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mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
, and especially
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
, a fundamental role is played by the space of continuous functions on a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ...
X with values in the real or
complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
. This space, denoted by \mathcal(X), is a
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
with respect to the pointwise addition of functions and scalar multiplication by constants. It is, moreover, a
normed space The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898. The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war p ...
with norm defined by \, f\, = \sup_ , f(x), , the uniform norm. The uniform norm defines the
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
of
uniform convergence In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E as the function domain i ...
of functions on X. The space \mathcal(X) is a Banach algebra with respect to this norm.


Properties

* By Urysohn's lemma, \mathcal(X) separates points of X: If x, y \in X are distinct points, then there is an f \in \mathcal(X) such that f(x) \neq f(y). * The space \mathcal(X) is infinite-dimensional whenever 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 \mathcal(X). Specifically, this dual space is the space of
Radon measure In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the -algebra of Borel sets of a Hausdorff topological space that is finite on all compact sets, outer regular on all Borel sets, and ...
s on X (regular
Borel measure In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. ...
s), denoted by \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 ...
of a measure, is also a Banach space belonging to the class of ba spaces. * Positive linear functionals on \mathcal(X) correspond to (positive) regular
Borel measure In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. ...
s on X, by a different form of the Riesz representation theorem. * If X is infinite, then \mathcal(X) is not reflexive, nor is it weakly complete. * The
Arzelà–Ascoli theorem The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded inte ...
holds: A subset K of \mathcal(X) is relatively compact if and only if it is bounded in the norm of \mathcal(X), and equicontinuous. * The
Stone–Weierstrass theorem In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval (mathematics), interval can be uniform convergence, uniformly approximated as closely as desired by a polynomial fun ...
holds for \mathcal(X). In the case of real functions, if A is a
subring In mathematics, a subring of a ring is a subset of that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and that shares the same multiplicative identity as .In general, not all s ...
of \mathcal(X) that contains all constants and separates points, then the closure of A is \mathcal(X). In the case of complex functions, the statement holds with the additional hypothesis that A is closed under
complex conjugation In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...
. * If X and Y are two compact Hausdorff spaces, and F : \mathcal(X) \to \mathcal(Y) is a
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
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 \to X. In particular, if C(X) and C(Y) are isomorphic as algebras, then X and Y are
homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...
topological spaces. * Let \Delta be the space of
maximal ideal In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals ...
s in \mathcal(X). 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(X) \to \Complex. Equip \Deltawith the initial topology with respect to this pairing with \mathcal(X) (that is, the Gelfand transform). Then X is homeomorphic to Δ equipped with this topology. * A sequence in \mathcal(X) is weakly Cauchy if and only if it is (uniformly) bounded in \mathcal(X) and pointwise convergent. In particular, \mathcal(X) is only weakly complete for X a finite set. * The vague topology is the weak* topology on the dual of \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 In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...
, to further refine this general definition by considering the special case when X is a locally compact 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 In mathematics, the support of a real-valued function f is the subset of the function domain of elements that are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest closed ...
. 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 \subseteq X such that , f(x), < r for all x \in X \backslash K. This is called the space of functions 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 Theory of continuous functions Functional analysis Real analysis Types of functions