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
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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 $\backslash mathcal(X),$ 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
$$\backslash ,\; f\backslash ,\; =\; \backslash sup\_\; ,\; f(x),\; ,$$
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 $\backslash mathcal(X)$ is a Banach algebra
Banach is a Polish-language surname
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with respect to this norm.
Properties

* ByUrysohn'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 ...

, $\backslash mathcal(X)$ separates points of $X$: If $x,\; y\; \backslash in\; X$ are distinct points, then there is an $f\; \backslash in\; \backslash mathcal(X)$ such that $f(x)\; \backslash neq\; f(y).$
* The space $\backslash mathcal(X)$ 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
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of $\backslash mathcal(X).$ 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 $\backslash operatorname(X).$ 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 $\backslash mathcal(X)$ 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 $\backslash mathcal(X)$ is not Reflexive space, reflexive, nor is it Weak topology, weakly Complete space, complete.
* The Arzelà–Ascoli theorem holds: A subset $K$ of $\backslash mathcal(X)$ is relatively compact if and only if it is Bounded set, bounded in the norm of $\backslash mathcal(X),$ and equicontinuous.
* The Stone–Weierstrass theorem holds for $\backslash mathcal(X).$ In the case of real functions, if $A$ is a subring of $\backslash mathcal(X)$ that contains all constants and separates points, then the Closure (topology), closure of $A$ is $\backslash mathcal(X).$ 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 $F\; :\; \backslash mathcal(X)\; \backslash to\; \backslash mathcal(Y)$ is a homomorphism of algebras which commutes with complex conjugation, then $F$ is continuous. Furthermore, $F$ has the form $F(h)(y)\; =\; h(f(y))$ for some continuous function $f\; :\; Y\; \backslash to\; X.$ In particular, if $C(X)$ and $C(Y)$ are isomorphic as algebras, then $X$ and $Y$ are homeomorphism, homeomorphic topological spaces.
* Let $\backslash Delta$ be the space of maximal ideals in $\backslash mathcal(X).$ Then there is a one-to-one correspondence between Δ and the points of $X.$ Furthermore, $\backslash Delta$ can be identified with the collection of all complex homomorphisms $\backslash mathcal(X)\; \backslash to\; \backslash Complex.$ Equip $\backslash Delta$with the initial topology with respect to this pairing with $\backslash mathcal(X)$ (that is, the Gelfand transform). Then $X$ is homeomorphic to Δ equipped with this topology.
* A sequence in $\backslash mathcal(X)$ is Weak topology, weakly Cauchy sequence, Cauchy if and only if it is (uniformly) bounded in $\backslash mathcal(X)$ and pointwise convergent. In particular, $\backslash mathcal(X)$ is only weakly complete for $X$ a finite set.
* The vague topology is the weak* topology on the dual of $\backslash mathcal(X).$
* The Banach–Alaoglu theorem implies that any normed space is isometrically isomorphic to a subspace of $C(X)$ for some $X.$
Generalizations

The space $C(X)$ of real or complex-valued continuous functions can be defined on any topological space $X.$ In the non-compact case, however, $C(X)$ 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 $C\_B(X)$ 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 alocally 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 $C\_B(X)$:
* $C\_(X),$ the subset of $C(X)$ consisting of functions with compact support. This is called the space of functions vanishing in a neighborhood of infinity.
* $C\_0(X),$ the subset of $C(X)$ consisting of functions such that for every $r\; >\; 0,$ there is a compact set $K\; \backslash subseteq\; X$ such that $,\; f(x),\; <\; r$ for all $x\; \backslash in\; X\; \backslash backslash\; K.$ This is called the space of functions vanish at infinity, vanishing at infinity.
The closure of $C\_(X)$ is precisely $C\_0(X).$ In particular, the latter is a Banach space.
References

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