function space

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In
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 of functions between two fixed sets. Often, the domain and/or
codomain In mathematics, the codomain or set of destination of a Function (mathematics), function is the Set (mathematics), set into which all of the output of the function is constrained to fall. It is the set in the notation . The term Range of a funct ...
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 as ...
which is inherited by the function space. For example, the set of functions from any set into a
vector space In 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 ...
has a natural 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 mathema ...
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 (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), 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 ...
(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 → 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 Set (mathematics), 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, ...
, the set of functions from ''X'' to ''Y'' may be denoted ''X'' → ''Y'' or ''Y''''X''. ** As a special case, the
power set In 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 ...
of a set ''X'' may be identified with the set of all functions from ''X'' to , denoted 2''X''. * The set of
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function (mathematics), function between the elements of two set (mathematics), sets, where each element of one set is pair ...
s 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, the same is seen for continuous 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 language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
; the best known examples include
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally ...
s and
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a Complete metric space, complete normed vector space. Thus, a Banach space is a vector space with a Metric (mathematics), metric that allows the computation ...
s. * In functional analysis, 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 n ...
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 language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, 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 Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
''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. Also available is the product topology 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
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 invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, the study of homotopy theory is essentially that of discrete invariants of function spaces; * In the theory of
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a Indexed family, family of random variables. Stochastic processes are widely used as mathematical models of systems and phen ...
es, 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 Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, categ ...
, the function space is called an exponential object or map object. It appears in one way as the representation canonical bifunctor; 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 Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied sc ...
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 (mathematics), set. A poset consists of a set toget ...
s that can model lambda calculus, by creating a well-behaved
Cartesian closed category In category theory, a Category (mathematics), category is Cartesian closed if, roughly speaking, any morphism defined on a product (category theory), product of two Object (category theory), objects can be naturally identified with a morphism defin ...
. * 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.

# Functional analysis

Functional analysis 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 $\Omega \subseteq \R^n$ *$C\left(\R\right)$ continuous functions endowed with the uniform norm topology *$C_c\left(\R\right)$ continuous functions with compact support * $B\left(\R\right)$ bounded functions * $C_0\left(\R\right)$ continuous functions which vanish at infinity * $C^r\left(\R\right)$ continuous functions that have continuous first ''r'' derivatives. * $C^\left(\R\right)$ smooth functions * $C^_c\left(\R\right)$ smooth functions with compact support *$C^\omega\left(\R\right)$ real analytic functions *$L^p\left(\R\right)$, 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 *$\mathcal\left(\R\right)$, the Schwartz space of rapidly decreasing smooth functions and its continuous dual, $\mathcal\text{'}\left(\R\right)$ tempered distributions *$D\left(\R\right)$ compact support in limit topology * $W^$ Sobolev space 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 * Càdlàg functions, also known as the Skorokhod space * $\text_0\left(\R\right)$, the space of all Lipschitz functions on $\R$ that vanish at zero.

# Norm

If is an element of the function space $\mathcal \left(a,b\right)$ of all
continuous function In mathematics, a continuous function is a function (mathematics), function such that a continuous variation (that is a change without jump) of the argument of a function, argument induces a continuous variation of the Value (mathematics), value ...
s that are defined on a closed interval , the norm $\, y\, _\infty$ defined on $\mathcal \left(a,b\right)$ is the maximum
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign (mathematics), sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative number, negative (in which cas ...
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.