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| {{DISPLAYTITLE:L<sup>''p''</sup> space}}
| | Friends call him Royal Seyler. The favorite hobby for him and his children is to drive and now he is trying to make cash with it. Kansas is our birth location and my mothers and fathers reside close by. Managing people is what I do in my working day job.<br><br>Here is my webpage: [http://Madavideos.com/SoCalderon-profil-78340 Madavideos.com] |
| In [[mathematics]], the '''L<sup>''p''</sup> spaces''' are [[function space]]s defined using a natural generalization of the ''p''-norm for finite-dimensional [[vector space]]s. They are sometimes called '''Lebesgue spaces''', named after [[Henri Lebesgue]] {{harv|Dunford|Schwartz|1958|loc=III.3}}, although according to the [[Nicolas Bourbaki|Bourbaki]] group {{harv|Bourbaki|1987}} they were first introduced by [[Frigyes Riesz]] {{harv|Riesz|1910}}.
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| '''L<sup>''p''</sup> spaces''' form an important class of [[Banach space]]s in [[functional analysis]], and of [[topological vector space]]s.
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| Lebesgue spaces have applications in physics, statistics, finance, engineering, and other disciplines.
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| ==The ''p''-norm in finite dimensions==
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| [[Image:Vector norms.svg|frame|right|Illustrations of [[unit circle]]s in 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'').]]
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| [[Image:Superellipse rounded diamond.svg|thumb|left|Unit circle ([[superellipse]]) in ''p'' = {{frac|3|2}} norm]]
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| The length of a vector ''x'' = (''x''<sub>1</sub>, ''x''<sub>2</sub>, …, ''x''<sub>''n''</sub>) in the ''n''-dimensional [[real number|real]] [[vector space]] '''R'''<sup>''n''</sup> is usually given by the [[Euclidean norm]]:
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| :<math>\ \|x\|_2=\left(x_1^2+x_2^2+\dotsb+x_n^2\right)^{\frac{1}{2}}</math>
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| The Euclidean distance between two points ''x'' and ''y'' is the length <math>\scriptstyle \|x \,-\, y\|_2</math> of the straight line between the two points. In many situations, the Euclidean distance is insufficient for capturing the actual distances in a given space. For example, taxi drivers in Manhattan should measure distance not in terms of the length of the straight line to their destination, but in terms of the [[Manhattan distance]], which takes into account that streets are either orthogonal or parallel to each other. The class of ''p''-norms generalizes these two examples and has an abundance of applications in many parts of [[mathematics]], [[physics]], and [[computer science]].
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| === Definition ===
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| For a [[real number]] ''p'' ≥ 1, the '''''p''-norm''' or '''''L''<sup>''p''</sup>-norm''' of ''x'' is defined by
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| :<math>\ \|x\|_p=\left(|x_1|^p+|x_2|^p+\dotsb+|x_n|^p\right)^{\frac{1}{p}}</math>
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| The Euclidean norm from above falls into this class and is the 2-norm, and the 1-norm is the norm that corresponds to the [[Manhattan distance]].
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| The '''''L''<sup>∞</sup>-norm''' or [[Chebyshev distance|maximum norm]] (or uniform norm) is the limit of the ''L''<sup>''p''</sup>-norms for <math>\scriptstyle p \,\to\, \infty</math>. It turns out that this limit is equivalent to the following definition:
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| :<math>\ \|x\|_\infty=\max \left\{|x_1|, |x_2|, \dotsc, |x_n|\right\}</math>
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| For all ''p'' ≥ 1, the p-norms and maximum norm as defined above indeed satisfy the properties of a "length function" (or [[norm (mathematics)|norm]]), which are that:
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| * only the zero vector has zero length,
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| * the length of the vector is positive homogeneous with respect to multiplication by a scalar, and
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| * the length of the sum of two vectors is no larger than the sum of lengths of the vectors ([[triangle inequality]]).
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| Abstractly speaking, this means that '''R'''<sup>''n''</sup> together with the ''p''-norm is a [[Banach space]]. This Banach space is the '''''L''<sup>''p''</sup>-space''' over '''R'''<sup>''n''</sup>.
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| ==== Relations between ''p''-norms ====
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| It is intuitively clear that the grid distance ("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).
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| Formally, this means that the Euclidean norm of any vector is bounded by its 1-norm:
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| :<math>\|x\|_2 \leq \|x\|_1</math>
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| This fact generalizes to ''p''-norms in that the ''p''-norm <math>\scriptstyle \|x\|_p</math> of any given vector ''x'' does not grow with ''p'':
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| :<math>\|x\|_{p+a} \leq \|x\|_{p}</math> for any vector ''x'' and real numbers ''p'' ≥ 1 and ''a'' ≥ 0. (In fact this remains true for 1>''p''>0 and ''a'' ≥ 0.)
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| For the opposite direction, the following relation between the 1-norm and the 2-norm is known:
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| :<math>\|x\|_1 \leq \sqrt{n}\|x\|_2</math>
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| This inequality depends on the dimension ''n'' of the underlying vector space and follows directly from the [[Cauchy–Schwarz inequality]].
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| In general, for vectors in <math>\mathbb{C}^n</math> where p > r > 0:
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| :<math>\|x\|_p\leq\|x\|_r\leq n^{\left(\frac{1}{r} - \frac{1}{p}\right)}\|x\|_p</math>
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| === When 0 < ''p'' < 1 ===
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| [[Image:Astroid.svg|thumb|right|[[Astroid]], unit circle in ''p'' = {{frac|2|3}} metric]]
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| In '''R'''<sup>''n''</sup> for ''n'' > 1, the formula
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| :<math>\ \|x\|_p=\left(|x_1|^p+|x_2|^p+\dotsb+|x_n|^p\right)^{\frac{1}{p}}</math>
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| defines an absolutely [[homogeneous function]] of degree 1 for 0 < ''p'' < 1; however, the resulting function does not define an [[F-norm]], because it is not [[Subadditivity|subadditive]]. In '''R'''<sup>''n''</sup> for ''n'' > 1, the formula for 0 < ''p'' < 1
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| :<math> |x_1|^p + |x_2|^p + \dotsb + |x_n|^p</math>
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| defines a subadditive function, which does define an F-norm. This F-norm is homogeneous of degree p.
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| However, the function
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| :<math>d_p(x,y) = \sum_{i=1}^n |x_i-y_i|^p</math>
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| defines a [[metric space|metric]]. The metric space ('''R'''<sup>''n''</sup>, ''d''<sub>''p''</sub>) is denoted by ℓ<sub>''n''</sub><sup>''p''</sup>.
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| Although the ''p''-unit ball ''B''<sub>''n''</sub><sup>''p''</sup> around the origin in this metric is "concave", the topology defined on '''R'''<sup>''n''</sup> by the metric ''d''<sub>''p''</sup> is the usual vector space topology of '''R'''<sup>''n''</sup>, hence ℓ<sub>''n''</sub><sup>''p''</sup> is a [[locally convex]] topological vector space. Beyond this qualitative statement, a quantitative way to measure the lack of convexity of ℓ<sub>''n''</sub><sup>''p''</sup> is to denote by ''C''<sub>''p''</sub>(''n'') the smallest constant ''C'' such that the multiple ''C'' ''B''<sub>''n''</sub><sup>''p''</sup> of the ''p''-unit ball contains the convex hull of ''B''<sub>''n''</sub><sup>''p''</sup>, equal to ''B''<sub>''n''</sub><sup>1</sup>. The fact that ''C''<sub>''p''</sub>(''n'') = ''n''<sup>1/''p'' – 1</sup> tends to infinity with ''n'' (for fixed ''p'' < 1) reflects the fact that the infinite-dimensional sequence space ℓ<sup>''p''</sup> defined below, is no longer locally convex.
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| ===When ''p'' = 0===
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| There is one l<sub>0</sub> norm and another function called the l<sub>0</sub> "norm" (with quotation marks).
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| The mathematical definition of the l<sub>0</sub> norm was established by [[Banach]]'s ''[[Theory of Linear Operations]]''. The [[F-space|space]] of sequences has a complete metric topology provided by the [[F-space|F-norm]] <math>\scriptstyle (x_n) \,\mapsto\, \sum_n{2^{-n} |x_n|/(1 \,+\, |x_n| )}</math>, which is discussed by Stefan Rolewicz in ''Metric Linear Spaces''.<ref name="RolewiczControl">{{Citation | title=Functional analysis and control theory: Linear systems|last=Rolewicz |first=Stefan|year=1987| isbn=90-277-2186-6| publisher=D. Reidel Publishing Co.; PWN—Polish Scientific Publishers|oclc=13064804|edition=Translated from the Polish by Ewa Bednarczuk|series=Mathematics and its Applications (East European Series)|location=Dordrecht; Warsaw|volume=29|pages=xvi+524| mr=920371}}</ref> The l<sub>0</sub>-normed space is studied in functional analysis, probability theory, and harmonic analysis.
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| Another function was called the l<sub>0</sub> "norm" by [[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 of terminology|abuse terminology]] by omitting the quotation marks. Defining 0<sup>0</sup> = 0, the zero "norm" of ''x'' is equal to <math>\scriptstyle |x_1|^0 \,+\, |x_2|^0 \,+\, \dotsb \,+\, |x_n|^0</math>. This is not a [[norm (mathematics)|norm]] (B-norm, with "B" for [[Banach]]) because it is not homogeneous. Despite these defects as a mathematical norm, the non-zero counting "norm" has uses in [[scientific computing]], [[information theory]], and [[statistics]] – notably in [[compressed sensing]] in [[signal processing]] and computational [[harmonic analysis]].
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| ==The ''p''-norm in countably infinite dimensions==
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| :{{Details|Sequence space}}
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| The ''p''-norm can be extended to vectors that have an infinite number of components, which yields the space <math>\scriptstyle \ell^p</math>. This contains as special cases:
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| * <math>\scriptstyle \ell^1</math>, the space of sequences whose series is [[Absolute convergence|absolutely convergent]],
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| * <math>\scriptstyle \ell^2</math>, the space of '''square-summable''' sequences, which is a [[Hilbert space]], and
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| * <math>\scriptstyle \ell^\infty</math>, the space of [[bounded sequence]]s.
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| The space of sequences has a natural vector space structure by applying addition and scalar multiplication coordinate by coordinate.
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| Explicitly, for <math>\scriptstyle \ x \;=\; (x_1,\, x_2,\, \dotsc,\, x_n,\, x_{n+1},\, \dotsc)</math> an infinite [[sequence]] of real (or [[complex number|complex]]) numbers, define the vector sum to be
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| :<math>\begin{align}
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| &(x_1, x_2, \dotsc, x_n, x_{n+1},\dotsc)+(y_1, y_2, \dotsc, y_n, y_{n+1},\dotsc) =\\
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| &(x_1+y_1, x_2+y_2, \dotsc, x_n+y_n, x_{n+1}+y_{n+1},\dotsc)
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| \end{align}</math>
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| while the scalar action is given by
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| :<math>\lambda(x_1, x_2, \dotsc, x_n, x_{n+1},\dotsc) = (\lambda x_1, \lambda x_2, \dotsc, \lambda x_n, \lambda x_{n+1},\dotsc)</math>
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| Define the ''p''-norm
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| :<math>\|x\|_p = \left(|x_1|^p + |x_2|^p + \dotsb+|x_n|^p + |x_{n+1}|^p + \dotsb\right)^{\frac{1}{p}}</math>
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| Here, a complication arises, namely that the [[series (mathematics)|series]] on the right is not always convergent, so for example, the sequence made up of only ones, (1, 1, 1, …), will have an infinite ''p''-norm (length) for every finite ''p'' ≥ 1. The space ℓ<sup>''p''</sup> is then defined as the set of all infinite sequences of real (or complex) numbers such that the ''p''-norm is finite.
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| One can check that as ''p'' increases, the set ℓ<sup>''p''</sup> grows larger. For example, the sequence
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| :<math>\left(1, \frac{1}{2}, \dotsc, \frac{1}{n}, \frac{1}{n+1},\dotsc\right)</math>
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| is not in ℓ<sup>1</sup>, but it is in ℓ<sup>''p''</sup> for ''p'' > 1, as the series
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| :<math>1^p + \frac{1}{2^p} + \dotsb + \frac{1}{n^p} + \frac{1}{(n+1)^p}+\dotsb</math>
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| diverges for ''p'' = 1 (the [[harmonic series (mathematics)|harmonic series]]), but is convergent for ''p'' > 1.
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| One also defines the ∞-norm using the [[supremum]]:
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| :<math>\ \|x\|_\infty=\sup(|x_1|, |x_2|, \dotsc, |x_n|,|x_{n+1}|, \dotsc)</math>
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| and the corresponding space ℓ<sup>∞</sup> of all bounded sequences. It turns out that<ref>{{Citation | last1=Maddox | first1=I.J. | author1-link=I.J. Maddox | title=Elements of Functional Analysis | publisher=CUP | location=Cambridge | edition=2nd | year=1988}}, page 16</ref>
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| :<math>\ \|x\|_\infty = \lim_{p\to\infty}\|x\|_p</math>
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| if the right-hand side is finite, or the left-hand side is infinite. Thus, we will consider ℓ<sup>''p''</sup> spaces for 1 ≤ ''p'' ≤ ∞.
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| The ''p''-norm thus defined on ℓ<sup>''p''</sup> is indeed a norm, and ℓ<sup>''p''</sup> together with this norm is a [[Banach space]]. The fully general ''L''<sup>''p''</sup> 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, [[function (mathematics)|functions]]. An [[integral]] instead of a sum is used to define the ''p''-norm.
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| == ''L<sup>p</sup>'' spaces ==
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| Let 1 ≤ ''p'' < ∞ and (''S'', ''Σ'', ''μ'') be a [[measure space]]. Consider the set of all [[measurable function]]s from ''S'' to '''C''' (or '''R''') whose [[absolute value]] raised to the ''p''-th power has finite integral, or equivalently, that
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| :<math>\|f\|_p \equiv \left({\int_S |f|^p\;\mathrm{d}\mu}\right)^{\frac{1}{p}}<\infty</math>
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| The set of such functions forms a vector space, with the following natural operations:
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| :<math>(f+g)(x) = f(x)+g(x), \ \ \ \text{and} \ \ \ (\lambda f)(x) = \lambda f(x) \,</math>
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| for every scalar ''λ''.
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| That the sum of two ''p''<sup>th</sup> power integrable functions is again ''p''<sup>th</sup> power integrable follows from the inequality |''f'' + ''g''|<sup>''p''</sup> ≤ 2<sup>''p-1''</sup> (|''f''|<sup>''p''</sup> + |''g''|<sup>''p''</sup>). In fact, more is true. [[Minkowski inequality|Minkowski's inequality]] says the [[triangle inequality]] holds for || · ||<sub>''p''</sub>. Thus the set of ''p''<sup>th</sup> power integrable functions, together with the function || · ||<sub>''p''</sub>, is a [[seminorm]]ed vector space, which is denoted by <math>\scriptstyle \mathcal{L}^p(S,\, \mu)</math>.
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| {{anchor|kernel}}
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| This can be made into a normed vector space in a standard way; one simply takes the [[Quotient Space|quotient space]] with respect to the [[kernel (set theory)|kernel]] of || · ||<sub>''p''</sub>. Since for any measurable function ''f'', we have that ||''f''||<sub>''p''</sub> = 0 if and only if ''f'' = 0 [[almost everywhere]], the kernel of || · ||<sub>''p''</sub> does not depend upon ''p'',
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| :<math>N \equiv \mathrm{ker}(\|\cdot\|_p) = \{f : f = 0 \ \mu\text{-almost everywhere} \}</math>
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| In the quotient space, two functions ''f'' and ''g'' are identified if ''f'' = ''g'' almost everywhere. The resulting normed vector space is, by definition,
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| :<math>L^p(S, \mu) \equiv \mathcal{L}^p(S, \mu) / N</math>
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| For ''p'' = ∞, the space ''L''<sup>∞</sup>(''S'', ''μ'') is defined as follows. We start with the set of all measurable functions from ''S'' to '''C''' (or '''R''') which are '''essentially bounded''', i.e. bounded up to a set of measure zero. Again two such functions are identified if they are equal almost everywhere. Denote this set by ''L''<sup>∞</sup>(''S'', ''μ''). For ''f'' in ''L''<sup>∞</sup>(''S'', ''μ''), its [[essential supremum]] serves as an appropriate norm:
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| :<math>\|f\|_\infty \equiv \inf \{ C\ge 0 : |f(x)| \le C \mbox{ for almost every } x\}.</math>
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| As before, we have
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| :<math>\|f\|_\infty=\lim_{p\to\infty}\|f\|_p</math>
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| if ''f'' ∈ ''L''<sup>∞</sup>(''S'', ''μ'') ∩ ''L''<sup>''q''</sup>(''S'', ''μ'') for some ''q'' < ∞.
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| For 1 ≤ ''p'' ≤ ∞, ''L''<sup>''p''</sup>(''S'', ''μ'') is a [[Banach space]]. The fact that ''L''<sup>''p''</sup> is ''complete'' is often referred to as the ''[[Riesz-Fischer theorem]]''. Completeness can be checked using the convergence theorems for Lebesgue integrals.
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| When the underlying measure space ''S'' is understood, ''L''<sup>''p''</sup>(''S'', ''μ'') is often abbreviated ''L''<sup>''p''</sup>(''μ''), or just ''L''<sup>''p''</sup>. The above definitions generalize to [[Bochner space]]s.
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| === Special cases ===
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| When ''p'' = 2; like the ℓ<sup>2</sup> space, the space ''L''<sup>2</sup> is the only [[Hilbert space]] of this class. In the complex case, the inner product on ''L''<sup>2</sup> is defined by
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| :<math> \langle f, g \rangle = \int_S f(x) \overline{g(x)} \, \mathrm{d}\mu(x)</math>
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| The additional inner product structure allows for a richer theory, with applications to, for instance, [[Fourier series]] and [[quantum mechanics]]. Functions in ''L''<sup>2</sup> are sometimes called '''[[quadratically integrable function]]s''', '''square-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]] {{harv|Titchmarsh|1976}}.
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| If we use complex-valued functions, the space ''L''<sup>∞</sup> is a [[commutative]] [[C*-algebra]] with pointwise multiplication and conjugation. For many measure spaces, including all sigma-finite ones, it is in fact a commutative [[von Neumann algebra]]. An element of ''L''<sup>∞</sup> defines a [[bounded operator]] on any ''L''<sup>''p''</sup> space by [[multiplication operator|multiplication]].
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| The ℓ<sup>''p''</sup> spaces (1 ≤ ''p'' ≤ ∞) are a special case of ''L<sup>p</sup>'' spaces, when ''S'' is the set '''N''' of positive [[integer]]s, and the measure ''μ'' is the [[counting measure]] on '''N'''. More generally, if one considers any set ''S'' with the counting measure, the resulting ''L<sup> p</sup>'' space is denoted ℓ<sup>''p''</sup>(''S''). For example, the space ℓ<sup>''p''</sup>('''Z''') 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. The space ℓ<sup>''p''</sup>(''n''), where ''n'' is the set with ''n'' elements, is '''R'''<sup>''n''</sup> with its ''p''-norm as defined above. As any Hilbert space, every space ''L''<sup>2</sup> is linearly isometric to a suitable ℓ<sup>2</sup>(''I''), where the cardinality of the set ''I'' is the cardinality of an arbitrary Hilbertian basis for this particular ''L''<sup>2</sup>.
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| ==Properties of ''L''<sup>''p''</sup> spaces==
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| ===Dual spaces===
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| The [[Dual_space#Continuous_dual_space|dual space]] (the space of all continuous linear functionals) of ''L''<sup>''p''</sup>(''μ'') for 1 < ''p'' < ∞ has a natural isomorphism with ''L''<sup>''q''</sup>(''μ''), where ''q''  is such that 1/''p'' + 1/''q'' = 1, which associates ''g'' ∈ ''L''<sup>''q''</sup>(''μ'') with the functional ''κ''<sub>''p''</sub>(''g'') ∈ ''L''<sup>''p''</sup>(''μ'')<sup>∗</sup> defined by
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| :<math>\kappa_p(g) \colon f \in L^p(\mu) \mapsto \int f g \, \mathrm{d}\mu</math>
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| The fact that ''κ''<sub>''p''</sub>(''g'') is well defined and continuous follows from [[Hölder's inequality]]. The mapping ''κ''<sub>''p''</sub> is a linear mapping from ''L''<sup>''q''</sup>(''μ'') into ''L''<sup>''p''</sup>(''μ'')<sup>∗</sup>, which is an [[isometry]] by the [[Hölder's inequality#Extremal equality|extremal case]] of Hölder's inequality. It is also possible to show (for example with the [[Radon–Nikodym theorem]], see<ref>{{Citation | last1=Rudin | first1=Walter | author1-link=Walter Rudin | title=Real and Complex Analysis | publisher=Tata McGraw-Hill | location=New Delhi | edition=2nd | year=1980 |isbn=9780070542341}}, Theorem 6.16</ref>) that any ''G'' ∈ ''L''<sup>''p''</sup>(''μ'')<sup>∗</sup> can be expressed this way: i.e., that ''κ''<sub>''p''</sub> is ''onto''. Since ''κ''<sub>''p''</sub> is onto and isometric, it is an [[isomorphism]] of [[Banach space]]s. With this (isometric) isomorphism in mind, it is usual to say simply that ''L''<sup>''q''</sup> "''is''" the dual of ''L''<sup>''p''</sup>.
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| When 1 < ''p'' < ∞, the space ''L''<sup>''p''</sup>(''μ'') is [[reflexive space|reflexive]]. Let ''κ''<sub>''p''</sub> be the above map and let ''κ''<sub>''q''</sub> be the corresponding linear isometry from ''L''<sup>''p''</sup>(''μ'') onto ''L''<sup>''q''</sup>(''μ'')<sup>∗</sup>. The map
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| :<math>j_p \colon L^p(\mu) \overset{\kappa_q}{\to} L^q(\mu)^* \overset{\,\,\left(\kappa_p^{-1}\right)^*}{\longrightarrow} L^p(\mu)^{**}</math>
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| from ''L''<sup>''p''</sup>(''μ'') to ''L''<sup>''p''</sup>(''μ'')<sup>∗∗</sup>, obtained by composing ''κ''<sub>''q''</sub> with the [[Dual space#Transpose of a continuous linear map|transpose]] (or adjoint) of the inverse of ''κ''<sub>''p''</sub>, coincides with the [[Reflexive space#Definitions|canonical embedding]] ''J''  of ''L''<sup>''p''</sup>(''μ'') into its bidual. Moreover, the map ''j''<sub>''p''</sub> is onto, as composition of two onto isometries, and this proves reflexivity.
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| If the measure ''μ'' on ''S'' is [[sigma-finite]], then the dual of ''L''<sup>1</sup>(''μ'') is isometrically isomorphic to ''L''<sup>∞</sup>(''μ'') (more precisely, the map ''κ''<sub>1</sub> corresponding to ''p'' = 1 is an isometry from ''L''<sup>∞</sup>(''μ'') onto ''L''<sup>1</sup>(''μ'')<sup>∗</sup>).
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| The dual of ''L''<sup>∞</sup> is subtler. Elements of (''L''<sup>∞</sup>(''μ''))<sup>∗</sup> can be identified with bounded signed ''finitely'' additive measures on ''S'' that are [[absolutely continuous]] with respect to ''μ''. See [[ba space]] for more details. If we assume the axiom of choice, this space is much bigger than ''L''<sup>1</sup>(''μ'') except in some trivial cases. However, [[Saharon Shelah]] proved that there are relatively consistent extensions of [[Zermelo-Fraenkel set theory]] (ZF + [[Axiom of dependent choice|DC]] + "Every subset of the real numbers has the [[Baire property]]") in which the dual of ''ℓ''<sup>∞</sup> is ''ℓ''<sup>1</sup>. <ref>{{Citation | title=Handbook of Analysis and its Foundations|last=Schechter |first=Eric|year=1997| publisher=Academic Press Inc.|location=London}} See Sections 14.77 and 27.44--47</ref>
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| ===Embeddings===
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| Colloquially, if 1 ≤ ''p'' < ''q'' ≤ ∞, ''L<sup>p</sup>''(''S'', ''μ'') contains functions that are more locally singular, while elements of ''L<sup>q</sup>''(''S'', ''μ'') can be more spread out. Consider the Lebesgue measure on the half line (0, ∞). A continuous function in ''L''<sup>1</sup> might blow up near 0 but must decay sufficiently fast toward infinity. On the other hand, continuous functions in ''L''<sup>∞</sup> need not decay at all but no blow-up is allowed. The precise technical result is the following:
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| #Let 0 ≤ ''p'' < ''q'' ≤ ∞. ''L<sup>q</sup>''(''S'', ''μ'') is contained in ''L<sup>p</sup>''(''S'', μ) iff ''S'' does not contain sets of arbitrarily large measure, and
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| #Let 0 ≤ ''p'' < ''q'' ≤ ∞. ''L<sup>p</sup>''(''S'', ''μ'') is contained in ''L<sup>q</sup>''(''S'', ''μ'') iff ''S'' does not contain sets of arbitrarily small non-zero measure.
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| In particular, if the domain ''S'' has finite measure, the bound (a consequence of [[Jensen's inequality]])
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| :<math>\ \|f\|_p \le \mu(S)^{\frac{1}{p} - \frac{1}{q}} \|f\|_q </math>
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| means the space ''L''<sup>''q''</sup> is continuously embedded in ''L''<sup>''p''</sup>. That is to say, the identity operator is a bounded linear map from ''L''<sup>''q''</sup> to ''L''<sup>''p''</sup>. The constant appearing in the above inequality is optimal, in the sense that the [[operator norm]] of the identity ''I'' : ''L<sup>q</sup>''(''S'', ''μ'') → ''L<sup>p</sup>''(''S'', ''μ'') is precisely
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| :<math>\|I\|_{q,p} = \mu(S)^{\frac{1}{p} - \frac{1}{q}}</math>
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| the case of equality being achieved exactly when ''f'' = 1 a.e.[μ].
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| === Dense subspaces ===
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| It is assumed that 1 ≤ ''p'' < ∞ throughout this section.<br />
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| Let (''S'', ''Σ'', ''μ'') be a measure space. An ''integrable simple function'' ''f''  on ''S''  is one of the form
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| :<math>f = \sum_{j=1}^n a_j \mathbf{1}_{A_j}</math>
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| where ''a<sub>j</sub>'' is scalar and ''A<sub>j</sub>'' ∈ ''Σ''  has finite measure, for ''j'' = 1, …, ''n''. By construction of the [[Lebesgue integration|integral]], the vector space of integrable simple functions is dense in ''L''<sup>''p''</sup>(''S'', ''Σ'', ''μ'').
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| More can be said when ''S''  is a [[Metrization theorem|metrizable]] [[topological space]] and ''Σ''  its [[Borel algebra|Borel ''σ''–algebra]], ''i.e.'', the smallest ''σ''–algebra of subsets of ''S''  containing the [[open set]]s.
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| Suppose that ''V'' ⊂ ''S''  is an open set with ''μ''(''V'') < ∞. It can be proved that for every Borel set ''A'' ∈ ''Σ''  contained in ''V'', and for every ''ε'' > 0, there exist a closed set ''F''  and an open set ''U''  such that
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| :<math>F \subset A \subset U \subset V \ \ \text{and} \ \ \mu(U) - \mu(F) = \mu(U \setminus F) < \varepsilon</math>
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| It follows that there exists ''φ'' continuous on ''S''  such that
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| :<math>0 \le \varphi \le \mathbf{1}_V \ \text{and} \ \int_S |\mathbf{1}_A - \varphi| \, \mathrm{d}\mu < \varepsilon</math>
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| If ''S''  can be covered by an increasing sequence (''V<sub>n</sub>'') of open sets that have finite measure, then the space of ''p''–integrable continuous functions is dense in ''L''<sup>''p''</sup>(''S'', ''Σ'', ''μ''). More precisely, one can use bounded continuous functions that vanish outside one of the open sets ''V<sub>n</sub>''.
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| This applies in particular when ''S'' = '''R'''<sup>''d''</sup> and when ''μ'' is the Lebesgue measure. The space of continuous and compactly supported functions is dense in ''L''<sup>''p''</sup>('''R'''<sup>''d''</sup>). Similarly, the space of integrable ''step functions''  is dense in ''L''<sup>''p''</sup>('''R'''<sup>''d''</sup>); 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.<br />
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| Several properties of general functions in ''L''<sup>''p''</sup>('''R'''<sup>''d''</sup>) are first proved for continuous and compactly supported functions (sometimes for step functions), then extended by density to all functions. For example, it is proved this way that translations are continuous on ''L''<sup>''p''</sup>('''R'''<sup>''d''</sup>), in the following sense: for every ''f'' ∈ ''L''<sup>''p''</sup>('''R'''<sup>''d''</sup>),
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| :<math>\|\tau_t f - f \|_p \rightarrow 0</math>
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| when ''t'' ∈ '''R'''<sup>''d''</sup> tends to 0, where <math>\scriptstyle \tau_t f</math> is the translated function defined by <math>\scriptstyle (\tau_t f)(x) \;=\; f(x \,-\, t)</math>.
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| ==Applications==
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| ''L<sup>p</sup>'' spaces are widely used in mathematics and applications.
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| ===Hausdorff–Young inequality===
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| The [[Fourier transform]] for the real line (resp. for periodic functions, cf. [[Fourier series]]) maps ''L<sup>p</sup>''('''R''') to ''L<sup>q</sup>''('''R''') (resp. ''L<sup>p</sup>''('''T''') to ℓ<sup>''q''</sup>), where 1 ≤ ''p'' ≤ 2 and 1/''p'' + 1/''q'' = 1. This is a consequence of the [[Riesz-Thorin theorem|Riesz-Thorin interpolation theorem]], and is made precise with the [[Hausdorff–Young inequality]].
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| By contrast, if ''p'' > 2, the Fourier transform does not map into ''L<sup>q</sup>''.
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| ===Hilbert spaces===
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| [[Hilbert space]]s are central to many applications, from [[quantum mechanics]] to [[stochastic calculus]]. The spaces ''L''<sup>2</sup> and ℓ<sup>2</sup> are both Hilbert spaces. In fact, by choosing a Hilbert basis, one sees that all Hilbert spaces are isometric to ℓ<sup>2</sup>(''E''), where ''E'' is a set with an appropriate cardinality.
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| ===Statistics===
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| In [[statistics]], measures of [[central tendency]] and [[statistical dispersion]], such as the [[mean]], [[median]], and [[standard deviation]], are defined in terms of ''L''<sup>''p''</sup> metrics, and measures of central tendency can be characterized as [[Average#Solutions to variational problems|solutions to variational problems]].
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| == ''L''<sup>''p''</sup> for 0 < ''p'' < 1 ==
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| Let (''S'', ''Σ'', ''μ'') be a measure space. If 0 < ''p'' < 1, then ''L<sup>p</sup>''(''μ'') can be defined as above: it is the vector space of those measurable functions ''f'' such that
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| :<math>N_p(f) = \int_S |f|^p\, d\mu < \infty</math>.
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| As before, we may introduce the ''p''-norm || ''f'' ||<sub>''p''</sub> = ''N''<sub>''p''</sup>(''f'')<sup>1/''p''</sup>,
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| but || · ||<sub>''p''</sub> does not satisfy the triangle inequality in this case, and defines only a [[quasi-norm]].
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| The inequality (''a'' + ''b'')<sup>''p''</sup> ≤ ''a''<sup>''p''</sup> + ''b''<sup>''p''</sup>, valid for ''a'' ≥ 0 and ''b'' ≥ 0 implies that {{harv|Rudin|1991|loc=§1.47}}
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| :<math>N_p(f+g)\le N_p(f) + N_p(g)</math>
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| and so the function
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| :<math>d_p(f,g) = N_p(f-g) = \|f - g\|_p^p</math>
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| is a metric on ''L''<sup>''p''</sup>(''μ''). The resulting metric space is [[complete space|complete]]; the verification is similar to the familiar case when ''p'' ≥ 1.
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| In this setting ''L''<sup>''p''</sup> satisfies a ''reverse Minkowski inequality'', that is for ''u'' and ''v'' in ''L<sup>p</sup>''
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| :<math>\|\,|u|+|v|\,\|_p\geq \|u\|_p+\|v\|_p</math> | |
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| This result may be used to prove Clarkson's inequalities, which are in turn used to establish the [[Uniformly convex space|uniform convexity]] of the spaces ''L''<sup>''p''</sup>
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| for 1 < ''p'' < ∞ {{harv|Adams|Fournier|2003}}.
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| The space ''L''<sup>''p''</sup> for 0 < ''p'' < 1 is an [[F-space]]: it admits a complete translation-invariant metric with respect to which the vector space operations are continuous. It is also [[locally bounded]], much like the case ''p'' ≥ 1. It is the prototypical example of an [[F-space]] that, for most reasonable measure spaces, is not [[locally convex]]: in ℓ<sup>''p''</sup> or
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| ''L''<sup>''p''</sup>([0, 1]), 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. Specifically, this is true if the measure space ''S'' contains an infinite family of disjoint measurable sets of finite positive measure.
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| The only nonempty convex open set in ''L''<sup>''p''</sup>([0, 1]) is the entire space {{harv|Rudin|1991|loc=§1.47}}. As a particular consequence, there are no nonzero linear functionals on ''L''<sup>''p''</sup>([0, 1]): the dual space is the zero space. In the case of the [[counting measure]] on the natural numbers (producing the sequence space ''L''<sup>''p''</sup>(''μ'') = ℓ<sup>''p''</sup>), the bounded linear functionals on ℓ<sup>''p''</sup> are exactly those that are bounded on ℓ<sup>1</sup>, namely those given by sequences in ℓ<sup>∞</sup>. Although ℓ<sup>''p''</sup> does contain non-trivial convex open sets, it fails to have enough of them to give a base for the topology.
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| The situation of having no linear functionals is highly undesirable for the purposes of doing analysis. In the case of the Lebesgue measure on '''R'''<sup>''n''</sup>, rather than work with ''L''<sup>''p''</sup> for 0 < ''p'' < 1, it is common to work with the [[Hardy space]] ''H''<sup>''p''</sup> whenever possible, as this has quite a few linear functionals: enough to distinguish points from one another. However, the [[Hahn–Banach theorem]] still fails in ''H''<sup>''p''</sup> for ''p'' < 1 {{harv|Duren|1970|loc=§7.5}}.
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| === ''L''<sup>0</sup>, the space of measurable functions ===
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| The vector space of (equivalence classes of) measurable functions on (''S'', ''Σ'', ''μ'') is denoted ''L''<sup>0</sup>(''S'', ''Σ'', ''μ'') {{harv|Kalton|Peck|Roberts|1984}}. By definition, it contains all the ''L''<sup>''p''</sup>, and is equipped with the topology of [[Convergence in measure|''convergence in measure'']]. When ''μ'' is a probability measure (i.e., ''μ''(''S'') = 1), this mode of convergence is named [[Convergence in probability|''convergence in probability'']].
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| The description is easier when ''μ'' is finite.
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| If ''μ'' is a finite measure on (''S'', ''Σ''), the 0 function admits for the convergence in measure the following fundamental system of neighborhoods
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| :<math>V_\varepsilon = \Bigl\{ f : \mu \bigl(\{x : |f(x)| > \varepsilon \} \bigr) < \varepsilon \Bigr\}, \ \ \varepsilon > 0</math>
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| The topology can be defined by any metric ''d''  of the form
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| :<math>d(f, g) = \int_S \varphi \bigl( |f(x) - g(x)| \bigr) \, \mathrm{d}\mu(x)</math>
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| where ''φ''  is bounded continuous concave and non-decreasing on [0, ∞), with ''φ''(0) = 0 and ''φ''(''t'') > 0 when ''t'' > 0 (for example, ''φ''(''t'') = min(''t'', 1)). Such a metric is called ''[[Paul Lévy (mathematician)|Lévy]]-metric for L<sup>0</sup>.'' Under this metric the space ''L''<sup>0</sup> is complete (it is again an F-space). The space ''L''<sup>0</sup> is in general not locally bounded, and not locally convex.
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| For the infinite Lebesgue measure ''λ'' on '''R'''<sup>''n''</sup>, the definition of the fundamental system of neighborhoods could be modified as follows
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| :<math>W_\varepsilon = \left\{ f : \lambda \left(\left\{ x : |f(x)| > \varepsilon \ \text{and} \ |x| < \frac{1}{\varepsilon}\right\} \right) < \varepsilon \right\}</math>
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| The resulting space ''L''<sup>0</sup>('''R'''<sup>''n''</sup>, ''λ'') coincides as topological vector space with ''L''<sup>0</sup>('''R'''<sup>''n''</sup>, ''g''(''x'') d''λ''(x)), for any positive ''λ''–integrable density ''g''.
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| ==Weak ''L<sup>p</sup>''==
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| Let (''S'', ''Σ'', ''μ'') be a measure space, and ''f'' a [[measurable function]] with real or complex values on ''S''. The [[cumulative distribution function|distribution function]] of ''f'' is defined for ''t'' > 0 by
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| :<math>\lambda_f(t) = \mu\left\{x\in S: |f(x)| > t\right\}</math>
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| If ''f'' is in ''L''<sup>''p''</sup>(''S'', ''μ'') for some ''p'' with 1 ≤ ''p'' < ∞, then by [[Markov's inequality]],
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| :<math>\lambda_f(t)\le \frac{\|f\|_p^p}{t^p}</math>
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| A function ''f'' is said to be in the space '''weak ''L<sup>p</sup>''(''S'', ''μ'')''', or ''L<sup>p,w</sup>''(''S'', ''μ''), if there is a constant ''C'' > 0 such that, for all ''t'' > 0,
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| :<math>\lambda_f(t) \le \frac{C^p}{t^p}</math>
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| The best constant ''C'' for this inequality is the ''L<sup>p,w</sup>''-norm of ''f'', and is denoted by
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| :<math>\|f\|_{p,w} = \sup_{t > 0} ~ t \lambda_f^{\frac{1}{p}}(t)</math>
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| The weak ''L''<sup>''p''</sup> coincide with the [[Lorentz space]]s ''L''<sup>''p'',∞</sup>, so this notation is also used to denote them.
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| The ''L<sup>p,w</sup>''-norm is not a true norm, since the [[triangle inequality]] fails to hold. Nevertheless, for ''f'' in ''L''<sup>p</sup>(''S'', ''μ''),
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| :<math>\|f\|_{p,w}\le \|f\|_p</math>
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| and in particular ''L<sup>p</sup>''(''S'', ''μ'') ⊂ ''L<sup>p,w</sup>''(''S'', ''μ''). Under the convention that two functions are equal if they are equal ''μ'' almost everywhere, then the spaces ''L''<sup>p,w</sup> are complete {{harv|Grafakos|2004}}.
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| For any 0 < ''r'' < ''p'' the expression
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| :<math>||| f |||_{L^{p,\infty}}=\sup_{0<\mu(E)<\infty} \mu(E)^{-\frac{1}{r}+\frac{1}{p}}\left(\int_E |f|^r\,d\mu\right)^{\frac{1}{r}}</math>
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| is comparable to the ''L<sup>p,w</sup>''-norm. Further in the case ''p'' > 1, this expression defines a norm if ''r'' = 1. Hence for ''p'' > 1 the weak ''L''<sup>''p''</sup> spaces are [[Banach space]]s {{harv|Grafakos|2004}}.
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| A major result that uses the ''L<sup>p,w</sup>''-spaces is the [[Marcinkiewicz interpolation|Marcinkiewicz interpolation theorem]], which has broad applications to [[harmonic analysis]] and the study of [[singular integrals]].
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| ==Weighted ''L<sup>p</sup>'' spaces==
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| As before, consider a [[measure space]] (''S'', ''Σ'', ''μ''). Let <math>\scriptstyle w :\; S \,\to\, [0,\, + \infty)</math> be a measurable function. The ''w''-'''weighted ''L<sup>p</sup>'' space''' is defined as ''L<sup>p</sup>''(''S'', ''w'' d''μ''), where ''w'' d''μ'' means the measure ''ν'' defined by
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| :<math>\ \nu (A) \equiv \int_{A} w(x) \, \mathrm{d} \mu (x), \ \ \ A \in \Sigma</math>
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| or, in terms of the [[Radon–Nikodym theorem|Radon–Nikodym derivative]],
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| :<math>\ w = \frac{\mathrm{d} \nu}{\mathrm{d} \mu}</math>
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| The [[norm (mathematics)|norm]] for ''L<sup>p</sup>''(''S'', ''w'' d''μ'') is explicitly
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| :<math>\ \| u \|_{L^{p} (S, w \, \mathrm{d} \mu)} \equiv \left( \int_{S} w(x) | u(x) |^{p} \, \mathrm{d} \mu (x) \right)^{\frac{1}{p}}</math>
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| As ''L''<sup>''p''</sub>-spaces, the weighted spaces have nothing special, since ''L<sup>p</sup>''(''S'', ''w'' d''μ'') is equal to ''L''<sup>''p''</sup>(''S'', d''ν''). But they are the natural framework for several results in harmonic analysis {{harv|Grafakos|2004}}<!--Please check this reference. Appears in Grafakos "Modern Fourier analysis", Chapter 9.-->; they appear for example in the [[Muckenhoupt weights|Muckenhoupt theorem]]: for 1 < ''p'' < ∞, the classical [[Hilbert transform]] is defined on ''L''<sup>''p''</sub>('''T''', ''λ'') where '''T''' denotes the unit circle and ''λ'' the Lebesgue measure; the (nonlinear) [[Hardy–Littlewood maximal operator]] is bounded on ''L''<sup>''p''</sub>('''R'''<sup>''n''</sup>, ''λ''). Muckenhoupt's theorem describes weights ''w'' such that the Hilbert transform remains bounded on ''L<sup>p</sup>''('''T''', ''w'' d''λ'') and the maximal operator on ''L<sup>p</sup>''('''R'''<sup>''n''</sup>, ''w'' d''λ'').
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| ==''L<sup>p</sup>'' spaces on manifolds==
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| One may also define spaces <math>\scriptstyle L^p(M)</math> on a manifold, called the '''intrinsic ''L<sup>p</sup>'' spaces''' of the manifold, using [[Density on a manifold|densities]].
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| ==See also==
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| * [[Birnbaum–Orlicz space]]
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| * [[Hardy space]]
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| * [[Riesz–Thorin theorem]]
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| * [[Hölder mean]]
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| * [[Hölder space]]
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| * [[Root mean square]]
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| * [[Locally integrable function]] <math>\left(\scriptstyle L^1_{\text{loc}}\right)</math>
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| * [[Pontryagin duality#Haar measure|<math>\scriptstyle L^p(G)</math> spaces over a locally compact group <math>G</math>]]
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| * [[Minkowski distance]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * {{citation|last1=Adams|first1=Robert A.|last2=Fournier|first2=John F.|title=Sobolev Spaces|edition=Second|publisher=Academic Press|year=2003|isbn=978-0-12-044143-3}}.
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| * {{citation|first=Nicolas|last=Bourbaki|authorlink=Nicolas Bourbaki|title=Topological vector spaces|series=Elements of mathematics|publisher= Springer-Verlag|publication-place=Berlin|year=1987|isbn=978-3-540-13627-9}}.
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| * {{citation | last=DiBenedetto|first=Emmanuele|title=Real analysis|publisher=Birkhäuser|year=2002|isbn=3-7643-4231-5}}.
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| * {{citation|last1=Dunford|first1=Nelson|last2=Schwartz|first2=Jacob T.|title=Linear operators, volume I|publisher=Wiley-Interscience|year=1958}}.
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| *{{citation
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| |last= Duren|first=P.|title=Theory of H<sup>p</sup>-Spaces|year=1970|publisher= Academic Press|publication-place= New York}}
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| * {{citation|title=Classical and Modern Fourier Analysis | last=Grafakos | first=Loukas | publisher=Pearson Education, Inc. | pages=253–257 | year=2004 | isbn=0-13-035399-X}}.
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| * {{citation|last1=Hewitt|first1=Edwin|last2=Stromberg|first2=Karl|title=Real and abstract analysis|publisher=Springer-Verlag|year=1965}}.
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| * {{citation
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| |last1=Kalton|first1=Nigel J.|author-link=Nigel Kalton
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| |last2=Peck|first2=N. Tenney
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| |last3=Roberts|first3=James W.
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| | title = An F-space sampler
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| | series = London Mathematical Society Lecture Note Series|volume=89
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| | publisher = Cambridge University Press| publication-place = Cambridge
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| | year = 1984 | isbn = 0-521-27585-7|mr=808777}}
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| * {{citation
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| |last=Riesz|first=Frigyes|authorlink=Frigyes Riesz
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| |title=Untersuchungen über Systeme integrierbarer Funktionen|journal=Mathematische Annalen|volume=69|year=1910|pages=449–497
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| |doi=10.1007/BF01457637
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| |issue=4}}
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| * {{Citation | last1=Rudin | first1=Walter | author1-link=Walter Rudin | title=Functional Analysis | publisher=McGraw-Hill Science/Engineering/Math | isbn=978-0-07-054236-5 | year=1991}}
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| * {{Citation | last1=Rudin | first1=Walter | author1-link=Walter Rudin | title=Real and complex analysis | publisher=[[McGraw-Hill]] | location=New York | edition=3rd | isbn=978-0-07-054234-1 | mr=924157 | year=1987}}
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| * {{citation|first=EC|last=Titchmarsh|authorlink=Edward Charles Titchmarsh|title=The theory of functions|publisher=Oxford University Press|year=1976|isbn=978-0-19-853349-8}}
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| ==External links==
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| * {{springer|title=Lebesgue space|id=p/l057910}}
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| * {{planetmath reference|id=6270|title=Proof that ''L''<sup>''p''</sup> spaces are complete }}
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| {{DEFAULTSORT:Lp Space}}
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| [[Category:Normed spaces]]
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| [[Category:Banach spaces]]
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| [[Category:Mathematical series]]
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| [[Category:Function spaces]]
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