Editing Function space (section)

==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>