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function space wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 in linear algebra 2 examples 3 functional analysis 4 uniform norm 5 bibliography 6 see also 7 references toggle the table of contents function space 23 languages العربية български català deutsch esperanto español فارسی suomi français magyar italiano 日本語 한국어 nederlands norsk bokmål polski português română simple english svenska türkçe українська 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia 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 𝔹 n x x ℤ ℤ x x ℝ ℝ x ℝ n x x ℂ ℂ x ℂ n 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 in mathematics a function space is a set of functions between two fixed sets often the domain and or codomain will have additional structure which is inherited by the function space for example the set of functions from any set x into a vector space has a natural vector space structure given by pointwise addition and scalar multiplication in other scenarios the function space might inherit a topological or metric 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 edit see also 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 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 or 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 x v form a vector space over f with pointwise operations often denoted hom x v one such space is the dual space of x the set of linear functionals x f with addition and scalar multiplication defined pointwise the cardinal dimension of a function space with no extra structure can be found by the erdős kaplansky theorem examples edit function spaces appear in various areas of mathematics in 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 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 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 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 e g loop space 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 displaystyle x it appears as an adjoint functor to a functor of type x displaystyle times x on objects in functional programming and lambda calculus function types are used to express the idea of higher order functions in programming more generally many higher order function concepts occur with or without explicit typing such as closures 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 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 1 functional analysis edit a main theme of 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 some are banach spaces some are hilbert spaces etc this allows mathematicians to apply intuitions from finite dimensional vector spaces the functional spaces have intricate interrelationships such as interpolation embedding representation banach space isomorphism etc many fundamental theorems and constructions in functional analysis deals with their relationships such as the riesz representation theorem the riesz thorin theorem the gagliardo nirenberg interpolation inequality the rellich kondrachov theorem the hardy littlewood maximal function etc let ω r n displaystyle omega subseteq mathbb r n be an open subset b ω displaystyle b omega bounded functions continuous ones c ω displaystyle c omega continuous functions endowed with the uniform norm topology c c ω displaystyle c_ c omega continuous functions with 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 c c ω displaystyle c_ c infty omega smooth functions with compact support i e the set of bump functions c ω ω displaystyle c omega omega real analytic functions l p ω displaystyle l p omega for 1 p displaystyle 1 leq p leq infty is the l p space of measurable 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 of rapidly decreasing smooth functions and its continuous dual s ω displaystyle mathcal s omega 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 functions on ω displaystyle omega that vanish at zero w k p displaystyle w k p sobolev space of functions whose weak derivatives 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 also called john nirenberg space linear functions piecewise linear functions continuous functions compact open topology all functions space of pointwise convergence hardy space hölder space skorokhod space the space of càdlàg functions besov space souček space triebel lizorkin space barron space uniform norm edit if y is an element of the function space c a b displaystyle mathcal c a b of all continuous functions that are defined on a closed interval a b the norm y displaystyle y _ infty defined on c a b displaystyle mathcal c a b is the maximum 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 or supremum norm sup norm bibliography edit 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 edit list of mathematical functions clifford algebra tensor field spectral theory functional determinant equicontinuity ascoli s theorem references edit fulton william harris joe 1991 representation theory a first course springer science business media p 4 isbn 9780387974958 conway john b 2007 a course in functional analysis vol 96 new york ny springer new york p 65 doi 10 1007 978 1 4757 4383 8 isbn 978 1 4419 3092 7 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 authority control databases national japan czech republic israel v t e lp spaces basic concepts banach hilbert spaces l p spaces measure lebesgue measure space measurable space function minkowski distance sequence spaces l 1 spaces integrable function lebesgue integral taxicab geometry l 2 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 l p 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 retrieved from https en wikipedia org w index php title function_space oldid 1354563238 categories function spaces topology of function spaces linear algebra hidden categories articles with short description short description is different from wikidata articles needing additional references from november 2017 all articles needing additional references pages that use a deprecated format of the math tags this page was last edited on 17 may 2026 at 04 24 utc text is available under the creative commons attribution sharealike 4 0 license additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about wikipedia disclaimers contact wikipedia legal safety contacts code of conduct developers statistics cookie statement mobile view search search toggle the table of contents function space 23 languages add topic
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