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displaystyle (759), the (421), and (152), space (149), for (106), infty (93), leq (79), that (78), #spaces (76), norm (72), measure (71), this (65), mathcal (63), are (55), function (54), vector (50), with (46), mathbb (45), ell (44), set (43), functions (42), inequality (39), edit (37), right (36), then (36), measurable (34), defined (34), which (33), left (33), all (32), finite (32), when (31), lebesgue (29), sigma (29), cdot (28), not (28), theorem (27), can (27), text (26), analysis (24), integrable (23), lambda (23), sup (22), every (22), def (22), locally (21), real (21), tfrac (21), only (21), where (21), convex (20), almost (20), any (20), mathrm (20), banach (19), mathbf (19), bounded (18), everywhere (18), its (18), one (18), from (17), distance (17), hilbert (17), has (17), theory (16), minkowski (16), continuous (16), sum (16), int (16), case (16), metric (15), sequence (15), but (15), such (15), geq (15), frac (15), non (14), topology (14), topological (14), algebra (14), 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q this inequality called hölder s inequality is in some sense optimal since if r 1 displaystyle r 1 and f displaystyle f is a measurable function such that sup g q 1 s f g d μ displaystyle sup _ g _ q leq 1 int _ s fg mathrm d mu infty where the supremum is taken over the closed unit ball of l q s μ displaystyle l q s mu then f l p s μ displaystyle f in l p s mu and f p sup g q 1 s f g d μ displaystyle f _ p sup _ g _ q leq 1 int _ s fg mathrm d mu generalized minkowski inequality edit minkowski inequality which states that p displaystyle cdot _ p satisfies the triangle inequality can be generalized if the measurable function f m n r displaystyle f m times n to mathbb r is non negative where m μ displaystyle m mu and n ν displaystyle n nu are measure spaces then for all 1 p q displaystyle 1 leq p leq q leq infty 11 f n l p m μ l q n ν f m l q n ν l p m μ displaystyle left left f cdot n right _ l p m mu right _ l q n nu leq left left f m cdot right _ l q n nu right _ l p m mu atomic decomposition edit if 1 p displaystyle 1 leq p infty then every non negative f l p μ displaystyle f in l p mu has an atomic decomposition 12 meaning that there exist a sequence r n n z displaystyle r_ n _ n in mathbb z of non negative real numbers and a sequence of non negative functions f n n z displaystyle f_ n _ n in mathbb z called the atoms whose supports supp f n n z displaystyle left operatorname supp f_ n right _ n in mathbb z are pairwise disjoint sets of measure μ supp f n 2 n 1 displaystyle mu left operatorname supp f_ n right leq 2 n 1 such that f n z r n f n displaystyle f sum _ n in mathbb z r_ n f_ n and for every integer n z displaystyle n in mathbb z f n 2 n p displaystyle f_ n _ infty leq 2 tfrac n p and 1 2 f p p n z r n p 2 f p p displaystyle tfrac 1 2 f _ p p leq sum _ n in mathbb z r_ n p leq 2 f _ p p and where moreover the sequence of functions r n f n n z displaystyle r_ n f_ n _ n in mathbb z depends only on f displaystyle f it is independent of p displaystyle p 12 these inequalities guarantee that f n p p 2 displaystyle f_ n _ p p leq 2 for all integers n displaystyle n while the supports of f n n z displaystyle f_ n _ n in mathbb z being pairwise disjoint implies 12 f p p n z r n p f n p p displaystyle f _ p p sum _ n in mathbb z r_ n p f_ n _ p p an atomic decomposition can be explicitly given by first defining for every integer n z displaystyle n in mathbb z 12 note 7 t n inf t r μ f t 2 n displaystyle t_ n inf t in mathbb r mu f t 2 n and then letting r n 2 n p t n and f n f r n 1 t n 1 f t n displaystyle r_ n 2 n p t_ n text and quad f_ n frac f r_ n mathbf 1 _ t_ n 1 f leq t_ n where μ f t μ s f s t displaystyle mu f t mu s f s t denotes the measure of the set f t s s f s t displaystyle f t s in s f s t and 1 t n 1 f t n displaystyle mathbf 1 _ t_ n 1 f leq t_ n denotes the indicator function of the set t n 1 f t n s s t n 1 f s t n displaystyle t_ n 1 f leq t_ n s in s t_ n 1 f s leq t_ n the sequence t n n z displaystyle t_ n _ n in mathbb z is decreasing and converges to 0 displaystyle 0 as n displaystyle n to infty 12 consequently if t n 0 displaystyle t_ n 0 then t n 1 0 displaystyle t_ n 1 0 and t n 1 f t n displaystyle t_ n 1 f leq t_ n varnothing so that f n 1 r n f 1 t n 1 f t n displaystyle f_ n frac 1 r_ n f mathbf 1 _ t_ n 1 f leq t_ n is identically equal to 0 displaystyle 0 in particular the division 1 r n displaystyle tfrac 1 r_ n by r n 0 displaystyle r_ n 0 causes no issues the complementary cumulative distribution function t r μ f t displaystyle t in mathbb r mapsto mu f t of f f displaystyle f f that was used to define the t n displaystyle t_ n also appears in the definition of the weak l p displaystyle l p norm given below and can be used to express the p displaystyle p norm p displaystyle cdot _ p for 1 p displaystyle 1 leq p infty of f l p s μ displaystyle f in l p s mu as the integral 12 f p p p 0 t p 1 μ f t d t displaystyle f _ p p p int _ 0 infty t p 1 mu f t mathrm d t where the integration is with respect to the usual lebesgue measure on 0 displaystyle 0 infty dual spaces edit the dual space of l p μ displaystyle l p mu for 1 p displaystyle 1 p infty has a natural isomorphism with l q μ displaystyle l q mu where q displaystyle q is such that 1 p 1 q 1 displaystyle tfrac 1 p tfrac 1 q 1 this isomorphism associates g l q μ displaystyle g in l q mu with the functional κ p g l p μ displaystyle kappa _ p g in l p mu defined by f κ p g f f g d μ displaystyle f mapsto kappa _ p g f int fg mathrm d mu for every f l p μ displaystyle f in l p mu κ p l q μ l p μ displaystyle kappa _ p l q mu to l p mu is a well defined continuous linear mapping which is an isometry by the extremal case of hölder s inequality if s σ μ displaystyle s sigma mu is a σ displaystyle sigma finite measure space one can use the radon nikodym theorem to show that any g l p μ displaystyle g in l p mu can be expressed this way i e κ p displaystyle kappa _ p is an isometric isomorphism of banach spaces 13 hence it is usual to say simply that l q μ displaystyle l q mu is the continuous dual space of l p μ displaystyle l p mu for 1 p displaystyle 1 p infty the space l p μ displaystyle l p mu is reflexive let κ p displaystyle kappa _ p be as above and let κ q l p μ l q μ displaystyle kappa _ q l p mu to l q mu be the corresponding linear isometry consider the map from l p μ displaystyle l p mu to l p μ displaystyle l p mu obtained by composing κ q displaystyle kappa _ q with the transpose or adjoint of the inverse of κ p displaystyle kappa _ p j p l p μ κ q l q μ κ p 1 l p μ displaystyle j_ p l p mu mathrel overset kappa _ q longrightarrow l q mu mathrel overset left kappa _ p 1 right longrightarrow l p mu this map coincides with the canonical embedding j displaystyle j of l p μ displaystyle l p mu into its bidual moreover the map j p displaystyle j_ p is onto as composition of two onto isometries and this proves reflexivity if the measure μ displaystyle mu on s displaystyle s is sigma finite then the dual of l 1 μ displaystyle l 1 mu is isometrically isomorphic to l μ displaystyle l infty mu more precisely the map κ 1 displaystyle kappa _ 1 corresponding to p 1 displaystyle p 1 is an isometry from l μ displaystyle l infty mu onto l 1 μ displaystyle l 1 mu the dual of l μ displaystyle l infty mu is subtler elements of l μ displaystyle l infty mu can be identified with bounded signed finitely additive measures on s displaystyle s that are absolutely continuous with respect to μ displaystyle mu see ba space for more details if we assume the axiom of choice this space is much bigger than l 1 μ displaystyle l 1 mu except in some trivial cases however saharon shelah proved that there are relatively consistent extensions of zermelo fraenkel set theory zf dc every subset of the real numbers has the baire property in which the dual of ℓ displaystyle ell infty is ℓ 1 displaystyle ell 1 14 embeddings edit colloquially if 1 p q displaystyle 1 leq p q leq infty then l p s μ displaystyle l p s mu contains functions that are more locally singular while elements of l q s μ displaystyle l q s mu can be more spread out consider the lebesgue measure on the half line 0 displaystyle 0 infty a continuous function in l 1 displaystyle l 1 might blow up near 0 displaystyle 0 but must decay sufficiently fast toward infinity on the other hand continuous functions in l displaystyle l infty need not decay at all but no blow up is allowed more formally 15 if 0 p q displaystyle 0 p q infty l q s μ l p s μ displaystyle l q s mu subseteq l p s mu if and only if s displaystyle s does not contain sets of finite but arbitrarily large measure e g any finite measure if 0 p q displaystyle 0 p q leq infty l p s μ l q s μ displaystyle l p s mu subseteq l q s mu if and only if s displaystyle s does not contain sets of non zero but arbitrarily small measure e g the counting measure neither condition holds for the lebesgue measure on the real line while both conditions holds for the counting measure on any finite set as a consequence of the closed graph theorem the embedding is continuous i e the identity operator is a bounded linear map from l q displaystyle l q to l p displaystyle l p in the first case and l p displaystyle l p to l q displaystyle l q in the second indeed if the domain s displaystyle s has finite measure one can make the following explicit calculation using hölder s inequality 1 f p 1 1 q q p f p q p displaystyle mathbf 1 f p _ 1 leq mathbf 1 _ q q p f p _ q p leading to f p μ s 1 p 1 q f q displaystyle f _ p leq mu s 1 p 1 q f _ q the constant appearing in the above inequality is optimal in the sense that the operator norm of the identity i l q s μ l p s μ displaystyle i l q s mu to l p s mu is precisely i q p μ s 1 p 1 q displaystyle i _ q p mu s 1 p 1 q the case of equality being achieved exactly when f 1 displaystyle f 1 μ displaystyle mu almost everywhere dense subspaces edit let 1 p displaystyle 1 leq p infty and s σ μ displaystyle s sigma mu be a measure space and consider an integrable simple function f displaystyle f on s displaystyle s given by f j 1 n a j 1 a j displaystyle f sum _ j 1 n a_ j mathbf 1 _ a_ j where a j displaystyle a_ j are scalars a j σ displaystyle a_ j in sigma has finite measure and 1 a j displaystyle mathbf 1 _ a_ j is the indicator function of the set a j displaystyle a_ j for j 1 n displaystyle j 1 dots n by construction of the integral the vector space of integrable simple functions is dense in l p s σ μ displaystyle l p s sigma mu more can be said when s displaystyle s is a normal topological space and σ displaystyle sigma its borel 𝜎 algebra suppose v s displaystyle v subseteq s is an open set with μ v displaystyle mu v infty then for every borel set a σ displaystyle a in sigma contained in v displaystyle v there exist a closed set f displaystyle f and an open set u displaystyle u such that f a u v and μ u f μ u μ f ε displaystyle f subseteq a subseteq u subseteq v quad text and quad mu u setminus f mu u mu f varepsilon for every ε 0 displaystyle varepsilon 0 subsequently there exists a urysohn function 0 φ 1 displaystyle 0 leq varphi leq 1 on s displaystyle s that is 1 displaystyle 1 on f displaystyle f and 0 displaystyle 0 on s u displaystyle s setminus u with s 1 a φ d μ ε displaystyle int _ s mathbf 1 _ a varphi mathrm d mu varepsilon if s displaystyle s can be covered by an increasing sequence v n displaystyle v_ n of open sets that have finite measure then the space of p displaystyle p integrable continuous functions is dense in l p s σ μ displaystyle l p s sigma mu more precisely one can use bounded continuous functions that vanish outside one of the open sets v n displaystyle v_ n this applies in particular when s r d displaystyle s mathbb r d and when μ displaystyle mu is the lebesgue measure for example the space of continuous and compactly supported functions as well as the space of integrable step functions are dense in l p r d displaystyle l p mathbb r d closed subspaces edit if 0 p displaystyle 0 p infty is any positive real number μ displaystyle mu is a probability measure on a measurable space s σ displaystyle s sigma so that l μ l p μ displaystyle l infty mu subseteq l p mu and v l μ displaystyle v subseteq l infty mu is a vector subspace then v displaystyle v is a closed subspace of l p μ displaystyle l p mu if and only if v displaystyle v is finite dimensional 16 v displaystyle v was chosen independent of p displaystyle p in this theorem which is due to alexander grothendieck 16 it is crucial that the vector space v displaystyle v be a subset of l displaystyle l infty since it is possible to construct an infinite dimensional closed vector subspace of l 1 s 1 1 2 π λ displaystyle l 1 left s 1 tfrac 1 2 pi lambda right which is even a subset of l 4 displaystyle l 4 where λ displaystyle lambda is lebesgue measure on the unit circle s 1 displaystyle s 1 and 1 2 π λ displaystyle tfrac 1 2 pi lambda is the probability measure that results from dividing it by its mass λ s 1 2 π displaystyle lambda s 1 2 pi 16 applications edit statistics edit in statistics measures of central tendency and statistical dispersion such as the mean median and standard deviation can be defined in terms of l p displaystyle l p metrics and measures of central tendency can be characterized as solutions to variational problems in penalized regression l1 penalty and l2 penalty refer to penalizing either the l 1 displaystyle l 1 norm of a solution s vector of parameter values i e the sum of its absolute values or its squared l 2 displaystyle l 2 norm its euclidean length techniques which use an l1 penalty like lasso encourage sparse solutions where the many parameters are zero 17 elastic net regularization uses a penalty term that is a combination of the l 1 displaystyle l 1 norm and the squared l 2 displaystyle l 2 norm of the parameter vector hausdorff young inequality edit the fourier transform for the real line or for periodic functions see fourier series maps l p r displaystyle l p mathbb r to l q r displaystyle l q mathbb r or l p t displaystyle l p mathbf t to ℓ q displaystyle ell q respectively where 1 p 2 displaystyle 1 leq p leq 2 and 1 p 1 q 1 displaystyle tfrac 1 p tfrac 1 q 1 this is a consequence of the riesz thorin interpolation theorem and is made precise with the hausdorff young inequality by contrast if p 2 displaystyle p 2 the fourier transform does not map into l q displaystyle l q hilbert spaces edit hilbert spaces are central to many applications from quantum mechanics to stochastic calculus the spaces l 2 displaystyle l 2 and ℓ 2 displaystyle ell 2 are both hilbert spaces in fact by choosing a hilbert basis e displaystyle e i e a maximal orthonormal subset of l 2 displaystyle l 2 or any hilbert space one sees that every hilbert space is isometrically isomorphic to ℓ 2 e displaystyle ell 2 e same e displaystyle e as above i e a hilbert space of type ℓ 2 displaystyle ell 2 generalizations and extensions edit weak l p edit let s σ μ displaystyle s sigma mu be a measure space and f displaystyle f a measurable function with real or complex values on s displaystyle s the distribution function of f displaystyle f is defined for t 0 displaystyle t geq 0 by λ f t μ x s f x t displaystyle lambda _ f t mu x in s f x t if f displaystyle f is in l p s μ displaystyle l p s mu for some p displaystyle p with 1 p displaystyle 1 leq p infty then by markov s inequality λ f t f p p t p displaystyle lambda _ f t leq frac f _ p p t p a function f displaystyle f is said to be in the space weak l p s μ displaystyle l p s mu or l p w s μ displaystyle l p w s mu if there is a constant c 0 displaystyle c 0 such that for all t 0 displaystyle t 0 λ f t c p t p displaystyle lambda _ f t leq frac c p t p the best constant c displaystyle c for this inequality is the l p w displaystyle l p w norm of f displaystyle f and is denoted by f p w sup t 0 t λ f 1 p t displaystyle f _ p w sup _ t 0 t lambda _ f 1 p t the weak l p displayst...