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Function space

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Set of functions between two fixed sets

Function
x ↦f (x)
History of the function concept
Types bydomain andcodomain
`X` →𝔹 𝔹 →`X` 𝔹ⁿ →`X` `X` →ℤ ℤ →`X` `X` →ℝ ℝ →`X` ℝⁿ →`X` `X` →ℂ ℂ →`X` ℂⁿ →`X`
Classes/properties
Constant Identity Linear Polynomial Rational Algebraic Analytic Smooth Continuous Measurable Injective Surjective Bijective
Constructions
Restriction Composition λ Inverse
Generalizations
Relation (Binary relation) Set-valued Multivalued Partial Implicit Space Higher-order Morphism Functor
List of specific functions
v t e

Inmathematics, afunction space is aset offunctions between two fixed sets. Often, thedomain and/orcodomain will have additionalstructure which is inherited by the function space. For example, the set of functions from any setX into avector space has anatural vector space structure given bypointwise addition and scalar multiplication. In other scenarios, the function space might inherit atopological ormetric structure, hence the name functionspace.

In linear algebra

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See also:Vector space § Function spaces

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LetF be afield and letX be any set. The functionsX →F can be given the structure of a vector space overF where the operations are defined pointwise, that is, for anyf,g :X →F, anyx inX, and anyc inF, define ${\begin{aligned}(f+g)(x)&=f(x)+g(x)\\(c\cdot f)(x)&=c\cdot f(x)\end{aligned}}$ When the domainX has additional structure, one might consider instead thesubset (orsubspace) of all such functions which respect that structure. For example, ifV and alsoX itself are vector spaces overF, the set oflinear mapsX →V form a vector space overF with pointwise operations (often denotedHom(X,V)). One such space is thedual space ofX: the set oflinear functionalsX →F with addition and scalar multiplication defined pointwise.

The cardinaldimension of a function space with no extra structure can be found by theErdős–Kaplansky theorem.

Examples

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Function spaces appear in various areas of mathematics:

Inset theory, the set of functions fromX toY may be denoted {X →Y} orY^X.
- As a special case, thepower set of a setX may be identified with the set of all functions fromX to {0, 1}, denoted 2^X.
The set ofbijections fromX toY is denoted $X\leftrightarrow Y$ . The factorial notationX! may be used for permutations of a single setX.
Infunctional analysis, the same is seen forcontinuous linear transformations, includingtopologies on the vector spaces in the above, and many of the major examples are function spaces carrying atopology; the best known examples includeHilbert spaces andBanach spaces.
Infunctional analysis, the set of all functions from thenatural numbers to some setX is called asequence space. It consists of the set of all possiblesequences of elements ofX.
Intopology, one may attempt to put a topology on the space of continuous functions from atopological spaceX to another oneY, with utility depending on the nature of the spaces. A commonly used example is thecompact-open topology, e.g.loop space. 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.
Inalgebraic topology, the study ofhomotopy theory is essentially that of discrete invariants of function spaces;
In the theory ofstochastic processes, the basic technical problem is how to construct aprobability measure on a function space ofpaths of the process (functions of time);
Incategory theory, the function space is called anexponential object ormap 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 $-\times X$ on objects;
Infunctional programming andlambda calculus,function types are used to express the idea ofhigher-order functions.
Indomain theory, the basic idea is to find constructions frompartial orders that can model lambda calculus, by creating a well-behavedCartesian closed category.
In therepresentation theory of finite groups, given two finite-dimensional representationsV andW of a groupG, one can form a representation ofG over the vector space of linear maps Hom(V,W) called theHom representation.^[1]

Functional analysis

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Functional analysis is organized around adequate techniques to bring function spaces astopological vector spaces within reach of the ideas that would apply tonormed spaces of finite dimension. Here we use the real line as an example domain, but the spaces below exist on suitable open subsets $\Omega \subseteq \mathbb {R} ^{n}$

$C(\mathbb {R} )$ continuous functions endowed with theuniform norm topology
$C_{c}(\mathbb {R} )$ continuous functions withcompact support
$B(\mathbb {R} )$ bounded functions
$C_{0}(\mathbb {R} )$ continuous functions which vanish at infinity
$C^{r}(\mathbb {R} )$ continuous functions that haver continuous derivatives.
$C^{\infty }(\mathbb {R} )$ smooth functions
$C_{c}^{\infty }(\mathbb {R} )$ smooth functions withcompact support (i.e. the set ofbump functions)
$C^{\omega }(\mathbb {R} )$ real analytic functions
$L^{p}(\mathbb {R} )$ , for $1\leq p\leq \infty$ , is theL^p space ofmeasurable functions whosep-norm ${\textstyle \|f\|_{p}=\left(\int _{\mathbb {R} }|f|^{p}\right)^{1/p}}$ is finite
${\mathcal {S}}(\mathbb {R} )$ , theSchwartz space ofrapidly decreasing smooth functions and its continuous dual, ${\mathcal {S}}'(\mathbb {R} )$ tempered distributions
$D(\mathbb {R} )$ compact support in limit topology
$W^{k,p}$ Sobolev space of functions whoseweak derivatives up to orderk are in $L^{p}$
${\mathcal {O}}_{U}$ 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 theSkorokhod space
${\text{Lip}}_{0}(\mathbb {R} )$ , the space of allLipschitz functions on $\mathbb {R}$ that vanish at zero.

Norm

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Ify is an element of the function space ${\mathcal {C}}(a,b)$ of allcontinuous functions that are defined on aclosed interval[a,b], thenorm $\|y\|_{\infty }$ defined on ${\mathcal {C}}(a,b)$ is the maximumabsolute value ofy (x) fora ≤x ≤b,^[2] $\|y\|_{\infty }\equiv \max _{a\leq x\leq b}|y(x)|\qquad {\text{where}}\ \ y\in {\mathcal {C}}(a,b)$

is called theuniform norm orsupremum norm ('sup norm').

Bibliography

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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.

References

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^Fulton, William; Harris, Joe (1991).Representation Theory: A First Course. Springer Science & Business Media. p. 4.ISBN 9780387974958.
^Gelfand, I. M.;Fomin, S. V. (2000). Silverman, Richard A. (ed.).Calculus of variations (Unabridged repr. ed.). Mineola, New York: Dover Publications. p. 6.ISBN 978-0486414485.

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Lp spaces

Basic concepts

L¹ spaces

L² spaces

L^{\infty }

spaces

Maps

Inequalities

Results

ForLebesgue measure	Isoperimetric inequality Brunn–Minkowski theorem Milman's reverse Minkowski–Steiner formula Prékopa–Leindler inequality Vitale's random Brunn–Minkowski inequality

Applications & related

Measure theory

Basic concepts

Sets

Types ofmeasures

Particular measures

Maps

Main results

Carathéodory's extension theorem
Convergence theorems
Decomposition theorems
Egorov's
Fatou's lemma
Fubini's
- Fubini–Tonelli
Hölder's inequality
Minkowski inequality
Radon–Nikodym
Riesz–Markov–Kakutani representation theorem

Other results

Disintegration theorem Lifting theory Lebesgue's density theorem Lebesgue differentiation theorem Sard's theorem Vitali–Hahn–Saks theorem
ForLebesgue measure	Isoperimetric inequality Brunn–Minkowski theorem Milman's reverse Minkowski–Steiner formula Prékopa–Leindler inequality Vitale's random Brunn–Minkowski inequality

Applications & related

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