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Concept
Continuous functions on a compact Hausdorff space
Summary
In mathematical analysis, and especially functional analysis, a fundamental role is played by the space of continuous functions on a compact Hausdorff space with values in the real or complex numbers. This space, denoted by 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
the uniform norm. The uniform norm defines the topology of uniform convergence of functions on The space is a Banach algebra with respect to this norm.
By Urysohn's lemma, separates points of : If are distinct points, then there is an such that
The space is infinite-dimensional whenever 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 Specifically, this dual space is the space of Radon measures on (regular Borel measures), denoted by 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 correspond to (positive) regular Borel measures on by a different form of the Riesz representation theorem.
If is infinite, then is not reflexive, nor is it weakly complete.
The Arzelà–Ascoli theorem holds: A subset of is relatively compact if and only if it is bounded in the norm of and equicontinuous.
The Stone–Weierstrass theorem holds for In the case of real functions, if is a subring of that contains all constants and separates points, then the closure of is In the case of complex functions, the statement holds with the additional hypothesis that is closed under complex conjugation.
If and are two compact Hausdorff spaces, and is a homomorphism of algebras which commutes with complex conjugation, then is continuous. Furthermore, has the form for some continuous function In particular, if and are isomorphic as algebras, then and are homeomorphic topological spaces.
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