continuous functions on a compact Hausdorff space

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In
mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ...
, and especially
functional analysis Functional analysis is a branch of mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), ...
, a fundamental role is played by the space of
continuous functions In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
on a
compact Compact as used in politics may refer broadly to a pact A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations International relations (IR), international affairs (IA) or internationa ...
Hausdorff space In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...

$X$ with values in the
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...

or
complex numbers In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. This space, denoted by $\mathcal\left(X\right),$ is a
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
with respect to the pointwise addition of functions and scalar multiplication by constants. It is, moreover, a
normed space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
with norm defined by $\, f\, = \sup_ , f(x), ,$ the
uniform norm In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathe ...
. The uniform norm defines the
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

of
uniform convergenceIn the mathematical field of analysis, uniform convergence is a mode Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to: Language * Grammatical mode or grammatical mood, a category of verbal inflections t ...
of functions on $X.$ The space $\mathcal\left(X\right)$ is a
Banach algebra Banach is a Polish-language surname Polish names have two main elements: the ''imię'', the first name, or given name; and the ''nazwisko'', the last name, surname, family name (surname). The usage of personal names in Poland is generally govern ...
with respect to this norm.

# Properties

* By
Urysohn's lemma In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...
, $\mathcal\left(X\right)$ separates points of $X$: If $x, y \in X$ are distinct points, then there is an $f \in \mathcal\left(X\right)$ such that $f\left(x\right) \neq f\left(y\right).$ * The space $\mathcal\left(X\right)$ is infinite-dimensional whenever $X$ is an infinite space (since it separates points). Hence, in particular, it is generally not
locally compact In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
. * The
Riesz–Markov–Kakutani representation theorem In mathematics, the Riesz–Markov–Kakutani representation theorem relates linear functionals on spaces of continuous functions on a locally compact space to Measure (mathematics), measures in measure theory. The theorem is named for who intr ...
gives a characterization of the
continuous dual space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of $\mathcal\left(X\right).$ Specifically, this dual space is the space of
Radon measure In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
s on $X$ (regular
Borel measure In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s), denoted by $\operatorname\left(X\right).$ This space, with the norm given by the
total variation In mathematics, the total variation identifies several slightly different concepts, related to the (local property, local or global) structure of the codomain of a Function (mathematics), function or a measure (mathematics), measure. For a real n ...

of a measure, is also a Banach space belonging to the class of ba spaces. *
Positive linear functionalIn mathematics, more specifically in functional analysis, a positive linear functional on an ordered vector space (V, \le) is a linear functional f on V so that for all positive element (ordered group), positive elements v\in V, that is v\ge0, it hol ...
s on $\mathcal\left(X\right)$ correspond to (positive) Regular measure, regular
Borel measure In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s on $X,$ by a different form of the Riesz representation theorem. * If $X$ is infinite, then $\mathcal\left(X\right)$ is not Reflexive space, reflexive, nor is it Weak topology, weakly Complete space, complete. * The Arzelà–Ascoli theorem holds: A subset $K$ of $\mathcal\left(X\right)$ is relatively compact if and only if it is Bounded set, bounded in the norm of $\mathcal\left(X\right),$ and equicontinuous. * The Stone–Weierstrass theorem holds for $\mathcal\left(X\right).$ In the case of real functions, if $A$ is a subring of $\mathcal\left(X\right)$ that contains all constants and separates points, then the Closure (topology), closure of $A$ is $\mathcal\left(X\right).$ In the case of complex functions, the statement holds with the additional hypothesis that $A$ is closed under complex conjugation. * If $X$ and $Y$ are two compact Hausdorff spaces, and $F : \mathcal\left(X\right) \to \mathcal\left(Y\right)$ is a homomorphism of algebras which commutes with complex conjugation, then $F$ is continuous. Furthermore, $F$ has the form $F\left(h\right)\left(y\right) = h\left(f\left(y\right)\right)$ for some continuous function $f : Y \to X.$ In particular, if $C\left(X\right)$ and $C\left(Y\right)$ are isomorphic as algebras, then $X$ and $Y$ are homeomorphism, homeomorphic topological spaces. * Let $\Delta$ be the space of maximal ideals in $\mathcal\left(X\right).$ Then there is a one-to-one correspondence between Δ and the points of $X.$ Furthermore, $\Delta$ can be identified with the collection of all complex homomorphisms $\mathcal\left(X\right) \to \Complex.$ Equip $\Delta$with the initial topology with respect to this pairing with $\mathcal\left(X\right)$ (that is, the Gelfand transform). Then $X$ is homeomorphic to Δ equipped with this topology. * A sequence in $\mathcal\left(X\right)$ is Weak topology, weakly Cauchy sequence, Cauchy if and only if it is (uniformly) bounded in $\mathcal\left(X\right)$ and pointwise convergent. In particular, $\mathcal\left(X\right)$ is only weakly complete for $X$ a finite set. * The vague topology is the weak* topology on the dual of $\mathcal\left(X\right).$ * The Banach–Alaoglu theorem implies that any normed space is isometrically isomorphic to a subspace of $C\left(X\right)$ for some $X.$

# Generalizations

The space $C\left(X\right)$ of real or complex-valued continuous functions can be defined on any topological space $X.$ In the non-compact case, however, $C\left(X\right)$ 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 $C_B\left(X\right)$ of bounded continuous functions on $X.$ This is a Banach space (in fact a commutative Banach algebra with identity) with respect to the uniform norm. It is sometimes desirable, particularly in measure theory, to further refine this general definition by considering the special case when $X$ is a
locally compact In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
Hausdorff space. In this case, it is possible to identify a pair of distinguished subsets of $C_B\left(X\right)$: * $C_\left(X\right),$ the subset of $C\left(X\right)$ consisting of functions with compact support. This is called the space of functions vanishing in a neighborhood of infinity. * $C_0\left(X\right),$ the subset of $C\left(X\right)$ consisting of functions such that for every $r > 0,$ there is a compact set $K \subseteq X$ such that $, f\left(x\right), < r$ for all $x \in X \backslash K.$ This is called the space of functions vanish at infinity, vanishing at infinity. The closure of $C_\left(X\right)$ is precisely $C_0\left(X\right).$ In particular, the latter is a Banach space.

# References

* . * . * * . {{Functional analysis Banach spaces Complex analysis Continuous mappings Functional analysis Real analysis Types of functions