# Function space > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Function_space > Markdown URL: https://mediated.wiki/source/Function_space.md > Source: https://en.wikipedia.org/wiki/Function_space > Source revision: 1354563238 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) Set of functions between two fixed sets Function x ↦ f (x) History of the function concept Types by domain and codomain 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 This article focuses on the space of scalar-valued functions. For the space of maps with a more general codomain, see also [mapping space](/source/Mapping_space). In [mathematics](/source/Mathematics), a **function space** is a [set](/source/Set_(mathematics)) of [functions](/source/Function_(mathematics)) between two fixed sets. Often, the [domain](/source/Domain_of_a_function) and/or [codomain](/source/Codomain) will have additional [structure](/source/Mathematical_structure) which is inherited by the function space. For example, the set of functions from any set X into a [vector space](/source/Vector_space) has a [natural](/source/List_of_mathematical_jargon#natural) vector space structure given by [pointwise](/source/Pointwise) addition and scalar multiplication. In other scenarios, the function space might inherit a [topological](/source/Topological_space) or [metric](/source/Metric_space) structure, hence the name function *space*. Often in mathematical jargon, especially in analysis or geometry, a function could refer to a map of the form X → R {\displaystyle X\to \mathbb {R} } or X → C {\displaystyle X\to \mathbb {C} } where X {\displaystyle X} is the space in question. Whilst other maps of the form X → Y {\displaystyle X\to Y} between any two spaces are simply referred to as maps. Example of this can be the space of compactly supported functions on a topological space. However in a larger context a function space could just consist of a set of functions (set theoretically) equipped with possibly some extra structure. ## In linear algebra See also: [Vector space § Function spaces](/source/Vector_space#Function_spaces) This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (November 2017) (Learn how and when to remove this message) Let F be a [field](/source/Field_(mathematics)) and let X be any set. The functions X → F can be given the structure of a vector space over F where the operations are defined pointwise, that is, for any f, g : X → F, any x in X, and any c in F, define ( f + g ) ( x ) = f ( x ) + g ( x ) ( c ⋅ f ) ( x ) = c ⋅ f ( x ) {\displaystyle {\begin{aligned}(f+g)(x)&=f(x)+g(x)\\(c\cdot f)(x)&=c\cdot f(x)\end{aligned}}} When the domain X has additional structure, one might consider instead the [subset](/source/Subset) (or [subspace](/source/Linear_subspace)) of all such functions which respect that structure. For example, if V and also X itself are vector spaces over F, the set of [linear maps](/source/Linear_map) X → V form a vector space over F with pointwise operations (often denoted [Hom](/source/Hom_set)(X,V)). One such space is the [dual space](/source/Dual_space) of X: the set of [linear functionals](/source/Linear_form) X → F with addition and scalar multiplication defined pointwise. The cardinal [dimension](/source/Dimension) of a function space with no extra structure can be found by the [Erdős–Kaplansky theorem](/source/Erd%C5%91s%E2%80%93Kaplansky_theorem). ## Examples Function spaces appear in various areas of mathematics: - In [set theory](/source/Set_theory), the set of functions from *X* to *Y* may be denoted {*X* → *Y*} or *Y**X*. - As a special case, the [power set](/source/Power_set) of a set *X* may be identified with the set of all functions from *X* to {0, 1}, denoted 2*X*. - The set of [bijections](/source/Bijection) from *X* to *Y* is denoted X ↔ Y {\displaystyle X\leftrightarrow Y} . The factorial notation *X*! may be used for permutations of a single set *X*. - In [functional analysis](/source/Functional_analysis), the same is seen for [continuous](/source/Continuous_function) linear transformations, including [topologies on the vector spaces](/source/Topological_vector_space) in the above, and many of the major examples are function spaces carrying a [topology](/source/Topology); the best known examples include [Hilbert spaces](/source/Hilbert_space) and [Banach spaces](/source/Banach_space). - In [functional analysis](/source/Functional_analysis), the set of all functions from the [natural numbers](/source/Natural_number) to some set *X* is called a *[sequence space](/source/Sequence_space)*. It consists of the set of all possible [sequences](/source/Sequences) of elements of *X*. - In [topology](/source/Topology), one may attempt to put a topology on the space of continuous functions from a [topological space](/source/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](/source/Compact-open_topology), e.g. [loop space](/source/Loop_space). Also available is the [product topology](/source/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](/source/Topology_of_pointwise_convergence). - In [algebraic topology](/source/Algebraic_topology), the study of [homotopy theory](/source/Homotopy_theory) is essentially that of discrete invariants of function spaces; - In the theory of [stochastic processes](/source/Stochastic_process), the basic technical problem is how to construct a [probability measure](/source/Probability_measure) on a function space of *paths of the process* (functions of time); - In [category theory](/source/Category_theory), the function space is called an [exponential object](/source/Exponential_object) or [map object](/source/Exponential_object). It appears in one way as the representation [canonical bifunctor](/source/Canonical_bifunctor); but as (single) functor, of type [ X , − ] {\displaystyle [X,-]} , it appears as an [adjoint functor](/source/Adjoint_functor) to a functor of type − × X {\displaystyle -\times X} on objects; - In [functional programming](/source/Functional_programming) and [lambda calculus](/source/Lambda_calculus), [function types](/source/Function_type) are used to express the idea of [higher-order functions](/source/Higher-order_function) - In programming more generally, many [higher-order function](/source/Higher-order_function) concepts occur with or without explicit typing, such as [closures](/source/Closure_(computer_programming)). - In [domain theory](/source/Domain_theory), the basic idea is to find constructions from [partial orders](/source/Partial_order) that can model lambda calculus, by creating a well-behaved [Cartesian closed category](/source/Cartesian_closed_category). - In the [representation theory of finite groups](/source/Representation_theory_of_finite_groups), given two finite-dimensional representations V and W of a group G, one can form a representation of G over the vector space of linear maps Hom(V,W) called the [Hom representation](/source/Hom_representation).[1] ## Functional analysis A main theme of [functional analysis](/source/Functional_analysis) is to study function spaces and vector spaces with more structure than the bare minimum of linear structure. Specifically, some are [topological vector spaces](/source/Topological_vector_space), some are [Banach spaces](/source/Normed_spaces), some are [Hilbert spaces](/source/Hilbert_space), etc. This allows mathematicians to apply intuitions from finite-dimensional vector spaces. The functional spaces have intricate interrelationships, such as [interpolation](/source/Interpolation_space), embedding, representation, Banach space isomorphism, etc. Many fundamental theorems and constructions in functional analysis deals with their relationships, such as the [Riesz representation theorem](/source/Riesz_representation_theorem), the [Riesz–Thorin theorem](/source/Riesz%E2%80%93Thorin_theorem), the [Gagliardo–Nirenberg interpolation inequality](/source/Gagliardo%E2%80%93Nirenberg_interpolation_inequality), the [Rellich–Kondrachov theorem](/source/Rellich%E2%80%93Kondrachov_theorem), the [Hardy–Littlewood maximal function](/source/Hardy%E2%80%93Littlewood_maximal_function), etc. Let Ω ⊆ R n {\displaystyle \Omega \subseteq \mathbb {R} ^{n}} be an open subset. - B ( Ω ) {\displaystyle B(\Omega )} [bounded functions](/source/Bounded_function) - continuous ones - C ( Ω ) {\displaystyle C(\Omega )} [continuous functions](/source/Continuous_functions) endowed with the [uniform norm](/source/Uniform_norm) topology - C c ( Ω ) {\displaystyle C_{c}(\Omega )} continuous functions with [compact support](/source/Support_(mathematics)#Compact_support) - C b ( Ω ) {\displaystyle C_{b}(\Omega )} continuous bounded functions - C 0 ( Ω ) {\displaystyle C_{0}(\Omega )} continuous functions which vanish at infinity; a closed subspace of C b ( Ω ) {\displaystyle C_{b}(\Omega )} [2] - C r ( Ω ) {\displaystyle C^{r}(\Omega )} continuous functions that have *r* continuous derivatives. - smooth ones - C ∞ ( Ω ) {\displaystyle C^{\infty }(\Omega )} [smooth functions](/source/Smooth_functions) - C c ∞ ( Ω ) {\displaystyle C_{c}^{\infty }(\Omega )} [smooth functions](/source/Smooth_functions) with [compact support](/source/Support_(mathematics)#Compact_support) (i.e. the set of [bump functions](/source/Bump_function)) - C ω ( Ω ) {\displaystyle C^{\omega }(\Omega )} [real analytic functions](/source/Analytic_function) - L p ( Ω ) {\displaystyle L^{p}(\Omega )} , for 1 ≤ p ≤ ∞ {\displaystyle 1\leq p\leq \infty } , is the [Lp space](/source/Lp_space) of [measurable](/source/Measurable_function) functions whose *p*-norm ‖ f ‖ p = ( ∫ Ω | f | p ) 1 / p {\textstyle \|f\|_{p}=\left(\int _{\Omega }|f|^{p}\right)^{1/p}} is finite - S ( Ω ) {\displaystyle {\mathcal {S}}(\Omega )} , the [Schwartz space](/source/Schwartz_space) of [rapidly decreasing](/source/Rapidly_decreasing) [smooth functions](/source/Smooth_functions) and its continuous dual, S ′ ( Ω ) {\displaystyle {\mathcal {S}}'(\Omega )} [tempered distributions](/source/Tempered_distributions) - D ( Ω ) {\displaystyle D(\Omega )} compact support in limit topology - Lip 0 ( Ω ) {\displaystyle {\text{Lip}}_{0}(\Omega )} , the space of all [Lipschitz](/source/Lipschitz_continuous) functions on Ω {\displaystyle \Omega } that vanish at zero. - W k , p {\displaystyle W^{k,p}} [Sobolev space](/source/Sobolev_space) of functions whose [weak derivatives](/source/Weak_derivative) up to order *k* are in L p {\displaystyle L^{p}} - O U {\displaystyle {\mathcal {O}}_{U}} holomorphic functions - B M O ( Ω ) {\displaystyle BMO(\Omega )} , space of [bounded mean oscillation](/source/Bounded_mean_oscillation). Also called John–Nirenberg space - linear functions - piecewise linear functions - continuous functions, compact open topology - all functions, space of pointwise convergence - [Hardy space](/source/Hardy_space) - [Hölder space](/source/H%C3%B6lder_space) - [Skorokhod space](/source/Skorokhod_space): the space of [càdlàg](/source/C%C3%A0dl%C3%A0g) functions. - [Besov space](/source/Besov_space) - [Souček space](/source/Sou%C4%8Dek_space) - [Triebel–Lizorkin space](/source/Triebel%E2%80%93Lizorkin_space) - [Barron space](/source/Barron_space) ## Uniform norm If *y* is an element of the function space C ( a , b ) {\displaystyle {\mathcal {C}}(a,b)} of all [continuous functions](/source/Continuous_function) that are defined on a [closed interval](/source/Closed_interval) [*a*, *b*], the **[norm](/source/Norm_(mathematics)) ‖ y ‖ ∞ {\displaystyle \|y\|_{\infty }}** defined on C ( a , b ) {\displaystyle {\mathcal {C}}(a,b)} is the maximum [absolute value](/source/Absolute_value) of *y* (*x*) for *a* ≤ *x* ≤ *b*,[3] ‖ y ‖ ∞ ≡ max a ≤ x ≤ b | y ( x ) | where y ∈ C ( a , b ) {\displaystyle \|y\|_{\infty }\equiv \max _{a\leq x\leq b}|y(x)|\qquad {\text{where}}\ \ y\in {\mathcal {C}}(a,b)} is called the *[uniform norm](/source/Uniform_norm)* or *supremum norm* ('sup norm'). ## Bibliography - [Kolmogorov, A. N.](/source/Andrey_Kolmogorov), & [Fomin, S. V.](/source/Sergei_Fomin) (1967). Elements of the theory of functions and functional analysis. Courier Dover Publications. - [Stein, Elias](/source/Elias_M._Stein); Shakarchi, R. (2011). Functional Analysis: An Introduction to Further Topics in Analysis. Princeton University Press. ## See also - [List of mathematical functions](/source/List_of_mathematical_functions) - [Clifford algebra](/source/Clifford_algebra) - [Tensor field](/source/Tensor_field) - [Spectral theory](/source/Spectral_theory) - [Functional determinant](/source/Functional_determinant) - [Equicontinuity](/source/Equicontinuity) - [Ascoli's theorem](/source/Ascoli's_theorem) ## References 1. **[^](#cite_ref-1)** Fulton, William; Harris, Joe (1991). [*Representation Theory: A First Course*](https://books.google.com/books?id=6GUH8ARxhp8C). Springer Science & Business Media. p. 4. [ISBN](/source/ISBN_(identifier)) [9780387974958](https://en.wikipedia.org/wiki/Special:BookSources/9780387974958). 1. **[^](#cite_ref-2)** [Conway, John B.](/source/John_B._Conway) (2007). [*A Course in Functional Analysis*](https://link.springer.com/book/10.1007/978-1-4757-4383-8). Vol. 96. New York, NY: Springer New York. p. 65. [doi](/source/Doi_(identifier)):[10.1007/978-1-4757-4383-8](https://doi.org/10.1007%2F978-1-4757-4383-8). [ISBN](/source/ISBN_(identifier)) [978-1-4419-3092-7](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4419-3092-7). 1. **[^](#cite_ref-GelfandFominP6_3-0)** [Gelfand, I. M.](/source/Israel_Gelfand); [Fomin, S. V.](/source/Sergei_Fomin) (2000). Silverman, Richard A. (ed.). [*Calculus of variations*](http://store.doverpublications.com/0486414485.html) (Unabridged repr. ed.). Mineola, New York: Dover Publications. p. 6. [ISBN](/source/ISBN_(identifier)) [978-0486414485](https://en.wikipedia.org/wiki/Special:BookSources/978-0486414485). Authority control databases: National Japan Czech Republic Israel v t e Lp spaces Basic concepts Banach & Hilbert spaces Lp spaces Measure Lebesgue Measure space Measurable space/function Minkowski distance Sequence spaces L1 spaces Integrable function Lebesgue integral Taxicab geometry L2 spaces Bessel's Cauchy–Schwarz Euclidean distance Hilbert space Parseval's identity Polarization identity Pythagorean theorem Square-integrable function L ∞ {\displaystyle L^{\infty }} spaces Bounded function Chebyshev distance Infimum and supremum Essential Uniform norm Maps Almost everywhere Convergence almost everywhere Convergence in measure Function space Integral transform Locally integrable function Measurable function Symmetric decreasing rearrangement Inequalities Babenko–Beckner Chebyshev's Clarkson's Hanner's Hausdorff–Young Hölder's Markov's Minkowski Young's convolution Results Marcinkiewicz interpolation theorem Plancherel theorem Riemann–Lebesgue Riesz–Fischer theorem Riesz–Thorin theorem For Lebesgue measure Isoperimetric inequality Brunn–Minkowski theorem Milman's reverse Minkowski–Steiner formula Prékopa–Leindler inequality Vitale's random Brunn–Minkowski inequality Applications & related Bochner space Fourier analysis Lorentz space Probability theory Quasinorm Real analysis Sobolev space *-algebra C*-algebra Von Neumann v t e Measure theory Basic concepts Absolute continuity of measures Lebesgue integration Lp spaces Measure Measure space Probability space Measurable space/function Sets Almost everywhere Atom Baire set Borel set equivalence relation Borel space Carathéodory's criterion Cylindrical σ-algebra Cylinder set 𝜆-system Essential range infimum/supremum Locally measurable π-system σ-algebra Non-measurable set Vitali set Null set Support Transverse measure Universally measurable Types of measures Atomic Baire Banach Besov Borel Brown Complex Complete Content (Logarithmically) Convex Decomposable Discrete Equivalent Finite Inner (Quasi-) Invariant Locally finite Maximising Metric outer Outer Perfect Pre-measure (Sub-) Probability Projection-valued Radon Random Regular Borel regular Inner regular Outer regular Saturated Set function σ-finite s-finite Signed Singular Spectral Strictly positive Tight Vector Particular measures Counting Dirac Euler Gaussian Haar Harmonic Hausdorff Intensity Lebesgue Infinite-dimensional Logarithmic Product Projections Pushforward Spherical measure Tangent Trivial Young Maps Measurable function Bochner Strongly Weakly Convergence: almost everywhere of measures in measure of random variables in distribution in probability Cylinder set measure Random: compact set element measure process variable vector Projection-valued measure Main results Carathéodory's extension theorem Convergence theorems Dominated Monotone Vitali Decomposition theorems Hahn Jordan Maharam's 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 For Lebesgue measure Isoperimetric inequality Brunn–Minkowski theorem Milman's reverse Minkowski–Steiner formula Prékopa–Leindler inequality Vitale's random Brunn–Minkowski inequality Applications & related Convex analysis Descriptive set theory Probability theory Real analysis Spectral theory --- Adapted from the Wikipedia article [Function space](https://en.wikipedia.org/wiki/Function_space) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Function_space?action=history)). 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