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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a function space is a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of functions between two fixed sets. Often, the
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
and/or codomain will have additional
structure A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such a ...
which is inherited by the function space. For example, the set of functions from any set into a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
has a
natural Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans ar ...
vector space structure given by
pointwise In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...
addition and scalar multiplication. In other scenarios, the function space might inherit a topological or
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathe ...
structure, hence the name function ''space''.


In linear algebra

Let be a vector space over a field and let be any set. The functions → can be given the structure of a vector space over where the operations are defined pointwise, that is, for any , : → , any in , and any in , define \begin (f+g)(x) &= f(x)+g(x) \\ (c\cdot f)(x) &= c\cdot f(x) \end When the domain has additional structure, one might consider instead the
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
(or subspace) of all such functions which respect that structure. For example, if is also a vector space over , the set of linear maps → form a vector space over with pointwise operations (often denoted Hom(,)). One such space is the dual space of : the set of
linear functionals In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the ...
→ with addition and scalar multiplication defined pointwise.


Examples

Function spaces appear in various areas of mathematics: * In
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
, the set of functions from ''X'' to ''Y'' may be denoted ''X'' → ''Y'' or ''Y''''X''. ** As a special case, the power set of a set ''X'' may be identified with the set of all functions from ''X'' to , denoted 2''X''. * The set of bijections from ''X'' to ''Y'' is denoted X \leftrightarrow Y. The factorial notation ''X''! may be used for permutations of a single set ''X''. * In
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 (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
, the same is seen for
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
linear transformations, including topologies on the vector spaces in the above, and many of the major examples are function spaces carrying a
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
; the best known examples include Hilbert spaces and Banach spaces. * In
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 (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
, the set of all functions from the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s to some set ''X'' is called a '' sequence space''. It consists of the set of all possible sequences of elements of ''X''. * In
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, one may attempt to put a topology on the space of continuous functions from a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
''X'' to another one ''Y'', with utility depending on the nature of the spaces. A commonly used example is the compact-open topology, e.g.
loop space In topology, a branch of mathematics, the loop space Ω''X'' of a pointed topological space ''X'' is the space of (based) loops in ''X'', i.e. continuous pointed maps from the pointed circle ''S''1 to ''X'', equipped with the compact-open topol ...
. Also available is the
product topology In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seem ...
on the space of set theoretic functions (i.e. not necessarily continuous functions) ''Y''''X''. In this context, this topology is also referred to as the
topology of pointwise convergence In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared. Definition Suppose that X is a set and ...
. * In
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, the study of homotopy theory is essentially that of discrete invariants of function spaces; * In the theory of stochastic processes, the basic technical problem is how to construct a probability measure on a function space of ''paths of the process'' (functions of time); * In category theory, the function space is called an exponential object or map object. It appears in one way as the representation
canonical bifunctor In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory a ...
; but as (single) functor, of type 'X'', - it appears as an adjoint functor to a functor of type (-×''X'') on objects; * In
functional programming In computer science, functional programming is a programming paradigm where programs are constructed by applying and composing functions. It is a declarative programming paradigm in which function definitions are trees of expressions tha ...
and lambda calculus, function types are used to express the idea of higher-order functions. * In domain theory, the basic idea is to find constructions from
partial order In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
s that can model lambda calculus, by creating a well-behaved Cartesian closed category. * In the representation theory of finite groups, given two finite-dimensional representations and of a group , one can form a representation of over the vector space of linear maps Hom(,) called the
Hom representation In mathematics, the tensor product of representations is a tensor product of vector spaces underlying representations together with the factor-wise group action on the product. This construction, together with the Clebsch–Gordan procedure, can be ...
.


Functional analysis

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 (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
is organized around adequate techniques to bring function spaces as topological vector spaces within reach of the ideas that would apply to normed spaces of finite dimension. Here we use the real line as an example domain, but the spaces below exist on suitable open subsets \Omega \subseteq \R^n *C(\R) continuous functions endowed with the uniform norm topology *C_c(\R) continuous functions with compact support * B(\R) bounded functions * C_0(\R) continuous functions which vanish at infinity * C^r(\R) continuous functions that have continuous first ''r'' derivatives. * C^(\R) smooth functions * C^_c(\R) smooth functions with compact support *C^\omega(\R) real analytic functions *L^p(\R), for 1\leq p \leq \infty, is the Lp space of measurable functions whose ''p''-norm \, f\, _p = \left( \int_\R , f, ^p \right)^ is finite *\mathcal(\R), the Schwartz space of rapidly decreasing smooth functions and its continuous dual, \mathcal'(\R)
tempered distributions Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives d ...
*D(\R) compact support in limit topology * W^
Sobolev space In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
of functions whose weak derivatives up to order ''k'' are in L^p * \mathcal_U holomorphic functions * linear functions * piecewise linear functions * continuous functions, compact open topology * all functions, space of pointwise convergence * Hardy space *
Hölder space Hölder: * ''Hölder, Hoelder'' as surname * Hölder condition * Hölder's inequality * Hölder mean * Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a mo ...
* Càdlàg functions, also known as the Skorokhod space * \text_0(\R), the space of all
Lipschitz Lipschitz, Lipshitz, or Lipchitz, is an Ashkenazi Jewish (Yiddish/German-Jewish) surname. The surname has many variants, including: Lifshitz ( Lifschitz), Lifshits, Lifshuts, Lefschetz; Lipschitz, Lipshitz, Lipshits, Lopshits, Lipschutz (Lip ...
functions on \R that vanish at zero.


Norm

If is an element of the function space \mathcal (a,b) of all
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
s that are defined on a closed interval , the norm \, y\, _\infty defined on \mathcal (a,b) is the maximum absolute value of for , \, y \, _\infty \equiv \max_ , y(x), \qquad \text \ \ y \in \mathcal (a,b) is called the '' uniform norm'' or ''supremum norm'' ('sup norm').


Bibliography

* Kolmogorov, A. N., & Fomin, S. V. (1967). Elements of the theory of functions and functional analysis. Courier Dover Publications. * Stein, Elias; Shakarchi, R. (2011). Functional Analysis: An Introduction to Further Topics in Analysis. Princeton University Press.


See also

* List of mathematical functions * Clifford algebra * Tensor field *
Spectral theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result ...
* Functional determinant


References

{{Authority control Topology of function spaces Linear algebra