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Continuous functions on a compact Hausdorff space

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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.(Rudin 1973, §11.3)

Properties

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  • 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. (Dunford & Schwartz 1958, §IV.6.3)
  • Positive linear functionals on correspond to (positive) regular Borel measures on by a different form of the Riesz representation theorem. (Rudin 1966, Chapter 2)
  • 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.
  • Let be the space of maximal ideals in Then there is a one-to-one correspondence between Δ and the points of Furthermore, can be identified with the collection of all complex homomorphisms Equip with the initial topology with respect to this pairing with (that is, the Gelfand transform). Then is homeomorphic to Δ equipped with this topology. (Rudin 1973, §11.13)
  • A sequence in is weakly Cauchy if and only if it is (uniformly) bounded in and pointwise convergent. In particular, is only weakly complete for a finite set.
  • The vague topology is the weak* topology on the dual of
  • The Banach–Alaoglu theorem implies that any normed space is isometrically isomorphic to a subspace of for some

Generalizations

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

  • the subset of consisting of functions with compact support. This is called the space of functions vanishing in a neighborhood of infinity.
  • the subset of consisting of functions such that for every there is a compact set such that for all This is called the space of functions vanishing at infinity.

The closure of is precisely In particular, the latter is a Banach space.

References

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  • 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.