <math>X.</math> In the non-compact case, however, <math>C(X)</math> 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 <math>C_B(X)</math> of bounded continuous functions on <math>X.</math> This is a Banach space (in fact a commutative Banach algebra with identity) with respect to the uniform norm. Шаблон:Harv
It is sometimes desirable, particularly in measure theory, to further refine this general definition by considering the special case when <math>X</math> is a locally compact Hausdorff space. In this case, it is possible to identify a pair of distinguished subsets of <math>C_B(X)</math>: Шаблон:Harv
- <math>C_{00}(X),</math> the subset of <math>C(X)</math> consisting of functions with compact support. This is called the space of functions vanishing in a neighborhood of infinity.
- <math>C_0(X),</math> the subset of <math>C(X)</math> consisting of functions such that for every <math>r > 0,</math> there is a compact set <math>K \subseteq X</math> such that <math>|f(x)| < r</math> for all <math>x \in X \backslash K.</math> This is called the space of functions vanishing at infinity.
The closure of <math>C_{00}(X)</math> is precisely <math>C_0(X).</math> In particular, the latter is a Banach space.
References
Шаблон:Banach spaces Шаблон:Functional analysis
