Editing Function space

{{Short description|Set of functions between two fixed sets}}
{{Functions}}
In [[mathematics]], a '''function space''' is a [[Set (mathematics)|set]] of [[function (mathematics)|function]]s between two fixed sets. Often, the [[Domain of a function|domain]] and/or [[codomain]] will have additional [[Mathematical structure|structure]] which is inherited by the function space. For example, the set of functions from any set {{var|X}} into a [[vector space]] has a [[List of mathematical jargon#natural|natural]] vector space structure given by [[pointwise]] addition and scalar multiplication. In other scenarios, the function space might inherit a [[Topological space|topological]] or [[Metric space|metric]] structure, hence the name function ''space''.

==In linear algebra==
{{See also|Vector space#Function spaces}}
{{Unreferenced section|date=November 2017}}

Let {{var|F}} be a [[Field (mathematics)|field]] and let {{var|X}} be any set. The functions {{var|X}} → {{var|F}} can be given the structure of a vector space over {{var|F}} where the operations are defined pointwise, that is, for any {{var|f}}, {{var|g}} : {{var|X}} → {{var|F}}, any {{var|x}} in {{var|X}}, and any {{var|c}} in {{var|F}}, define
<math display="block">
\begin{align}
  (f+g)(x) &= f(x)+g(x) \\
  (c\cdot f)(x) &= c\cdot f(x)
\end{align}
</math>
When the domain {{var|X}} has additional structure, one might consider instead the [[subset]] (or [[Linear subspace|subspace]]) of all such functions which respect that structure. For example, if {{var|V}} and also {{var|X}} itself are vector spaces over {{var|F}}, the set of [[Linear map|linear maps]] {{var|X}} → {{var|V}} form a vector space over {{var|F}} with pointwise operations (often denoted [[Hom set|Hom]]({{var|X}},{{var|V}})). One such space is the [[dual space]] of {{var|X}}: the set of [[Linear form|linear functionals]] {{var|X}} → {{var|F}} with addition and scalar multiplication defined pointwise.

The cardinal [[dimension]] of a function space with no extra structure can be found by the [[Erdős–Kaplansky theorem]].

==Examples==
Function spaces appear in various areas of mathematics:

* In [[set theory]], the set of functions from ''X'' to ''Y'' may be denoted {''X'' → ''Y''} or ''Y''<sup>''X''</sup>.
** As a special case, the [[power set]] of a set ''X'' may be identified with the set of all functions from ''X'' to {0, 1}, denoted 2<sup>''X''</sup>. 
* The set of [[bijection]]s from ''X'' to ''Y'' is denoted <math>X \leftrightarrow Y</math>.  The factorial notation ''X''! may be used for permutations of a single set ''X''.
* In [[functional analysis]], the same is seen for [[continuous function|continuous]] linear transformations, including [[topological vector space|topologies on the vector spaces]] in the above, and many of the major examples are function spaces carrying a [[topology]]; the best known examples include [[Hilbert space]]s and [[Banach space]]s.
* In [[functional analysis]], the set of all functions from the [[natural number]]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]], one may attempt to put a topology on the space of continuous functions from a [[topological space]] ''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''<sup>''X''</sup>.  In this context, this topology is also referred to as the [[topology of pointwise convergence]].
* In [[algebraic topology]], the study of [[homotopy theory]] is essentially that of discrete invariants of function spaces;
* In the theory of [[stochastic process]]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]], the function space is called an [[exponential object]] or [[exponential object|map object]]. It appears in one way as the representation [[canonical bifunctor]]; but as (single) functor, of type <math>[X,-]</math>, it appears as an [[adjoint functor]] to a functor of type <math> - \times X</math> on objects;
* In [[functional programming]] and [[lambda calculus]], [[function type]]s are used to express the idea of [[higher-order function]]s
* In programming more generally, many [[higher-order function]] concepts occur with or without explicit typing, such as [[Closure_(computer_programming)|closures]].
* In [[domain theory]], the basic idea is to find constructions from [[partial order]]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 {{var|V}} and {{var|W}} of a group {{var|G}}, one can form a representation of {{var|G}} over the vector space of linear maps Hom({{var|V}},{{var|W}}) called the [[Hom representation]].<ref>{{Cite book|url=https://books.google.com/books?id=6GUH8ARxhp8C|title=Representation Theory: A First Course|last1=Fulton|first1=William|last2=Harris|first2=Joe|date=1991|publisher=Springer Science & Business Media|isbn=9780387974958|language=en|page=4}}</ref>

==Functional analysis==
[[Functional analysis]] is organized around adequate techniques to bring function spaces as [[topological vector space]]s within reach of the ideas that would apply to [[normed space]]s of finite dimension. Here we use the real line as an example domain, but the spaces below exist on suitable open subsets <math>\Omega \subseteq \R^n</math>

*<math>C(\R)</math> [[continuous functions]] endowed with the [[uniform norm]] topology
*<math>C_c(\R)</math> continuous functions with [[Support (mathematics)#Compact support|compact support]]
* <math>B(\R)</math> [[bounded function]]s
* <math>C_0(\R)</math> continuous functions which vanish at infinity
* <math>C^r(\R)</math> continuous functions that have ''r'' continuous derivatives.
* <math>C^{\infty}(\R)</math> [[smooth functions]]
* <math>C^{\infty}_c(\R)</math> [[smooth functions]] with [[Support (mathematics)#Compact support|compact support]] (i.e. the set of [[bump function]]s)
*<math>C^\omega(\R)</math> [[Analytic function|real analytic functions]]
*<math>L^p(\R)</math>, for <math>1\leq p \leq \infty</math>, is the [[Lp space|L<sup>p</sup> space]] of [[Measurable function|measurable]] functions whose ''p''-norm <math display="inline">\|f\|_p = \left( \int_\R |f|^p \right)^{1/p}</math> is finite
*<math>\mathcal{S}(\R)</math>, the [[Schwartz space]] of [[rapidly decreasing]] [[smooth functions]] and its continuous dual, <math>\mathcal{S}'(\R)</math> [[tempered distributions]]
*<math>D(\R)</math> compact support in limit topology
* <math>W^{k,p}</math> [[Sobolev space]] of functions whose [[Weak_derivative|weak derivatives]] up to order ''k'' are in <math>L^p</math>
* <math>\mathcal{O}_U</math> 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 [[Anatoliy Skorokhod|Skorokhod]] space
* <math>\text{Lip}_0(\R)</math>, the space of all [[Lipschitz continuous|Lipschitz]] functions on <math>\R</math> that vanish at zero.

==Norm==
If {{math|''y''}} is an element of the function space <math> \mathcal {C}(a,b) </math> of all [[continuous function]]s that are defined on a [[closed interval]] {{closed-closed|''a'', ''b''}}, the '''[[Norm (mathematics)|norm]]  <math>\|y\|_\infty</math>''' defined on <math> \mathcal {C}(a,b) </math> is the maximum [[absolute value]] of  {{math|''y'' (''x'')}} for {{math|''a'' ≤ ''x'' ≤ ''b''}},<ref name='GelfandFominP6'>{{cite book|last1=Gelfand |first1=I. M. |authorlink1=Israel Gelfand |last2=Fomin |first2=S. V. |authorlink2=Sergei Fomin |title=Calculus of variations |year=2000 |page=6 |publisher=Dover Publications |location=Mineola, New York |isbn=978-0486414485 |url=http://store.doverpublications.com/0486414485.html |edition=Unabridged repr. |editor1-last=Silverman |editor1-first=Richard A.}}</ref>
<math display="block"> \| y \|_\infty \equiv \max_{a \le x \le b} |y(x)| \qquad \text{where} \ \ y \in \mathcal {C}(a,b) </math>

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]]
*[[Functional determinant]]

== References ==
{{Reflist}}

{{Authority control}}

{{Lp spaces}}
{{Measure theory}}

[[Category:Function spaces| ]]
[[Category:Topology of function spaces]]
[[Category:Linear algebra]]