function space
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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, a function space is a set of
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
s between two fixed sets. Often, the
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Doma ...
and/or
codomain In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

codomain
will have additional
structure A structure is an arrangement and organization of interrelated elements in a material object or system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A ...
which is inherited by the function space. For example, the set of functions from any set into a
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
has a
natural Nature, in the broadest sense, is the natural, physical, material world or universe The universe ( la, universus) is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and ...

natural
vector space structure given by
pointwiseIn 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 o ...
addition and scalar multiplication. In other scenarios, the function space might inherit a
topological s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...
or
metric METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...
structure, hence the name function ''space''.


In linear algebra

Let be a vector space over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

subset
(or subspace) of all such functions which respect that structure. For example, if is also a vector space over , the set of
linear maps
linear maps
→ form a vector space over with pointwise operations (often denoted Hom(,)). One such space is the
dual space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of : the set of
linear functionals In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

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 that studies , 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 , is mostly concerned with those that are relevant to ...
, 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 (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
of a set ''X'' may be identified with the set of all functions from ''X'' to , denoted 2''X''. * The set of
bijection In , a bijection, bijective function, one-to-one correspondence, or invertible function, is a between the elements of two , where each element of one set is paired with exactly one element of the other set, and each element of the other set is p ...

bijection
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 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat ...
the same is seen for
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
linear transformations, including topologies on the vector spaces in the above, and many of the major examples are function spaces carrying a
topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

topology
; the best known examples include
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s and
Banach space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s. * In
functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat ...
the set of all functions from the
natural number File:Three Baskets.svg, Natural numbers can be used for counting (one apple, two apples, three apples, ...) In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, o ...
s to some set ''X'' is called a
sequence space In functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functiona ...
. It consists of the set of all possible
sequences In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of elements of ''X''. * In
topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

topology
, one may attempt to put a topology on the space of continuous functions from a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
''X'' to another one ''Y'', with utility depending on the nature of the spaces. A commonly used example is the
compact-open topologyIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, e.g.
loop spaceIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...
. 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 exponential object, 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 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 orders 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 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(\R) continuous functions endowed with the uniform norm topology *C_c(\R) continuous functions with Support (mathematics)#Compact support, compact support * B(\R) bounded functions * C_0(\R) continuous functions which vanish at infinity * C^r(\R) continuous functions that have continuous first ''r'' derivatives. * C^(\R) smooth functions * C^_c(\R) smooth functions with Support (mathematics)#Compact support, compact support *C^\omega(\R) Analytic function, real analytic functions *L^p(\R), for 1\leq p \leq \infty, is the Lp space, Lp space of Measurable function, measurable functions whose ''p''-norm \, f\, _p = \left( \int_\R , f, ^p \right)^ is finite *\mathcal(\R), the Schwartz space of rapidly decreasing smooth functions and its continuous dual, \mathcal'(\R) tempered distributions *D(\R) compact support in limit topology * W^ Sobolev space of functions whose Weak_derivative, 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 Anatoliy Skorokhod, Skorokhod space * \text_0(\R), the space of all Lipschitz continuous, Lipschitz functions on \R that vanish at zero.


Norm

If is an element of the function space \mathcal (a,b) of all continuous functions that are defined on a closed interval , the Norm (mathematics), norm \, y\, _\infty defined on \mathcal (a,b) is the maximum absolute value 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.


See also

*List of mathematical functions *Clifford algebra *Tensor field *Spectral theory *Functional determinant


References

{{Authority control Function spaces, Topology of function spaces Linear algebra