Function space

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 a topological space or a vector space or both.


Function spaces appear in various areas of mathematics:

* in set theory, the power 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 all linear transformations from a vector space "V" to another one, "W", over the same field, is itself a vector space;

* 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; the best known examples include Hilbert spaces and Banach spaces.

* in functional analysis the set of all functions from the natural numbers 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. 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, 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; but as (single) functor, of type ["X", -] , it appears as an adjoint functor to a functor of type (-×"X") on objects;

* in lambda calculus and functional programming, function space types are used to express the idea of higher-order function.

* in domain theory, the basic idea is to find constructions from partial orders that can model lambda calculus, by creating a well-behaved cartesian closed category.

Functional analysis

The whole subject of 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.

* Schwartz space of smooth 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

Wikimedia Foundation. 2010.

Игры ⚽ Поможем решить контрольную работу

Look at other dictionaries:

  • function space — Math. a linear space, the elements of which are functions. [1930 35] * * * …   Universalium

  • function space — noun Any metric space whose elements are functions …   Wiktionary

  • function space — Math. a linear space, the elements of which are functions. [1930 35] …   Useful english dictionary

  • Space (mathematics) — This article is about mathematical structures called spaces. For space as a geometric concept, see Euclidean space. For all other uses, see space (disambiguation). A hierarchy of mathematical spaces: The inner product induces a norm. The norm… …   Wikipedia

  • Function (mathematics) — f(x) redirects here. For the band, see f(x) (band). Graph of example function, In mathematics, a function associates one quantity, the a …   Wikipedia

  • space — 1. noun /speɪs/ a) The intervening contents of a volume. If it be only a Single Letter or two that drops, he thruſts the end of his Bodkin between every Letter of that Word, till he comes to a Space: and then perhaps by forcing thoſe Letters… …   Wiktionary

  • space station — space station, adj. an orbiting manned structure that can be used for a variety of purposes, as to assemble or service satellites, refuel spacecraft, etc. Also called space platform. [1940 45] * * * Manned artificial structure designed to revolve …   Universalium

  • Space charge — is a concept in which excess electric charge is treated as a continuum of charge distributed over a region of space (either a volume or an area) rather than distinct point like charges. This model typically applies when charge carriers have been… …   Wikipedia

  • Space debris — populations seen from outside geosynchronous orbit (GEO). Note the two primary debris fields, the ring of objects in GEO, and the cloud of objects in low earth orbit (LEO) …   Wikipedia

  • Space Station Freedom — was the name given to NASA s project to construct a permanently manned Earth orbiting space station. Although approved by then president Ronald Reagan and announced in the 1984 State of the Union Address, Freedom was never constructed or… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”