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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, 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 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 gen ...
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 mathematical logic that studies Set (mathematics), sets, 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 mathematics, i ...
, 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 mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

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 In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

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 Product may refer to: Business * Product (business) In marketing, a product is an object or system made available for consumer use; it is anything that can be offered to a Market (economics), market to satisfy the desire or need of a customer ...
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 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 ...
. * In
algebraic topology Algebraic topology is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...
, the study of
homotopy theoryIn 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 ...
is essentially that of discrete invariants of function spaces; * In the theory of
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a Indexed family, family of random variables. Stochastic processes are widely used as mathematical models of systems and phen ...

stochastic process
es, the basic technical problem is how to construct a
probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability meas ...
on a function space of ''paths of the process'' (functions of time); * In
category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
the function space is called an
exponential object 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 ...
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 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 ...
to a functor of type (-×''X'') on objects; * In
functional programming In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of , ...
and
lambda calculus Lambda calculus (also written as ''λ''-calculus) is a formal system A formal system is an used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, ar ...
,
function typeIn computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of Algorith ...
s are used to express the idea of
higher-order function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s. * In
domain theory Domain theory is a branch of mathematics that studies special kinds of partially ordered sets (posets) commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer s ...
, the basic idea is to find constructions from
partial order upright=1.15, Fig.1 The set of all subsets of a three-element set \, ordered by set inclusion">inclusion Inclusion or Include may refer to: Sociology * Social inclusion, affirmative action to change the circumstances and habits that leads to s ...
s that can model lambda calculus, by creating a well-behaved
cartesian closed category In category theory, a category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being *Categories (Aristotle), ''Categories' ...
. * In the
representation theory of finite groups The representation theory of group (mathematics), groups is a part of mathematics which examines how groups act on given structures. Here the focus is in particular on Group action (mathematics), operations of groups on vector spaces. Nevertheless, ...
, 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 Functional analysis is a branch of mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), ...
is organized around adequate techniques to bring function spaces as
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space (an Abstra ...
s within reach of the ideas that would apply to
normed space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
s 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 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 ...
endowed with the uniform norm topology *C_c(\R) continuous functions with
compact support Compact as used in politics may refer broadly to a pact A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations International relations (IR), international affairs (IA) or internationa ...
* B(\R)
bounded function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s * C_0(\R) continuous functions which vanish at infinity * C^r(\R) continuous functions that have continuous first ''r'' derivatives. * C^(\R)
smooth functions In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mat ...
* C^_c(\R)
smooth functions In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mat ...
with
compact support Compact as used in politics may refer broadly to a pact A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations International relations (IR), international affairs (IA) or internationa ...
*C^\omega(\R) real analytic functions *L^p(\R), for 1\leq p \leq \infty, is the Lp space of
measurable 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). ...
functions whose ''p''-norm \, f\, _p = \left( \int_\R , f, ^p \right)^ is finite *\mathcal(\R), the
Schwartz space Schwartz may refer to: *Schwartz (surname), a surname (and list of people with the name) *Schwartz (brand), a spice brand *Schwartz's, a delicatessen in Montreal, Quebec, Canada *Schwartz Publishing, an Australian publishing house *"Danny Schwartz", ...
of rapidly decreasing
smooth functions In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mat ...
and its continuous dual, \mathcal'(\R) tempered distributions *D(\R) compact support in limit topology * W^
Sobolev space In mathematics, a Sobolev space is a vector space of functions equipped with a normed space, norm that is a combination of Lp norm, ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a s ...
of functions whose 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 In complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related ...
* Hölder space *
CàdlàgIn 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 ...
functions, also known as the Skorokhod space * \text_0(\R), the space of all
LipschitzLipschitz, Lipshitz, or Lipchitz is an Ashkenazi Jewish surname. The surname has many variants, including: Lifshitz (Lifschitz), Lifshits, Lifshuts, Lefschetz; Lipschitz, Lipshitz, Lipshits, Lopshits, Lipschutz (Lipschütz), Lipshutz, Lüpschütz; ...
functions on \R that vanish at zero.


Norm

If is an element of the function space \mathcal (a,b) of all
continuous function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s that are defined on a
closed interval In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, the
norm Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy) Norms are concepts ( sentences) of practical import, oriented to effecting an action, rat ...
\, y\, _\infty defined on \mathcal (a,b) is the maximum
absolute value In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

absolute value
of for , \, y \, _\infty \equiv \max_ , y(x), \qquad \text \ \ y \in \mathcal (a,b) is called the ''
uniform norm frame, The perimeter of the square is the set of points in R2 where the sup norm equals a fixed positive constant. In mathematical analysis, the uniform norm (or sup norm) assigns to real- or complex-valued bounded functions ''f'' defined on ...
'' 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 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 ...
*
Clifford algebra In mathematics, a Clifford algebra is an algebra over a field, algebra generated by a vector space with a quadratic form, and is a Unital algebra, unital associative algebra. As algebra over a field, ''K''-algebras, they generalize the real nu ...
*
Tensor field In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
*
Spectral theoryIn 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 ...
*
Functional determinantIn 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 functional an ...


References

{{Authority control Topology of function spaces Linear algebra