In
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 ...
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
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
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
The space
is a
Banach algebra with respect to this norm.
Properties
* 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 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
(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
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
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
by a different form of the Riesz representation theorem.
* If
is infinite, then
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
of
is
relatively compact if and only if it is
bounded in the norm of
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
In the case of real functions, if
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
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
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
and
are two compact Hausdorff spaces, and
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
is continuous. Furthermore,
has the form
for some continuous function
In particular, if
and
are isomorphic as algebras, then
and
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
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
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.
* 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
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.
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
is a
locally compact Hausdorff space. In this case, it is possible to identify a pair of distinguished subsets of
:
*
the subset of
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.
*
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
* .
* .
*
* .
{{Functional analysis
Banach spaces
Complex analysis
Theory of continuous functions
Functional analysis
Real analysis
Types of functions