- Function space
In

mathematics , a**function space**is a set of functions of a given kind from a set "X" to a set "Y". It is called a space because in many applications, it is atopological space or avector space or both.**Examples**Function spaces appear in various areas of mathematics:

* in

set theory , thepower set of a set "X" may be identified with the set of all functions from "X" to {0,1};, denoted 2^{"X"}. More generally, the set of functions "X" → "Y" is denoted "Y"^{"X"}.* in

linear algebra the set of alllinear transformation s from avector space "V" to another one, "W", over the same field, is itself a vector space;* in

functional analysis the same is seen forcontinuous linear transformations, including topologies on the vector spaces in the above, and many of the major examples are function spaces carrying atopology ; the best known examples includeHilbert space s andBanach space s.* in

functional analysis the set of all functions from thenatural number s to some set "X" is called asequence space . It consists of the set of all possiblesequences of elements of "X".* in

topology , one may attempt to put a topology on the space of continuous functions from atopological space "X" to another one "Y", with utility depending on the nature of the spaces. A commonly used example is thecompact-open topology . Also available is theproduct 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 thetopology of pointwise convergence .* in

algebraic topology , the study ofhomotopy 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 aprobability measure on a function space of "paths of the process" (functions of time);* in

category theory the function space is called anexponential object or map object. It appears in one way as the representationcanonical bifunctor ; but as (single) functor, of type ["X", -] , it appears as anadjoint functor to a functor of type (-×"X") on objects;* in

lambda calculus andfunctional programming , function space types are used to express the idea ofhigher-order function .* in

domain theory , the basic idea is to find constructions frompartial order s that can model lambda calculus, by creating a well-behavedcartesian closed category .**Functional analysis**The whole subject of

functional analysis is organized around adequate techniques to bring function spaces astopological vector space s within reach of the ideas that would apply tonormed space s of finite dimension.*

Schwartz space ofsmooth functions of rapid decrease and its dual, tempered distributions

*Lp space

* κ(**R**) continuous functions with compact support endowed with the uniform norm topology

* "B"(**R**) bounded continuous (Bounded function )

* "C"_{∞}(**R**) continuous functions which vanish at infinity

* "C"^{r}(**R**) continuous function that has continuous first r derivatives.

* "C"^{∞}(**R**)Smooth functions

* "C"^{∞}_{0}smooth functions with compact support

* "D"(**R**) compact support in limit topology

* "W"^{"k","p"}Sobolev space

***O**_{"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**See also***

List of mathematical functions

*Linear algebra

*Vector space

*Banach space

*Hilbert space

*Clifford algebra

*Tensor field

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