{"history":[],"article":{"id":"5789bec7-2e6b-447c-8804-b2f04c7ffda7","title":"Lp space","slug":"lp-space","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the '''\nLp\nspaces''' are [function spaces](https://bliptext.com/articles/function-space) defined using a natural generalization of the [[Norm (mathematics)#p-norm|\np\n-norm]] for finite-dimensional [vector spaces](https://bliptext.com/articles/vector-space).\nThey are sometimes called **Lebesgue spaces**, named after [Henri Lebesgue](https://bliptext.com/articles/henri-lebesgue), although according to the [Bourbaki](https://bliptext.com/articles/nicolas-bourbaki) group they were first introduced by [Frigyes Riesz](https://bliptext.com/articles/frigyes-riesz).\nLp\nspaces form an important class of [Banach spaces](https://bliptext.com/articles/banach-space) in [functional analysis](https://bliptext.com/articles/functional-analysis), and of [topological vector spaces](https://bliptext.com/articles/topological-vector-space).\nBecause of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, economics, finance, engineering, and other disciplines.\n\n# Applications\n\n## Statistics\nIn statistics, measures of [central tendency](https://bliptext.com/articles/central-tendency) and [statistical dispersion](https://bliptext.com/articles/statistical-dispersion), such as the [mean](https://bliptext.com/articles/mean), [median](https://bliptext.com/articles/median), and [standard deviation](https://bliptext.com/articles/standard-deviation), can be defined in terms of L^p metrics, and measures of [central tendency](https://bliptext.com/articles/central-tendency) can be characterized as [solutions to variational problems](https://bliptext.com/articles/central-tendency).\nIn [penalized regression](https://bliptext.com/articles/penalized-regression), \"L1 penalty\" and \"L2 penalty\" refer to penalizing either the [L^1 norm](https://bliptext.com/articles/taxicab-geometry) of a solution's vector of parameter values (i.e. the sum of its absolute values), or its squared L^2 norm (its [Euclidean length](https://bliptext.com/articles/euclidean-norm)).\nTechniques which use an L1 penalty, like [LASSO](https://bliptext.com/articles/lasso), encourage sparse solutions (where the many parameters are zero).\n[Elastic net regularization](https://bliptext.com/articles/elastic-net-regularization) uses a penalty term that is a combination of the L^1 norm and the squared L^2 norm of the parameter vector.\n\n## Hausdorff–Young inequality\nThe [Fourier transform](https://bliptext.com/articles/fourier-transform) for the real line (or, for [periodic functions](https://bliptext.com/articles/periodic-functions), see [Fourier series](https://bliptext.com/articles/fourier-series)), maps L^p(\\Reals) to L^q(\\Reals) (or to \\ell^q) respectively, where and This is a consequence of the [Riesz–Thorin interpolation theorem](https://bliptext.com/articles/riesz-thorin-theorem), and is made precise with the [Hausdorff–Young inequality](https://bliptext.com/articles/hausdorff-young-inequality).\nBy contrast, if p > 2, the Fourier transform does not map into L^q.\n\n## Hilbert spaces\n[Hilbert spaces](https://bliptext.com/articles/hilbert-space) are central to many applications, from [quantum mechanics](https://bliptext.com/articles/quantum-mechanics) to [stochastic calculus](https://bliptext.com/articles/stochastic-calculus).\nThe spaces L^2 and \\ell^2 are both Hilbert spaces.\nIn fact, by choosing a Hilbert basis E, i.e., a maximal orthonormal subset of L^2 or any Hilbert space, one sees that every Hilbert space is isometrically isomorphic to \\ell^2(E) (same E as above), i.e., a Hilbert space of type \\ell^2.\n\n# The \n\np\n\n-norm in finite dimensions\nThe Euclidean length of a vector in the n-dimensional [real](https://bliptext.com/articles/real-number) [vector space](https://bliptext.com/articles/vector-space) \\Reals^n is given by the [Euclidean norm](https://bliptext.com/articles/euclidean-norm):\nThe Euclidean distance between two points x and y is the length \\|x - y\\|_2 of the straight line between the two points.\nIn many situations, the Euclidean distance is appropriate for capturing the actual distances in a given space.\nIn contrast, consider taxi drivers in a grid street plan who should measure distance not in terms of the length of the straight line to their destination, but in terms of the [rectilinear distance](https://bliptext.com/articles/taxicab-geometry), which takes into account that streets are either orthogonal or parallel to each other.\nThe class of p-norms generalizes these two examples and has an abundance of applications in many parts of [mathematics](https://bliptext.com/articles/mathematics), [physics](https://bliptext.com/articles/physics), and [computer science](https://bliptext.com/articles/computer-science).\n# [Illustrations of unit circles (see also superellipse) in \\Reals^2 based on different p-norms (every vector from the origin to the unit circle has a length of one, the length being calculated with length-formula of the corresponding p). | upload.wikimedia.org/wikipedia/commons/d/d4/Vector-p-Norms///qtl1.svg]\n\n## Definition\nFor a [real number](https://bliptext.com/articles/real-number) p \\geq 1, the **p-norm** or L^**p-norm** of x is defined by\nThe absolute value bars can be dropped when p is a rational number with an even numerator in its reduced form, and x is drawn from the set of real numbers, or one of its subsets.\nThe Euclidean norm from above falls into this class and is the 2-norm, and the 1-norm is the norm that corresponds to the [rectilinear distance](https://bliptext.com/articles/taxicab-geometry).\nThe **L^\\infty-norm** or [maximum norm](https://bliptext.com/articles/chebyshev-distance) (or uniform norm) is the limit of the L^p-norms for It turns out that this limit is equivalent to the following definition:\nSee [[L-infinity|\nL\n-infinity]].\nFor all p \\geq 1, the p-[norm](https://bliptext.com/articles/norm-mathematics)s and maximum [norm](https://bliptext.com/articles/norm-mathematics) as defined above indeed satisfy the properties of a \"length function\" (or [norm](https://bliptext.com/articles/norm-mathematics)), which are that:\nAbstractly speaking, this means that \\Reals^n together with the p-norm is a [normed vector space](https://bliptext.com/articles/normed-vector-space).\nMoreover, it turns out that this space is complete, thus making it a [Banach space](https://bliptext.com/articles/banach-space).\nThis Banach space is the L^p-space over\n\n### Relations between \n\np\n\n-norms\nThe grid distance or rectilinear distance (sometimes called the \"[Manhattan distance](https://bliptext.com/articles/manhattan-distance)\") between two points is never shorter than the length of the line segment between them (the Euclidean or \"as the crow flies\" distance).\nFormally, this means that the Euclidean norm of any vector is bounded by its 1-norm:\nThis fact generalizes to p-norms in that the p-norm \\|x\\|_p of any given vector x does not grow with p:\nFor the opposite direction, the following relation between the 1-norm and the 2-norm is known:\nThis inequality depends on the dimension n of the underlying vector space and follows directly from the [Cauchy–Schwarz inequality](https://bliptext.com/articles/cauchy-schwarz-inequality).\nIn general, for vectors in \\Complex^n where 0 < r < p:\nThis is a consequence of [Hölder's inequality](https://bliptext.com/articles/h-lder-s-inequality).\n\n## When \n\n0 < p < 1\nIn \\Reals^n for n > 1, the formula\ndefines an absolutely [homogeneous function](https://bliptext.com/articles/homogeneous-function) for 0 < p < 1; however, the resulting function does not define a norm, because it is not [subadditive](https://bliptext.com/articles/subadditivity).\nOn the other hand, the formula\ndefines a subadditive function at the cost of losing absolute homogeneity.\nIt does define an [F-norm](https://bliptext.com/articles/f-space), though, which is homogeneous of degree p.\nHence, the function\ndefines a [metric](https://bliptext.com/articles/metric-space).\nThe [metric space](https://bliptext.com/articles/metric-space) is denoted by \\ell_n^p.\nAlthough the p-unit ball B_n^p around the origin in this metric is \"concave\", the topology defined on \\Reals^n by the metric B_p is the usual vector space topology of \\Reals^n, hence \\ell_n^p is a [locally convex](https://bliptext.com/articles/locally-convex) topological vector space.\nBeyond this qualitative statement, a quantitative way to measure the lack of convexity of \\ell_n^p is to denote by C_p(n) the smallest constant C such that the scalar multiple C \\, B_n^p of the p-unit ball contains the convex hull of B_n^p, which is equal to B_n^1.\nThe fact that for fixed p < 1 we have\nshows that the infinite-dimensional sequence space \\ell^p defined below, is no longer locally convex.\n# [Astroid, unit circle in metric | upload.wikimedia.org/wikipedia/commons/0/03/Astroid.svg]\n\n## When \n\np = 0\nThere is one \\ell_0 norm and another function called the \\ell_0 \"norm\" (with quotation marks).\nThe mathematical definition of the \\ell_0 norm was established by [Banach](https://bliptext.com/articles/stefan-banach)'s [Theory of Linear Operations](https://bliptext.com/articles/theory-of-linear-operations).\nThe [space](https://bliptext.com/articles/f-space) of sequences has a complete metric topology provided by the [F-norm](https://bliptext.com/articles/f-space)\nwhich is discussed by Stefan Rolewicz in Metric Linear Spaces.\nThe \\ell_0-normed space is studied in functional analysis, probability theory, and harmonic analysis.\nAnother function was called the \\ell_0 \"norm\" by [David Donoho](https://bliptext.com/articles/david-donoho)—whose quotation marks warn that this function is not a proper norm—is the number of non-zero entries of the vector x. Many authors [abuse terminology](https://bliptext.com/articles/abuse-of-terminology) by omitting the quotation marks.\nDefining [0^0 = 0,](https://bliptext.com/articles/zero-to-the-power-of-zero) the zero \"norm\" of x is equal to\nThis is not a [norm](https://bliptext.com/articles/norm-mathematics) because it is not [homogeneous](https://bliptext.com/articles/homogeneous-function).\nFor example, scaling the vector x by a positive constant does not change the \"norm\".\nDespite these defects as a mathematical norm, the non-zero counting \"norm\" has uses in [scientific computing](https://bliptext.com/articles/scientific-computing), [information theory](https://bliptext.com/articles/information-theory), and [statistics](https://bliptext.com/articles/statistics)–notably in [compressed sensing](https://bliptext.com/articles/compressed-sensing) in [signal processing](https://bliptext.com/articles/signal-processing) and computational [harmonic analysis](https://bliptext.com/articles/harmonic-analysis).\nDespite not being a norm, the associated metric, known as [Hamming distance](https://bliptext.com/articles/hamming-distance), is a valid distance, since homogeneity is not required for distances.\n\n# The \n\np\n\n-norm in infinite dimensions and \n\nℓ{{i sup|p}}\n\n spaces\n\n## The sequence space \n\nℓ{{i sup|p}}\nThe p-norm can be extended to vectors that have an infinite number of components ([sequences](https://bliptext.com/articles/sequence)), which yields the space \\ell^p.\nThis contains as special cases:\nThe space of sequences has a natural vector space structure by applying addition and scalar multiplication coordinate by coordinate.\nExplicitly, the vector sum and the scalar action for infinite [sequences](https://bliptext.com/articles/sequence) of real (or [complex](https://bliptext.com/articles/complex-number)) numbers are given by:\nDefine the p-norm:\nHere, a complication arises, namely that the [series](https://bliptext.com/articles/series-mathematics) on the right is not always convergent, so for example, the sequence made up of only ones, will have an infinite p-norm for The space \\ell^p is then defined as the set of all infinite sequences of real (or complex) numbers such that the p-norm is finite.\nOne can check that as p increases, the set \\ell^p grows larger.\nFor example, the sequence\nis not in \\ell^1, but it is in \\ell^p for p > 1, as the series\ndiverges for p = 1 (the [harmonic series](https://bliptext.com/articles/harmonic-series-mathematics)), but is convergent for p > 1.\nOne also defines the \\infty-norm using the [supremum](https://bliptext.com/articles/supremum):\nand the corresponding space \\ell^\\infty of all bounded sequences.\nIt turns out that\nif the right-hand side is finite, or the left-hand side is infinite.\nThus, we will consider \\ell^p spaces for\nThe p-norm thus defined on \\ell^p is indeed a norm, and \\ell^p together with this norm is a [Banach space](https://bliptext.com/articles/banach-space).\nThe fully general L^p space is obtained—as seen below—by considering vectors, not only with finitely or countably-infinitely many components, but with \"arbitrarily many components\"; in other words, [functions](https://bliptext.com/articles/function-mathematics).\nAn [integral](https://bliptext.com/articles/integral) instead of a sum is used to define the p-norm.\n\n## General ℓp-space\nIn complete analogy to the preceding definition one can define the space \\ell^p(I) over a general [index set](https://bliptext.com/articles/index-set) I (and ) as\nwhere convergence on the right means that only countably many summands are nonzero (see also [Unconditional convergence](https://bliptext.com/articles/unconditional-convergence)).\nWith the norm\nthe space \\ell^p(I) becomes a Banach space.\nIn the case where I is finite with n elements, this construction yields \\Reals^n with the p-norm defined above.\nIf I is countably infinite, this is exactly the sequence space \\ell^p defined above.\nFor uncountable sets I this is a non-[separable](https://bliptext.com/articles/separable-space) Banach space which can be seen as the [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) [direct limit](https://bliptext.com/articles/direct-limit) of \\ell^p-sequence spaces.\nFor p = 2, the -norm is even induced by a canonical [inner product](https://bliptext.com/articles/inner-product) called the , which means that holds for all vectors \\mathbf{x}.\nThis inner product can expressed in terms of the norm by using the [polarization identity](https://bliptext.com/articles/polarization-identity).\nOn \\ell^2, it can be defined by\nwhile for the space L^2(X, \\mu) associated with a [measure space](https://bliptext.com/articles/measure-mathematics) which consists of all [square-integrable functions](https://bliptext.com/articles/square-integrable-function), it is\nNow consider the case p = \\infty.\nDefine\nwhere for all x\nThe index set I can be turned into a [measure space](https://bliptext.com/articles/measure-space) by giving it the [discrete σ-algebra](https://bliptext.com/articles/algebra) and the [counting measure](https://bliptext.com/articles/counting-measure).\nThen the space \\ell^p(I) is just a special case of the more general L^p-space (defined below).\n\n# Lp spaces and Lebesgue integrals\nAn L^p space may be defined as a space of measurable functions for which the p-th power of the [absolute value](https://bliptext.com/articles/absolute-value) is [Lebesgue integrable](https://bliptext.com/articles/lebesgue-integrable), where functions which agree almost everywhere are identified.\nMore generally, let be a [measure space](https://bliptext.com/articles/measure-space) and\nWhen, consider the set of all [measurable functions](https://bliptext.com/articles/measurable-function) f from S to \\Complex or \\Reals whose [absolute value](https://bliptext.com/articles/absolute-value) raised to the p-th power has a finite integral, or in symbols:\nTo define the set for p = \\infty, recall that two functions f and g defined on S are said to be, written , if the set is measurable and has measure zero.\nSimilarly, a measurable function f (and its [absolute value](https://bliptext.com/articles/absolute-value)) is (or ) by a real number C, written, if the (necessarily) measurable set has measure zero.\nThe space is the set of all measurable functions f that are bounded almost everywhere (by some real C) and is defined as the [infimum](https://bliptext.com/articles/infimum) of these bounds:\nWhen then this is the same as the [essential supremum](https://bliptext.com/articles/essential-supremum) of the absolute value of f:\nFor example, if f is a measurable function that is equal to 0 almost everywhere then \\|f\\|_p = 0 for every p and thus for all p.\nFor every positive p, the value under of a measurable function f and its absolute value are always the same (that is, for all p) and so a measurable function belongs to if and only if its absolute value does.\nBecause of this, many formulas involving p-norms are stated only for non-negative real-valued functions.\nConsider for example the identity which holds whenever f \\geq 0 is measurable, r > 0 is real, and (here when p = \\infty).\nThe non-negativity requirement f \\geq 0 can be removed by substituting |f| in for f, which gives\nNote in particular that when p = r is finite then the formula relates the p-norm to the 1-norm.\n**Seminormed space of p-th power integrable functions**\nEach set of functions forms a [vector space](https://bliptext.com/articles/vector-space) when addition and scalar multiplication are defined pointwise.\nThat the sum of two p-th power integrable functions f and g is again p-th power integrable follows from\nalthough it is also a consequence of [Minkowski's inequality](https://bliptext.com/articles/minkowski-inequality)\nwhich establishes that \\|\\cdot\\|_p satisfies the [triangle inequality](https://bliptext.com/articles/triangle-inequality) for (the [triangle inequality](https://bliptext.com/articles/triangle-inequality) does not hold for 0 < p < 1).\nThat is closed under scalar multiplication is due to \\|\\cdot\\|_p being [absolutely homogeneous](https://bliptext.com/articles/absolute-homogeneity), which means that for every scalar s and every function f.\n[Absolute homogeneity](https://bliptext.com/articles/absolute-homogeneity), the [triangle inequality](https://bliptext.com/articles/triangle-inequality), and non-negativity are the defining properties of a [seminorm](https://bliptext.com/articles/seminorm).\nThus \\|\\cdot\\|_p is a seminorm and the set of p-th power integrable functions together with the function \\|\\cdot\\|_p defines a [seminormed vector space](https://bliptext.com/articles/seminormed-vector-space).\nIn general, the [seminorm](https://bliptext.com/articles/semi[norm](https://bliptext.com/articles/norm-mathematics)) \\|\\cdot\\|_p is not a [norm](https://bliptext.com/articles/norm-mathematics) because there might exist measurable functions f that satisfy \\|f\\|_p = 0 but are not equal to 0 (\\|\\cdot\\|_p is a [norm](https://bliptext.com/articles/norm-mathematics) if and only if no such f exists).\n**Zero sets of p-seminorms**\nIf f is measurable and equals 0 a.e. then \\|f\\|_p = 0 for all positive\nOn the other hand, if f is a measurable function for which there exists some such that \\|f\\|_p = 0 then f = 0 almost everywhere.\nWhen p is finite then this follows from the p = 1 case and the formula mentioned above.\nThus if is positive and f is any measurable function, then \\|f\\|_p = 0 if and only if f = 0 [almost everywhere](https://bliptext.com/articles/almost-everywhere).\nSince the right hand side (f = 0 a.e.) does not mention p, it follows that all \\|\\cdot\\|_p have the same [zero set](https://bliptext.com/articles/zero-set) (it does not depend on p).\nSo denote this common set by\nThis set is a vector subspace of for every positive\n**Quotient vector space**\nLike every [seminorm](https://bliptext.com/articles/semi[norm](https://bliptext.com/articles/norm-mathematics)), the [seminorm](https://bliptext.com/articles/semi[norm](https://bliptext.com/articles/norm-mathematics)) \\|\\cdot\\|_p induces a [norm](https://bliptext.com/articles/norm-mathematics) (defined shortly) on the canonical [quotient vector space](https://bliptext.com/articles/quotient-space-linear-algebra) of by its vector subspace\nThis normed quotient space is called and it is the subject of this article.\nWe begin by defining the quotient vector space.\nGiven any the [coset](https://bliptext.com/articles/coset) consists of all measurable functions g that are equal to f [almost everywhere](https://bliptext.com/articles/almost-everywhere).\nThe set of all cosets, typically denoted by\nforms a vector space with origin when vector addition and scalar multiplication are defined by and\nThis particular quotient vector space will be denoted by\nTwo cosets are equal if and only if (or equivalently, ), which happens if and only if f = g almost everywhere; if this is the case then f and g are identified in the quotient space.\n**The p-norm on the quotient vector space**\nGiven any the value of the seminorm \\|\\cdot\\|_p on the [coset](https://bliptext.com/articles/coset) is constant and equal to \\|f\\|_p; denote this unique value by so that:\nThis assignment defines a map, which will also be denoted by on the [quotient vector space](https://bliptext.com/articles/quotient-space-linear-algebra)\nThis map is a [norm](https://bliptext.com/articles/norm-mathematics) on L^p(S, \\mu) called the.\nThe value of a coset is independent of the particular function f that was chosen to represent the coset, meaning that if is any coset then for every (since for every ).\n**The Lebesgue L^p space**\nThe [normed vector space](https://bliptext.com/articles/normed-vector-space) is called or the of p-th power integrable functions and it is a [Banach space](https://bliptext.com/articles/banach-space) for every (meaning that it is a [complete metric space](https://bliptext.com/articles/complete-metric-space), a result that is sometimes called the [Riesz–Fischer theorem](https://bliptext.com/articles/riesz-fischer-theorem)).\nWhen the underlying measure space S is understood then L^p(S, \\mu) is often abbreviated L^p(\\mu), or even just L^p.\nDepending on the author, the subscript notation L_p might denote either L^p(S, \\mu) or\nIf the seminorm \\|\\cdot\\|_p on happens to be a norm (which happens if and only if ) then the normed space will be [linearly](https://bliptext.com/articles/linear-map) [isometrically isomorphic](https://bliptext.com/articles/isometrically-isomorphic) to the normed quotient space via the canonical map (since ); in other words, they will be, [up to](https://bliptext.com/articles/up-to) a [linear isometry](https://bliptext.com/articles/linear-isometry), the same normed space and so they may both be called \"L^p space\".\nThe above definitions generalize to [Bochner spaces](https://bliptext.com/articles/bochner-space).\nIn general, this process cannot be reversed: there is no consistent way to define a \"canonical\" representative of each coset of \\mathcal{N} in L^p.\nFor L^\\infty, however, there is a [theory of lifts](https://bliptext.com/articles/lifting-theory) enabling such recovery.\n\n## Special cases\nSimilar to the \\ell^p spaces, L^2 is the only [Hilbert space](https://bliptext.com/articles/hilbert-space) among L^p spaces.\nIn the complex case, the inner product on L^2 is defined by\nThe additional inner product structure allows for a richer theory, with applications to, for instance, [Fourier series](https://bliptext.com/articles/fourier-series) and [quantum mechanics](https://bliptext.com/articles/quantum-mechanics).\nFunctions in L^2 are sometimes called **[square-integrable functions](https://bliptext.com/articles/square-integrable-function)**, **quadratically integrable functions** or **square-summable functions**, but sometimes these terms are reserved for functions that are square-integrable in some other sense, such as in the sense of a [Riemann integral](https://bliptext.com/articles/riemann-integral).\nIf we use complex-valued functions, the space L^\\infty is a [commutative](https://bliptext.com/articles/commutative) [C*-algebra](https://bliptext.com/articles/c-algebra) with pointwise multiplication and conjugation.\nFor many measure spaces, including all sigma-finite ones, it is in fact a commutative [von Neumann algebra](https://bliptext.com/articles/von-neumann-algebra).\nAn element of L^\\infty defines a [bounded operator](https://bliptext.com/articles/bounded-operator) on any L^p space by [multiplication](https://bliptext.com/articles/multiplication-operator).\nFor the \\ell^p spaces are a special case of L^p spaces, when consists of the [natural numbers](https://bliptext.com/articles/natural-number) and \\mu is the [counting measure](https://bliptext.com/articles/counting-measure) on \\mathbb{N}.\nMore generally, if one considers any set S with the counting measure, the resulting L^p space is denoted \\ell^p(S).\nFor example, the space is the space of all sequences indexed by the integers, and when defining the p-norm on such a space, one sums over all the integers.\nThe space \\ell^p(n), where n is the set with n elements, is \\Reals^n with its p-norm as defined above.\nAs any Hilbert space, every space L^2 is linearly isometric to a suitable \\ell^2(I), where the cardinality of the set I is the cardinality of an arbitrary Hilbertian basis for this particular L^2.\n\n# Properties of Lp spaces\nAs in the discrete case, if there exists q < \\infty such that then\n**Hölder's inequality**\nSuppose satisfy (where ).\nIf and then and\nThis inequality, called [Hölder's inequality](https://bliptext.com/articles/h-lder-s-inequality), is in some sense optimal since if r = 1 (so ) and f is a measurable function such that\nwhere the [supremum](https://bliptext.com/articles/supremum) is taken over the closed unit ball of then and\n**Minkowski inequality**\n[Minkowski inequality](https://bliptext.com/articles/minkowski-inequality), which states that \\|\\cdot\\|_p satisfies the [triangle inequality](https://bliptext.com/articles/triangle-inequality), can be generalized:\nIf the measurable function is non-negative (where (M, \\mu) and (N, \\nu) are measure spaces) then for all\n\n## Atomic decomposition\nIf then every non-negative has an, meaning that there exist a sequence of non-negative real numbers and a sequence of non-negative functions called , whose supports are [pairwise disjoint sets](https://bliptext.com/articles/disjoint-sets) of measure such that\nand for every integer n \\in \\Z,\nand\nand where moreover, the sequence of functions depends only on f (it is independent of p).\nThese inequalities guarantee that for all integers n while the supports of being pairwise disjoint implies\nAn atomic decomposition can be explicitly given by first defining for every integer n \\in \\Z,\n(this [infimum](https://bliptext.com/articles/infimum) is attained by t_n; that is, holds) and then letting\nwhere denotes the measure of the set and denotes the [indicator function](https://bliptext.com/articles/indicator-function) of the set\nThe sequence is decreasing and converges to 0 as Consequently, if t_n = 0 then t_{n+1} = 0 and so that is identically equal to 0 (in particular, the division by r_n = 0 causes no issues).\nThe [complementary cumulative distribution function](https://bliptext.com/articles/complementary-cumulative-distribution-function) of |f| = f that was used to define the t_n also appears in the definition of the weak L^p-norm (given below) and can be used to express the p-norm \\|\\cdot\\|_p (for ) of as the integral\nwhere the integration is with respect to the usual Lebesgue measure on\n\n## Dual spaces\nThe [dual space](https://bliptext.com/articles/continuous-dual) (the Banach space of all continuous linear functionals) of L^p(\\mu) for has a natural isomorphism with L^q(\\mu), where q is such that (i.e. ). This isomorphism associates with the functional defined by for every\nThe fact that \\kappa_p(g) is well defined and continuous follows from [Hölder's inequality](https://bliptext.com/articles/h-lder-s-inequality).\nis a linear mapping which is an [isometry](https://bliptext.com/articles/isometry) by the [extremal case](https://bliptext.com/articles/h-lder-s-inequality) of Hölder's inequality.\nIt is also possible to show (for example with the [Radon–Nikodym theorem](https://bliptext.com/articles/radon-nikodym-theorem), see ) that any can be expressed this way: i.e., that \\kappa_p is onto.\nSince \\kappa_p is onto and isometric, it is an [isomorphism](https://bliptext.com/articles/isomorphism) of [Banach spaces](https://bliptext.com/articles/banach-space).\nWith this (isometric) isomorphism in mind, it is usual to say simply that L^q(\\mu) is the [continuous dual space](https://bliptext.com/articles/continuous-dual-space) of L^p(\\mu).\nFor the space L^p(\\mu) is [reflexive](https://bliptext.com/articles/reflexive-space).\nLet \\kappa_p be as above and let be the corresponding linear isometry.\nConsider the map from L^p(\\mu) to obtained by composing \\kappa_q with the [transpose](https://bliptext.com/articles/dual-space) (or adjoint) of the inverse of \\kappa_p:\nThis map coincides with the [canonical embedding](https://bliptext.com/articles/reflexive-space) J of L^p(\\mu) into its bidual.\nMoreover, the map j_p is onto, as composition of two onto isometries, and this proves reflexivity.\nIf the measure \\mu on S is [sigma-finite](https://bliptext.com/articles/sigma-finite), then the dual of L^1(\\mu) is isometrically isomorphic to (more precisely, the map \\kappa_1 corresponding to p = 1 is an isometry from onto L^1(\\mu)^*.\nThe dual of is subtler.\nElements of can be identified with bounded signed finitely additive measures on S that are [absolutely continuous](https://bliptext.com/articles/absolutely-continuous) with respect to \\mu.\nSee [ba space](https://bliptext.com/articles/ba-space) for more details.\nIf we assume the axiom of choice, this space is much bigger than L^1(\\mu) except in some trivial cases.\nHowever, [Saharon Shelah](https://bliptext.com/articles/saharon-shelah) proved that there are relatively consistent extensions of [Zermelo–Fraenkel set theory](https://bliptext.com/articles/zermelo-fraenkel-set-theory) (ZF + [DC](https://bliptext.com/articles/axiom-of-dependent-choice) + \"Every subset of the real numbers has the [Baire property](https://bliptext.com/articles/baire-property)\") in which the dual of \\ell^\\infty is \\ell^1.\n\n## Embeddings\nColloquially, if then L^p(S, \\mu) contains functions that are more locally singular, while elements of L^q(S, \\mu) can be more spread out.\nConsider the Lebesgue measure on the half line A continuous function in L^1 might blow up near 0 but must decay sufficiently fast toward infinity.\nOn the other hand, continuous functions in L^\\infty need not decay at all but no blow-up is allowed.\nThe precise technical result is the following.\nSuppose that Then:\nNeither condition holds for the real line with the Lebesgue measure while both conditions holds for the [counting measure](https://bliptext.com/articles/counting-measure) on any finite set.\nIn both cases the embedding is continuous, in that the identity operator is a bounded linear map from L^q to L^p in the first case, and L^p to L^q in the second.\n(This is a consequence of the [closed graph theorem](https://bliptext.com/articles/closed-graph-theorem) and properties of L^p spaces.)\nIndeed, if the domain S has finite measure, one can make the following explicit calculation using [Hölder's inequality](https://bliptext.com/articles/h-lder-s-inequality)\nleading to\nThe constant appearing in the above inequality is optimal, in the sense that the [operator norm](https://bliptext.com/articles/operator-norm) of the identity is precisely\nthe case of equality being achieved exactly when f = 1 \\mu-almost-everywhere.\n\n## Dense subspaces\nThroughout this section we assume that\nLet be a measure space.\nAn integrable simple function f on S is one of the form\nwhere a_j are scalars, has finite measure and is the [indicator function](https://bliptext.com/articles/indicator-function) of the set A_j, for By construction of the [integral](https://bliptext.com/articles/lebesgue-integration), the vector space of integrable simple functions is dense in\nMore can be said when S is a [normal](https://bliptext.com/articles/normal-space) [topological space](https://bliptext.com/articles/topological-space) and \\Sigma its [Borel 𝜎–algebra](https://bliptext.com/articles/borel-algebra), i.e., the smallest 𝜎–algebra of subsets of S containing the [open sets](https://bliptext.com/articles/open-set).\nSuppose is an open set with It can be proved that for every Borel set contained in V, and for every there exist a closed set F and an open set U such that\nIt follows that there exists a continuous [Urysohn function](https://bliptext.com/articles/urysohn-s-lemma) on S that is 1 on F and 0 on with\nIf S can be covered by an increasing sequence (V_n) of open sets that have finite measure, then the space of p–integrable continuous functions is dense in More precisely, one can use bounded continuous functions that vanish outside one of the open sets V_n.\nThis applies in particular when and when \\mu is the Lebesgue measure.\nThe space of continuous and compactly supported functions is dense in Similarly, the space of integrable step functions is dense in this space is the linear span of indicator functions of bounded intervals when d = 1, of bounded rectangles when d = 2 and more generally of products of bounded intervals.\nSeveral properties of general functions in are first proved for continuous and compactly supported functions (sometimes for step functions), then extended by density to all functions.\nFor example, it is proved this way that translations are continuous on in the following sense:\nwhere\n\n## Closed subspaces\nIf is any positive real number, \\mu is a [probability measure](https://bliptext.com/articles/probability-measure) on a measurable space (S, \\Sigma) (so that ), and is a vector subspace, then V is a closed subspace of L^p(\\mu) if and only if V is finite-dimensional (V was chosen independent of p).\nIn this theorem, which is due to [Alexander Grothendieck](https://bliptext.com/articles/alexander-grothendieck), it is crucial that the vector space V be a subset of L^\\infty since it is possible to construct an infinite-dimensional closed vector subspace of (that is even a subset of L^4), where \\lambda is [Lebesgue measure](https://bliptext.com/articles/lebesgue-measure) on the [unit circle](https://bliptext.com/articles/unit-circle) S^1 and is the probability measure that results from dividing it by its mass\n\n# L{{sup|p}} (0 < p < 1)\nLet be a measure space.\nIf 0 < p < 1, then L^p(\\mu) can be defined as above: it is the quotient vector space of those measurable functions f such that\nAs before, we may introduce the p-norm but \\|\\cdot\\|_p does not satisfy the triangle inequality in this case, and defines only a [quasi-norm](https://bliptext.com/articles/quasi-norm).\nThe inequality valid for implies that\nand so the function\nis a metric on L^p(\\mu).\nThe resulting metric space is [complete](https://bliptext.com/articles/complete-metric-space); the verification is similar to the familiar case when p \\geq 1.\nThe balls\nform a local base at the origin for this topology, as r > 0 ranges over the positive reals.\nThese balls satisfy for all real r > 0, which in particular shows that B_1 is a [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space) neighborhood of the origin; in other words, this space is locally [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space), just like every [normed space](https://bliptext.com/articles/normed-space), despite \\|\\cdot\\|_p not being a norm.\nIn this setting L^p satisfies a reverse Minkowski inequality, that is for\nThis result may be used to prove [Clarkson's inequalities](https://bliptext.com/articles/clarkson-s-inequalities), which are in turn used to establish the [uniform convexity](https://bliptext.com/articles/uniformly-convex-space) of the spaces L^p for.\nThe space L^p for 0 < p < 1 is an [F-space](https://bliptext.com/articles/f-space): it admits a complete translation-invariant metric with respect to which the vector space operations are continuous.\nIt is the prototypical example of an [F-space](https://bliptext.com/articles/f-space) that, for most reasonable measure spaces, is not [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space): in \\ell^p or every open convex set containing the 0 function is unbounded for the p-quasi-norm; therefore, the 0 vector does not possess a fundamental system of convex neighborhoods.\nSpecifically, this is true if the measure space S contains an infinite family of disjoint measurable sets of finite positive measure.\nThe only nonempty convex open set in L^p([0, 1]) is the entire space.\nAs a particular consequence, there are no nonzero continuous linear functionals on the [continuous dual space](https://bliptext.com/articles/continuous-dual-space) is the zero space.\nIn the case of the [counting measure](https://bliptext.com/articles/counting-measure) on the natural numbers (producing the sequence space ), the bounded linear functionals on \\ell^p are exactly those that are bounded on \\ell^1, namely those given by sequences in Although \\ell^p does contain non-trivial convex open sets, it fails to have enough of them to give a base for the topology.\nThe situation of having no linear functionals is highly undesirable for the purposes of doing analysis.\nIn the case of the Lebesgue measure on \\Reals^n, rather than work with L^p for 0 < p < 1, it is common to work with the [Hardy space](https://bliptext.com/articles/hardy-space)\nH\nwhenever possible, as this has quite a few linear functionals: enough to distinguish points from one another.\nHowever, the [Hahn–Banach theorem](https://bliptext.com/articles/hahn-banach-theorem) still fails in\nH\nfor p < 1.\n\n## L{{sup|0}}\n, the space of measurable functions\nThe vector space of (equivalence classes of) measurable functions on is denoted.\nBy definition, it contains all the L^p, and is equipped with the topology of [convergence in measure](https://bliptext.com/articles/convergence-in-measure).\nWhen \\mu is a probability measure (i.e., \\mu(S) = 1), this mode of convergence is named [convergence in probability](https://bliptext.com/articles/convergence-in-probability).\nThe space L^0 is always a [topological abelian group](https://bliptext.com/articles/topological-abelian-group) but is only a [topological vector space](https://bliptext.com/articles/topological-vector-space) if This is because scalar multiplication is continuous if and only if If is \\sigma-finite then the [weaker topology](https://bliptext.com/articles/weaker-topology) of [local convergence in measure](https://bliptext.com/articles/local-convergence-in-measure) is an [F-space](https://bliptext.com/articles/f-space), i.e. a [completely](https://bliptext.com/articles/complete-topological-vector-space) metrizable [topological vector space](https://bliptext.com/articles/topological-vector-space).\nMoreover, this topology is isometric to global convergence in measure for a suitable choice of [probability measure](https://bliptext.com/articles/probability-measure) \\nu.\nThe description is easier when \\mu is finite.\nIf \\mu is a [finite measure](https://bliptext.com/articles/finite-measure) on the 0 function admits for the convergence in measure the following [fundamental system of neighborhoods](https://bliptext.com/articles/fundamental-system-of-neighborhoods)\nThe topology can be defined by any metric d of the form\nwhere \\varphi is bounded continuous concave and non-decreasing on with and when t > 0 (for example, Such a metric is called [Lévy](https://bliptext.com/articles/paul-l-vy-mathematician)-metric for L^0. Under this metric the space L^0 is complete. However, as mentioned above, scalar multiplication is continuous with respect to this metric only if . To see this, consider the Lebesgue measurable function defined by f(x)=x. Then clearly . The space L^0 is in general not locally bounded, and not locally convex.\nFor the infinite Lebesgue measure \\lambda on \\Reals^n, the definition of the fundamental system of neighborhoods could be modified as follows\nThe resulting space, with the topology of local convergence in measure, is isomorphic to the space for any positive \\lambda–integrable density g.\n\n# Generalizations and extensions\n\n## Weak \n\nL{{sup|p}}\nLet be a measure space, and f a [measurable function](https://bliptext.com/articles/measurable-function) with real or complex values on S. The [distribution function](https://bliptext.com/articles/cumulative-distribution-function) of f is defined for t \\geq 0 by\nIf f is in L^p(S, \\mu) for some p with then by [Markov's inequality](https://bliptext.com/articles/markov-s-inequality),\nA function f is said to be in the space **weak L^p(S, \\mu)**, or if there is a constant C > 0 such that, for all t > 0,\nThe best constant C for this inequality is the L^{p,w}-norm of f, and is denoted by\nThe weak L^p coincide with the [Lorentz spaces](https://bliptext.com/articles/lorentz-space) so this notation is also used to denote them.\nThe L^{p,w}-norm is not a true norm, since the [triangle inequality](https://bliptext.com/articles/triangle-inequality) fails to hold.\nNevertheless, for f in\nand in particular\nIn fact, one has\nand raising to power 1/p and taking the supremum in t one has\nUnder the convention that two functions are equal if they are equal \\mu almost everywhere, then the spaces L^{p,w} are complete.\nFor any 0 < r < p the expression\nis comparable to the L^{p,w}-norm.\nFurther in the case p > 1, this expression defines a norm if r = 1.\nHence for p > 1 the weak L^p spaces are [Banach spaces](https://bliptext.com/articles/banach-space).\nA major result that uses the L^{p,w}-spaces is the [Marcinkiewicz interpolation theorem](https://bliptext.com/articles/marcinkiewicz-interpolation), which has broad applications to [harmonic analysis](https://bliptext.com/articles/harmonic-analysis) and the study of [singular integrals](https://bliptext.com/articles/singular-integrals).\n\n## Weighted \n\nL{{sup|p}}\n\n spaces\nAs before, consider a [measure space](https://bliptext.com/articles/measure-space) Let be a measurable function.\nThe w-**weighted L^p space** is defined as where means the measure \\nu defined by\nor, in terms of the [Radon–Nikodym derivative](https://bliptext.com/articles/radon-nikodym-theorem), the [norm](https://bliptext.com/articles/norm-mathematics) for is explicitly\nAs L^p-spaces, the weighted spaces have nothing special, since is equal to But they are the natural framework for several results in harmonic analysis ; they appear for example in the [Muckenhoupt theorem](https://bliptext.com/articles/muckenhoupt-weights): for the classical [Hilbert transform](https://bliptext.com/articles/hilbert-transform) is defined on where \\mathbf{T} denotes the [unit circle](https://bliptext.com/articles/unit-circle) and \\lambda the Lebesgue measure; the (nonlinear) [Hardy–Littlewood maximal operator](https://bliptext.com/articles/hardy-littlewood-maximal-operator) is bounded on Muckenhoupt's theorem describes weights w such that the [Hilbert transform](https://bliptext.com/articles/hilbert-transform) remains bounded on and the maximal operator on\n\n## L{{sup|p}}\n\n spaces on manifolds\nOne may also define spaces L^p(M) on a manifold, called the **intrinsic L^p spaces** of the manifold, using [densities](https://bliptext.com/articles/density-on-a-manifold).\n\n## Vector-valued \n\nL{{sup|p}}\n\n spaces\nGiven a measure space and a [locally convex space](https://bliptext.com/articles/locally-convex-topological-vector-space) E (here assumed to be [complete](https://bliptext.com/articles/complete-topological-vector-space)), it is possible to define spaces of p-integrable E-valued functions on \\Omega in a number of ways.\nOne way is to define the spaces of [Bochner integrable](https://bliptext.com/articles/bochner-integral) and [Pettis integrable](https://bliptext.com/articles/pettis-integral) functions, and then endow them with [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) [TVS-topologies](https://bliptext.com/articles/vector-topology) that are (each in their own way) a natural generalization of the usual L^p topology.\nAnother way involves [topological tensor products](https://bliptext.com/articles/topological-tensor-product) of with E. Element of the vector space are finite sums of simple tensors where each simple tensor f \\times e may be identified with the function that sends This [tensor product](https://bliptext.com/articles/tensor-product) is then endowed with a locally convex topology that turns it into a topological [tensor product](https://bliptext.com/articles/tensor-product), the most common of which are the projective [tensor product](https://bliptext.com/articles/tensor-product), denoted by and the injective [tensor product](https://bliptext.com/articles/tensor-product), denoted by In general, neither of these space are complete so their [completions](https://bliptext.com/articles/complete-topological-vector-space) are constructed, which are respectively denoted by and (this is analogous to how the space of scalar-valued [simple functions](https://bliptext.com/articles/simple-function) on \\Omega, when seminormed by any is not complete so a completion is constructed which, after being quotiented by is isometrically isomorphic to the Banach space ).\n[Alexander Grothendieck](https://bliptext.com/articles/alexander-grothendieck) showed that when E is a [nuclear space](https://bliptext.com/articles/nuclear-space) (a concept he introduced), then these two constructions are, respectively, canonically TVS-isomorphic with the spaces of Bochner and Pettis integral functions mentioned earlier; in short, they are indistinguishable.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T17:37:15.901Z","updated_at":"2025-01-25T17:37:15.901Z","search_vector":"'lp':1 'space':2","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the '''\nLp\nspaces''' are [function spaces](https://bliptext.com/articles/function-space) defined using a natural generalization of the [[Norm (mathematics)#p-norm|\np\n-norm]] for finite-dimensional [vector spaces](https://bliptext.com/articles/vector-space).\nThey are sometimes called **Lebesgue spaces**, named after [Henri Lebesgue](https://bliptext.com/articles/henri-lebesgue), although according to the [Bourbaki](https://bliptext.com/articles/nicolas-bourbaki) group they were first introduced by [Frigyes Riesz](https://bliptext.com/articles/frigyes-riesz).\nLp\nspaces form an important class of [Banach spaces](https://bliptext.com/articles/banach-space) in [functional analysis](https://bliptext.com/articles/functional-analysis), and of [topological vector spaces](https://bliptext.com/articles/topological-vector-space).\nBecause of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, economics, finance, engineering, and other disciplines.\n\n# Applications\n\n## Statistics\nIn statistics, measures of [central tendency](https://bliptext.com/articles/central-tendency) and [statistical dispersion](https://bliptext.com/articles/statistical-dispersion), such as the [mean](https://bliptext.com/articles/mean), [median](https://bliptext.com/articles/median), and [standard deviation](https://bliptext.com/articles/standard-deviation), can be defined in terms of L^p metrics, and measures of [central tendency](https://bliptext.com/articles/central-tendency) can be characterized as [solutions to variational problems](https://bliptext.com/articles/central-tendency).\nIn [penalized regression](https://bliptext.com/articles/penalized-regression), \"L1 penalty\" and \"L2 penalty\" refer to penalizing either the [L^1 norm](https://bliptext.com/articles/taxicab-geometry) of a solution's vector of parameter values (i.e. the sum of its absolute values), or its squared L^2 norm (its [Euclidean length](https://bliptext.com/articles/euclidean-norm)).\nTechniques which use an L1 penalty, like [LASSO](https://bliptext.com/articles/lasso), encourage sparse solutions (where the many parameters are zero).\n[Elastic net regularization](https://bliptext.com/articles/elastic-net-regularization) uses a penalty term that is a combination of the L^1 norm and the squared L^2 norm of the parameter vector.\n\n## Hausdorff–Young inequality\nThe [Fourier transform](https://bliptext.com/articles/fourier-transform) for the real line (or, for [periodic functions](https://bliptext.com/articles/periodic-functions), see [Fourier series](https://bliptext.com/articles/fourier-series)), maps L^p(\\Reals) to L^q(\\Reals) (or to \\ell^q) respectively, where and This is a consequence of the [Riesz–Thorin interpolation theorem](https://bliptext.com/articles/riesz-thorin-theorem), and is made precise with the [Hausdorff–Young inequality](https://bliptext.com/articles/hausdorff-young-inequality).\nBy contrast, if p > 2, the Fourier transform does not map into L^q.\n\n## Hilbert spaces\n[Hilbert spaces](https://bliptext.com/articles/hilbert-space) are central to many applications, from [quantum mechanics](https://bliptext.com/articles/quantum-mechanics) to [stochastic calculus](https://bliptext.com/articles/stochastic-calculus).\nThe spaces L^2 and \\ell^2 are both Hilbert spaces.\nIn fact, by choosing a Hilbert basis E, i.e., a maximal orthonormal subset of L^2 or any Hilbert space, one sees that every Hilbert space is isometrically isomorphic to \\ell^2(E) (same E as above), i.e., a Hilbert space of type \\ell^2.\n\n# The \n\np\n\n-norm in finite dimensions\nThe Euclidean length of a vector in the n-dimensional [real](https://bliptext.com/articles/real-number) [vector space](https://bliptext.com/articles/vector-space) \\Reals^n is given by the [Euclidean norm](https://bliptext.com/articles/euclidean-norm):\nThe Euclidean distance between two points x and y is the length \\|x - y\\|_2 of the straight line between the two points.\nIn many situations, the Euclidean distance is appropriate for capturing the actual distances in a given space.\nIn contrast, consider taxi drivers in a grid street plan who should measure distance not in terms of the length of the straight line to their destination, but in terms of the [rectilinear distance](https://bliptext.com/articles/taxicab-geometry), which takes into account that streets are either orthogonal or parallel to each other.\nThe class of p-norms generalizes these two examples and has an abundance of applications in many parts of [mathematics](https://bliptext.com/articles/mathematics), [physics](https://bliptext.com/articles/physics), and [computer science](https://bliptext.com/articles/computer-science).\n# [Illustrations of unit circles (see also superellipse) in \\Reals^2 based on different p-norms (every vector from the origin to the unit circle has a length of one, the length being calculated with length-formula of the corresponding p). | upload.wikimedia.org/wikipedia/commons/d/d4/Vector-p-Norms///qtl1.svg]\n\n## Definition\nFor a [real number](https://bliptext.com/articles/real-number) p \\geq 1, the **p-norm** or L^**p-norm** of x is defined by\nThe absolute value bars can be dropped when p is a rational number with an even numerator in its reduced form, and x is drawn from the set of real numbers, or one of its subsets.\nThe Euclidean norm from above falls into this class and is the 2-norm, and the 1-norm is the norm that corresponds to the [rectilinear distance](https://bliptext.com/articles/taxicab-geometry).\nThe **L^\\infty-norm** or [maximum norm](https://bliptext.com/articles/chebyshev-distance) (or uniform norm) is the limit of the L^p-norms for It turns out that this limit is equivalent to the following definition:\nSee [[L-infinity|\nL\n-infinity]].\nFor all p \\geq 1, the p-[norm](https://bliptext.com/articles/norm-mathematics)s and maximum [norm](https://bliptext.com/articles/norm-mathematics) as defined above indeed satisfy the properties of a \"length function\" (or [norm](https://bliptext.com/articles/norm-mathematics)), which are that:\nAbstractly speaking, this means that \\Reals^n together with the p-norm is a [normed vector space](https://bliptext.com/articles/normed-vector-space).\nMoreover, it turns out that this space is complete, thus making it a [Banach space](https://bliptext.com/articles/banach-space).\nThis Banach space is the L^p-space over\n\n### Relations between \n\np\n\n-norms\nThe grid distance or rectilinear distance (sometimes called the \"[Manhattan distance](https://bliptext.com/articles/manhattan-distance)\") between two points is never shorter than the length of the line segment between them (the Euclidean or \"as the crow flies\" distance).\nFormally, this means that the Euclidean norm of any vector is bounded by its 1-norm:\nThis fact generalizes to p-norms in that the p-norm \\|x\\|_p of any given vector x does not grow with p:\nFor the opposite direction, the following relation between the 1-norm and the 2-norm is known:\nThis inequality depends on the dimension n of the underlying vector space and follows directly from the [Cauchy–Schwarz inequality](https://bliptext.com/articles/cauchy-schwarz-inequality).\nIn general, for vectors in \\Complex^n where 0 1, the formula\ndefines an absolutely [homogeneous function](https://bliptext.com/articles/homogeneous-function) for 0 1, as the series\ndiverges for p = 1 (the [harmonic series](https://bliptext.com/articles/harmonic-series-mathematics)), but is convergent for p > 1.\nOne also defines the \\infty-norm using the [supremum](https://bliptext.com/articles/supremum):\nand the corresponding space \\ell^\\infty of all bounded sequences.\nIt turns out that\nif the right-hand side is finite, or the left-hand side is infinite.\nThus, we will consider \\ell^p spaces for\nThe p-norm thus defined on \\ell^p is indeed a norm, and \\ell^p together with this norm is a [Banach space](https://bliptext.com/articles/banach-space).\nThe fully general L^p space is obtained—as seen below—by considering vectors, not only with finitely or countably-infinitely many components, but with \"arbitrarily many components\"; in other words, [functions](https://bliptext.com/articles/function-mathematics).\nAn [integral](https://bliptext.com/articles/integral) instead of a sum is used to define the p-norm.\n\n## General ℓp-space\nIn complete analogy to the preceding definition one can define the space \\ell^p(I) over a general [index set](https://bliptext.com/articles/index-set) I (and ) as\nwhere convergence on the right means that only countably many summands are nonzero (see also [Unconditional convergence](https://bliptext.com/articles/unconditional-convergence)).\nWith the norm\nthe space \\ell^p(I) becomes a Banach space.\nIn the case where I is finite with n elements, this construction yields \\Reals^n with the p-norm defined above.\nIf I is countably infinite, this is exactly the sequence space \\ell^p defined above.\nFor uncountable sets I this is a non-[separable](https://bliptext.com/articles/separable-space) Banach space which can be seen as the [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) [direct limit](https://bliptext.com/articles/direct-limit) of \\ell^p-sequence spaces.\nFor p = 2, the -norm is even induced by a canonical [inner product](https://bliptext.com/articles/inner-product) called the , which means that holds for all vectors \\mathbf{x}.\nThis inner product can expressed in terms of the norm by using the [polarization identity](https://bliptext.com/articles/polarization-identity).\nOn \\ell^2, it can be defined by\nwhile for the space L^2(X, \\mu) associated with a [measure space](https://bliptext.com/articles/measure-mathematics) which consists of all [square-integrable functions](https://bliptext.com/articles/square-integrable-function), it is\nNow consider the case p = \\infty.\nDefine\nwhere for all x\nThe index set I can be turned into a [measure space](https://bliptext.com/articles/measure-space) by giving it the [discrete σ-algebra](https://bliptext.com/articles/algebra) and the [counting measure](https://bliptext.com/articles/counting-measure).\nThen the space \\ell^p(I) is just a special case of the more general L^p-space (defined below).\n\n# Lp spaces and Lebesgue integrals\nAn L^p space may be defined as a space of measurable functions for which the p-th power of the [absolute value](https://bliptext.com/articles/absolute-value) is [Lebesgue integrable](https://bliptext.com/articles/lebesgue-integrable), where functions which agree almost everywhere are identified.\nMore generally, let be a [measure space](https://bliptext.com/articles/measure-space) and\nWhen, consider the set of all [measurable functions](https://bliptext.com/articles/measurable-function) f from S to \\Complex or \\Reals whose [absolute value](https://bliptext.com/articles/absolute-value) raised to the p-th power has a finite integral, or in symbols:\nTo define the set for p = \\infty, recall that two functions f and g defined on S are said to be, written , if the set is measurable and has measure zero.\nSimilarly, a measurable function f (and its [absolute value](https://bliptext.com/articles/absolute-value)) is (or ) by a real number C, written, if the (necessarily) measurable set has measure zero.\nThe space is the set of all measurable functions f that are bounded almost everywhere (by some real C) and is defined as the [infimum](https://bliptext.com/articles/infimum) of these bounds:\nWhen then this is the same as the [essential supremum](https://bliptext.com/articles/essential-supremum) of the absolute value of f:\nFor example, if f is a measurable function that is equal to 0 almost everywhere then \\|f\\|_p = 0 for every p and thus for all p.\nFor every positive p, the value under of a measurable function f and its absolute value are always the same (that is, for all p) and so a measurable function belongs to if and only if its absolute value does.\nBecause of this, many formulas involving p-norms are stated only for non-negative real-valued functions.\nConsider for example the identity which holds whenever f \\geq 0 is measurable, r > 0 is real, and (here when p = \\infty).\nThe non-negativity requirement f \\geq 0 can be removed by substituting |f| in for f, which gives\nNote in particular that when p = r is finite then the formula relates the p-norm to the 1-norm.\n**Seminormed space of p-th power integrable functions**\nEach set of functions forms a [vector space](https://bliptext.com/articles/vector-space) when addition and scalar multiplication are defined pointwise.\nThat the sum of two p-th power integrable functions f and g is again p-th power integrable follows from\nalthough it is also a consequence of [Minkowski's inequality](https://bliptext.com/articles/minkowski-inequality)\nwhich establishes that \\|\\cdot\\|_p satisfies the [triangle inequality](https://bliptext.com/articles/triangle-inequality) for (the [triangle inequality](https://bliptext.com/articles/triangle-inequality) does not hold for 0 0 ranges over the positive reals.\nThese balls satisfy for all real r > 0, which in particular shows that B_1 is a [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space) neighborhood of the origin; in other words, this space is locally [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space), just like every [normed space](https://bliptext.com/articles/normed-space), despite \\|\\cdot\\|_p not being a norm.\nIn this setting L^p satisfies a reverse Minkowski inequality, that is for\nThis result may be used to prove [Clarkson's inequalities](https://bliptext.com/articles/clarkson-s-inequalities), which are in turn used to establish the [uniform convexity](https://bliptext.com/articles/uniformly-convex-space) of the spaces L^p for.\nThe space L^p for 0 0 (for example, Such a metric is called [Lévy](https://bliptext.com/articles/paul-l-vy-mathematician)-metric for L^0. Under this metric the space L^0 is complete. However, as mentioned above, scalar multiplication is continuous with respect to this metric only if . To see this, consider the Lebesgue measurable function defined by f(x)=x. Then clearly . The space L^0 is in general not locally bounded, and not locally convex.\nFor the infinite Lebesgue measure \\lambda on \\Reals^n, the definition of the fundamental system of neighborhoods could be modified as follows\nThe resulting space, with the topology of local convergence in measure, is isomorphic to the space for any positive \\lambda–integrable density g.\n\n# Generalizations and extensions\n\n## Weak \n\nL{{sup|p}}\nLet be a measure space, and f a [measurable function](https://bliptext.com/articles/measurable-function) with real or complex values on S. The [distribution function](https://bliptext.com/articles/cumulative-distribution-function) of f is defined for t \\geq 0 by\nIf f is in L^p(S, \\mu) for some p with then by [Markov's inequality](https://bliptext.com/articles/markov-s-inequality),\nA function f is said to be in the space **weak L^p(S, \\mu)**, or if there is a constant C > 0 such that, for all t > 0,\nThe best constant C for this inequality is the L^{p,w}-norm of f, and is denoted by\nThe weak L^p coincide with the [Lorentz spaces](https://bliptext.com/articles/lorentz-space) so this notation is also used to denote them.\nThe L^{p,w}-norm is not a true norm, since the [triangle inequality](https://bliptext.com/articles/triangle-inequality) fails to hold.\nNevertheless, for f in\nand in particular\nIn fact, one has\nand raising to power 1/p and taking the supremum in t one has\nUnder the convention that two functions are equal if they are equal \\mu almost everywhere, then the spaces L^{p,w} are complete.\nFor any 0 1, this expression defines a norm if r = 1.\nHence for p > 1 the weak L^p spaces are [Banach spaces](https://bliptext.com/articles/banach-space).\nA major result that uses the L^{p,w}-spaces is the [Marcinkiewicz interpolation theorem](https://bliptext.com/articles/marcinkiewicz-interpolation), which has broad applications to [harmonic analysis](https://bliptext.com/articles/harmonic-analysis) and the study of [singular integrals](https://bliptext.com/articles/singular-integrals).\n\n## Weighted \n\nL{{sup|p}}\n\n spaces\nAs before, consider a [measure space](https://bliptext.com/articles/measure-space) Let be a measurable function.\nThe w-**weighted L^p space** is defined as where means the measure \\nu defined by\nor, in terms of the [Radon–Nikodym derivative](https://bliptext.com/articles/radon-nikodym-theorem), the [norm](https://bliptext.com/articles/norm-mathematics) for is explicitly\nAs L^p-spaces, the weighted spaces have nothing special, since is equal to But they are the natural framework for several results in harmonic analysis ; they appear for example in the [Muckenhoupt theorem](https://bliptext.com/articles/muckenhoupt-weights): for the classical [Hilbert transform](https://bliptext.com/articles/hilbert-transform) is defined on where \\mathbf{T} denotes the [unit circle](https://bliptext.com/articles/unit-circle) and \\lambda the Lebesgue measure; the (nonlinear) [Hardy–Littlewood maximal operator](https://bliptext.com/articles/hardy-littlewood-maximal-operator) is bounded on Muckenhoupt's theorem describes weights w such that the [Hilbert transform](https://bliptext.com/articles/hilbert-transform) remains bounded on and the maximal operator on\n\n## L{{sup|p}}\n\n spaces on manifolds\nOne may also define spaces L^p(M) on a manifold, called the **intrinsic L^p spaces** of the manifold, using [densities](https://bliptext.com/articles/density-on-a-manifold).\n\n## Vector-valued \n\nL{{sup|p}}\n\n spaces\nGiven a measure space and a [locally convex space](https://bliptext.com/articles/locally-convex-topological-vector-space) E (here assumed to be [complete](https://bliptext.com/articles/complete-topological-vector-space)), it is possible to define spaces of p-integrable E-valued functions on \\Omega in a number of ways.\nOne way is to define the spaces of [Bochner integrable](https://bliptext.com/articles/bochner-integral) and [Pettis integrable](https://bliptext.com/articles/pettis-integral) functions, and then endow them with [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) [TVS-topologies](https://bliptext.com/articles/vector-topology) that are (each in their own way) a natural generalization of the usual L^p topology.\nAnother way involves [topological [tensor product](https://bliptext.com/articles/tensor-product)s](https://bliptext.com/articles/topological-tensor-product) of with E. Element of the vector space are finite sums of simple tensors where each simple tensor f \\times e may be identified with the function that sends This [tensor product](https://bliptext.com/articles/tensor-product) is then endowed with a locally convex topology that turns it into a topological [tensor product](https://bliptext.com/articles/tensor-product), the most common of which are the projective [tensor product](https://bliptext.com/articles/tensor-product), denoted by and the injective [tensor product](https://bliptext.com/articles/tensor-product), denoted by In general, neither of these space are complete so their [completions](https://bliptext.com/articles/complete-topological-vector-space) are constructed, which are respectively denoted by and (this is analogous to how the space of scalar-valued [simple functions](https://bliptext.com/articles/simple-function) on \\Omega, when seminormed by any is not complete so a completion is constructed which, after being quotiented by is isometrically isomorphic to the Banach space ).\n[Alexander Grothendieck](https://bliptext.com/articles/alexander-grothendieck) showed that when E is a [nuclear space](https://bliptext.com/articles/nuclear-space) (a concept he introduced), then these two constructions are, respectively, canonically TVS-isomorphic with the spaces of Bochner and Pettis integral functions mentioned earlier; in short, they are indistinguishable.","revisionCount":0},"hasMore":false,"nextCursor":null}

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