Locally Compact Hausdorff Space Continuous Function Uniformly Bounded

In mathematical analysis, and especially functional analysis, a fundamental role is played by the space of continuous functions on a compact Hausdorff space X {\displaystyle X} with values in the real or complex numbers. This space, denoted by C ( X ) , {\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

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

the uniform norm. The uniform norm defines the topology of uniform convergence of functions on X . {\displaystyle X.} The space C ( X ) {\displaystyle {\mathcal {C}}(X)} is a Banach algebra with respect to this norm.(Rudin 1973, §11.3)

Properties [edit]

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

Generalizations [edit]

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

The closure of C 00 ( X ) {\displaystyle C_{00}(X)} is precisely C 0 ( X ) . {\displaystyle 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. ISBN978-0-07-054236-5. OCLC 21163277.
  • Rudin, Walter (1966), Real and complex analysis, McGraw-Hill, ISBN0-07-054234-1 .

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Source: https://en.wikipedia.org/wiki/Continuous_functions_on_a_compact_Hausdorff_space

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