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Inmathematics, theL^p spaces arefunction spaces defined using a natural generalization of thep-norm for finite-dimensionalvector spaces. They are sometimes calledLebesgue spaces, named afterHenri Lebesgue (Dunford & Schwartz 1958, III.3), although according to theBourbaki group (Bourbaki 1987) they were first introduced byFrigyes Riesz (Riesz 1910).

L^p spaces form an important class ofBanach spaces infunctional analysis, and oftopological vector spaces. Because 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.

The Euclidean length of a vector${\displaystyle x=(x_1,x_2,\dots ,x_n)}$ in the${\displaystyle n}$-dimensionalrealvector space${\displaystyle {\mathbb{R}}^n}$ is given by theEuclidean norm:$${\displaystyle \Vert x{\Vert }_2={\left({x_1}^2+{x_2}^2+\cdots +{x_n}^2\right)}^{1/2}.}$$

The Euclidean distance between two points${\displaystyle x}$ and${\displaystyle y}$ is the length${\displaystyle \Vert x-y{\Vert }_2}$ of the straight line between the two points. In many situations, the Euclidean distance is appropriate for capturing the actual distances in a given space. In 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 therectilinear distance, which takes into account that streets are either orthogonal or parallel to each other. The class of${\displaystyle p}$-norms generalizes these two examples and has an abundance of applications in many parts ofmathematics,physics, andcomputer science.

For areal number${\displaystyle p\ge 1,}$ the${\displaystyle p}$-norm or${\displaystyle L^p}$-norm of${\displaystyle x}$ is defined by$${\displaystyle \Vert x{\Vert }_p={\left(|x_1{|}^p+|x_2{|}^p+\cdots +|x_n{|}^p\right)}^{1/p}.}$$The absolute value bars can be dropped when${\displaystyle p}$ is a rational number with an even numerator in its reduced form, and${\displaystyle x}$ is drawn from the set of real numbers, or one of its subsets.

The Euclidean norm from above falls into this class and is the${\displaystyle 2}$-norm, and the${\displaystyle 1}$-norm is the norm that corresponds to therectilinear distance.

The${\displaystyle L^{\mathrm{\infty }}}$-norm ormaximum norm (or uniform norm) is the limit of the${\displaystyle L^p}$-norms for${\displaystyle p\to \mathrm{\infty }}$, given by:$${\displaystyle \Vert x{\Vert }_{\mathrm{\infty }}=max\left\{|x_1|,|x_2|,\dots ,|x_n|\right\}}$$

For all${\displaystyle p\ge 1,}$ the${\displaystyle p}$-norms and maximum norm satisfy the properties of a "length function" (ornorm), that is:

• only the zero vector has zero length,
• the length of the vector is positive homogeneous with respect to multiplication by a scalar (positive homogeneity), and
• the length of the sum of two vectors is no larger than the sum of lengths of the vectors (triangle inequality).

Abstractly speaking, this means that${\displaystyle {\mathbb{R}}^n}$ together with the${\displaystyle p}$-norm is anormed vector space. Moreover, it turns out that this space iscomplete, thus making it aBanach space.

The grid distance or rectilinear distance (sometimes called the "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). Formally, this means that the Euclidean norm of any vector is bounded by its 1-norm:$${\displaystyle \Vert x{\Vert }_2\le \Vert x{\Vert }_1.}$$

This fact generalizes to${\displaystyle p}$-norms in that the${\displaystyle p}$-norm${\displaystyle \Vert x{\Vert }_p}$ of any given vector${\displaystyle x}$ does not grow with${\displaystyle p}$:

${\displaystyle \Vert x{\Vert }_{p+a}\le \Vert x{\Vert }_p}$ for any vector${\displaystyle x}$ and real numbers${\displaystyle p\ge 1}$ and${\displaystyle a\ge 0.}$ (In fact this remains true for and${\displaystyle a\ge 0}$ .)

For the opposite direction, the following relation between the${\displaystyle 1}$-norm and the${\displaystyle 2}$-norm is known:$${\displaystyle \Vert x{\Vert }_1\le \sqrt{n}\Vert x{\Vert }_2.}$$

This inequality depends on the dimension${\displaystyle n}$ of the underlying vector space and follows directly from theCauchy–Schwarz inequality.

In general, for vectors in${\displaystyle {\mathbb{C}}^n}$ where$${\displaystyle \Vert x{\Vert }_p\le \Vert x{\Vert }_r\le n^{\frac{1}{r}-\frac{1}{p}}\Vert x{\Vert }_p.}$$

This is a consequence ofHölder's inequality.

In${\displaystyle {\mathbb{R}}^n}$ for${\displaystyle n>1,}$ the formula$${\displaystyle \Vert x{\Vert }_p={\left(|x_1{|}^p+|x_2{|}^p+\cdots +|x_n{|}^p\right)}^{1/p}}$$defines an absolutelyhomogeneous function for however, the resulting function does not define a norm, because it is notsubadditive. On the other hand, the formula$${\displaystyle |x_1{|}^p+|x_2{|}^p+\cdots +|x_n{|}^p}$$defines a subadditive function at the cost of losing absolute homogeneity. It does define anF-norm, though, which is homogeneous of degree${\displaystyle p.}$

Hence, the function$${\displaystyle d_p(x,y)=\sum _{i=1}^n|x_i-y_i{|}^p}$$defines ametric. Themetric space${\displaystyle ({\mathbb{R}}^n,d_p)}$ is denoted by${\displaystyle {\ell }_n^p.}$

Although the${\displaystyle p}$-unit ball${\displaystyle B_n^p}$ around the origin in this metric is "concave", the topology defined on${\displaystyle {\mathbb{R}}^n}$ by the metric${\displaystyle B_p}$ is the usual vector space topology of${\displaystyle {\mathbb{R}}^n,}$ hence${\displaystyle {\ell }_n^p}$ is alocally convex topological vector space. Beyond this qualitative statement, a quantitative way to measure the lack of convexity of${\displaystyle {\ell }_n^p}$ is to denote by${\displaystyle C_p(n)}$ the smallest constant${\displaystyle C}$ such that the scalar multiple${\displaystyle C{B}_n^p}$ of the${\displaystyle p}$-unit ball contains the convex hull of${\displaystyle B_n^p,}$ which is equal to${\displaystyle B_n^1.}$ The fact that for fixed we have$${\displaystyle C_p(n)=n^{\frac{1}{p}-1}\to \mathrm{\infty },\text{as }n\to \mathrm{\infty }}$$shows that the infinite-dimensional sequence space${\displaystyle {\ell }^p}$ defined below, is no longer locally convex.^{[citation needed]}

There is one${\displaystyle {\ell }_0}$ norm and another function called the${\displaystyle {\ell }_0}$ "norm" (with quotation marks).

The mathematical definition of the${\displaystyle {\ell }_0}$ norm was established byBanach'sTheory of Linear Operations. Thespace of sequences has a complete metric topology provided by theF-norm on theproduct metric:^{[citation needed]}$${\displaystyle (x_n)\mapsto \Vert x\Vert :=d(0,x)=\sum _n{2}^{-n}\frac{|x_n|}{1+|x_n|}.}$$ The${\displaystyle {\ell }_0}$-normed space is studied in functional analysis, probability theory, and harmonic analysis.

Another function was called the${\displaystyle {\ell }_0}$ "norm" byDavid Donoho—whose quotation marks warn that this function is not a proper norm—is the number of non-zero entries of the vector${\displaystyle x.}$^{[citation needed]} Many authorsabuse terminology by omitting the quotation marks. Defining${\displaystyle 0^0=0,}$ the zero "norm" of${\displaystyle x}$ is equal to$${\displaystyle |x_1{|}^0+|x_2{|}^0+\cdots +|x_n{|}^0.}$$

This is not anorm because it is nothomogeneous. For example, scaling the vector${\displaystyle x}$ by a positive constant does not change the "norm". Despite these defects as a mathematical norm, the non-zero counting "norm" has uses inscientific computing,information theory, andstatistics–notably incompressed sensing insignal processing and computationalharmonic analysis. Despite not being a norm, the associated metric, known asHamming distance, is a valid distance, since homogeneity is not required for distances.

The${\displaystyle p}$-norm can be extended to vectors that have an infinite number of components (sequences), which yields the space${\displaystyle {\ell }^p.}$ This contains as special cases:

• ${\displaystyle {\ell }^1,}$ the space of sequences whose series areabsolutely convergent,
• ${\displaystyle {\ell }^2,}$ the space ofsquare-summable sequences, which is aHilbert space, and
• ${\displaystyle {\ell }^{\mathrm{\infty }},}$ the space ofbounded sequences.

The space of sequences has a natural vector space structure by applying scalar addition and multiplication. Explicitly, the vector sum and the scalar action for infinitesequences of real (orcomplex) numbers are given by:

Define the${\displaystyle p}$-norm:$${\displaystyle \Vert x{\Vert }_p={\left(|x_1{|}^p+|x_2{|}^p+\cdots +|x_n{|}^p+|x_{n+1}{|}^p+\cdots \right)}^{1/p}}$$

Here, a complication arises, namely that theseries on the right is not always convergent, so for example, the sequence made up of only ones,${\displaystyle (1,1,1,\dots ),}$ will have an infinite${\displaystyle p}$-norm for The space${\displaystyle {\ell }^p}$ is then defined as the set of all infinite sequences of real (or complex) numbers such that the${\displaystyle p}$-norm is finite.

One can check that as${\displaystyle p}$ increases, the set${\displaystyle {\ell }^p}$ grows larger. For example, the sequence$${\displaystyle \left(1,\frac{1}{2},\dots ,\frac{1}{n},\frac{1}{n+1},\dots \right)}$$is not in${\displaystyle {\ell }^1,}$ but it is in${\displaystyle {\ell }^p}$ for${\displaystyle p>1,}$ as the series$${\displaystyle 1^p+\frac{1}{2^p}+\cdots +\frac{1}{n^p}+\frac{1}{(n+1{)}^p}+\cdots ,}$$diverges for${\displaystyle p=1}$ (theharmonic series), but is convergent for${\displaystyle p>1.}$

One also defines the${\displaystyle \mathrm{\infty }}$-norm using thesupremum:$${\displaystyle \Vert x{\Vert }_{\mathrm{\infty }}=sup(|x_1|,|x_2|,\dots ,|x_n|,|x_{n+1}|,\dots )}$$and the corresponding space${\displaystyle {\ell }^{\mathrm{\infty }}}$ of all bounded sequences. It turns out that^[1]$${\displaystyle \Vert x{\Vert }_{\mathrm{\infty }}=\underset{p\to \mathrm{\infty }}{lim}\Vert x{\Vert }_p}$$if the right-hand side is finite, or the left-hand side is infinite. Thus, we will consider${\displaystyle {\ell }^p}$ spaces for${\displaystyle 1\le p\le \mathrm{\infty }.}$

The${\displaystyle p}$-norm thus defined on${\displaystyle {\ell }^p}$ is indeed a norm, and${\displaystyle {\ell }^p}$ together with this norm is aBanach space.

In complete analogy to the preceding definition one can define the space${\displaystyle {\ell }^p(I)}$ over a generalindex set${\displaystyle I}$ (and) aswhere convergence on the right requires that only countably many summands are nonzero (see alsoAbsolute convergence over sets).With the norm$${\displaystyle \Vert x{\Vert }_p={\left(\sum _{i\in I}|x_i{|}^p\right)}^{1/p}}$$the space${\displaystyle {\ell }^p(I)}$ becomes a Banach space.In the case where${\displaystyle I}$ is finite with${\displaystyle n}$ elements, this construction yields${\displaystyle {\mathbb{R}}^n}$ with the${\displaystyle p}$-norm defined above.If${\displaystyle I}$ is countably infinite, this is exactly the sequence space${\displaystyle {\ell }^p}$ defined above.For uncountable sets${\displaystyle I}$ this is a non-separable Banach space which can be seen as thelocally convexdirect limit of${\displaystyle {\ell }^p}$-sequence spaces.^[2]

For${\displaystyle p=2,}$ the${\displaystyle \Vert \cdot {\Vert }_2}$-norm is even induced by a canonicalinner product${\displaystyle ⟨\cdot ,\cdot ⟩,}$ called theEuclidean inner product, which means that${\displaystyle \Vert \mathbf{x}{\Vert }_2=\sqrt{⟨\mathbf{x},\mathbf{x}⟩}}$ holds for all vectors${\displaystyle \mathbf{x}.}$ This inner product can be expressed in terms of the norm by using thepolarization identity. On${\displaystyle {\ell }^2,}$ it can be defined by$${\displaystyle ⟨{\left(x_i\right)}_i,{\left(y_n\right)}_i{⟩}_{{\ell }^2}=\sum _i{x}_i\overline{y_i}.}$$Now consider the case${\displaystyle p=\mathrm{\infty }.}$ Define^{[note 1]}where for all${\displaystyle x}$^{[3][note 2]}

The index set${\displaystyle I}$ can be turned into ameasure space by giving it thediscrete σ-algebra and thecounting measure. Then the space${\displaystyle {\ell }^p(I)}$ is just a special case of the more general${\displaystyle L^p}$-space (defined below).

An${\displaystyle L^p}$ space may be defined as a space of measurable functions for which the${\displaystyle p}$-th power of theabsolute value isLebesgue integrable, where functions which agree almost everywhere are identified. More generally, let${\displaystyle (S,\mathrm{\Sigma },\mu )}$ be ameasure space and${\displaystyle 1\le p\le \mathrm{\infty }.}$^{[note 3]} When${\displaystyle p\ne \mathrm{\infty }}$, consider the set${\displaystyle {\mathcal{L}}^p(S,\mu )}$ of allmeasurable functions${\displaystyle f}$ from${\displaystyle S}$ to${\displaystyle \mathbb{C}}$ or${\displaystyle \mathbb{R}}$ whoseabsolute value raised to the${\displaystyle p}$-th power has a finite integral, or in symbols:^[4]

To define the set for${\displaystyle p=\mathrm{\infty },}$ recall that two functions${\displaystyle f}$ and${\displaystyle g}$ defined on${\displaystyle S}$ are said to beequalalmost everywhere, written${\displaystyle f=g}$ a.e., if the set${\displaystyle \{s\in S:f(s)\ne g(s)\}}$ is measurable and has measure zero. Similarly, a measurable function${\displaystyle f}$ (and itsabsolute value) isbounded (ordominated)almost everywhere by a real number${\displaystyle C,}$ written${\displaystyle |f|\le C}$ a.e., if the (necessarily) measurable set${\displaystyle \{s\in S:|f(s)|>C\}}$ has measure zero. The space${\displaystyle {\mathcal{L}}^{\mathrm{\infty }}(S,\mu )}$ is the set of all measurable functions${\displaystyle f}$ that are bounded almost everywhere (by some real${\displaystyle C}$) and${\displaystyle \Vert f{\Vert }_{\mathrm{\infty }}}$ is defined as theinfimum of these bounds:$${\displaystyle \Vert f{\Vert }_{\mathrm{\infty }}\stackrel{\text{def}}{=}inf\{C\in {\mathbb{R}}_{\ge 0}:|f(s)|\le C\text{ for almost every }s\}.}$$ When${\displaystyle \mu (S)\ne 0}$ then this is the same as theessential supremum of the absolute value of${\displaystyle f}$:^{[note 4]}

For example, if${\displaystyle f}$ is a measurable function that is equal to${\displaystyle 0}$ almost everywhere^{[note 5]} then${\displaystyle \Vert f{\Vert }_p=0}$ for every${\displaystyle p}$ and thus${\displaystyle f\in {\mathcal{L}}^p(S,\mu )}$ for all${\displaystyle p.}$

For every positive${\displaystyle p,}$ the value under${\displaystyle \Vert \cdot {\Vert }_p}$ of a measurable function${\displaystyle f}$ and its absolute value${\displaystyle |f|:S\to [0,\mathrm{\infty }]}$ are always the same (that is,${\displaystyle \Vert f{\Vert }_p=\Vert |f|{\Vert }_p}$ for all${\displaystyle p}$) and so a measurable function belongs to${\displaystyle {\mathcal{L}}^p(S,\mu )}$ if and only if its absolute value does. Because of this, many formulas involving${\displaystyle p}$-norms are stated only for non-negative real-valued functions. Consider for example the identity${\displaystyle \Vert f{\Vert }_p^r=\Vert f^r{\Vert }_{p/r},}$ which holds whenever${\displaystyle f\ge 0}$ is measurable,${\displaystyle r>0}$ is real, and (here${\displaystyle \mathrm{\infty }/r\stackrel{\text{def}}{=}\mathrm{\infty }}$ when${\displaystyle p=\mathrm{\infty }}$). The non-negativity requirement${\displaystyle f\ge 0}$ can be removed by substituting${\displaystyle |f|}$ in for${\displaystyle f,}$ which gives${\displaystyle \Vert |f|{\Vert }_p^r=\Vert |f{|}^r{\Vert }_{p/r}.}$ Note in particular that when${\displaystyle p=r}$ is finite then the formula${\displaystyle \Vert f{\Vert }_p^p=\Vert |f{|}^p{\Vert }_1}$ relates the${\displaystyle p}$-norm to the${\displaystyle 1}$-norm.

Seminormed space of${\displaystyle p}$-th power integrable functions

Each set of functions${\displaystyle {\mathcal{L}}^p(S,\mu )}$ forms avector space when addition and scalar multiplication are defined pointwise.^{[note 6]} That the sum of two${\displaystyle p}$-th power integrable functions${\displaystyle f}$ and${\displaystyle g}$ is again${\displaystyle p}$-th power integrable follows from$\Vert f+g{\Vert }_p^p\le 2^{p-1}\left(\Vert f{\Vert }_p^p+\Vert g{\Vert }_p^p\right),$^{[proof 1]} although it is also a consequence ofMinkowski's inequality$${\displaystyle \Vert f+g{\Vert }_p\le \Vert f{\Vert }_p+\Vert g{\Vert }_p}$$ which establishes that${\displaystyle \Vert \cdot {\Vert }_p}$ satisfies thetriangle inequality for${\displaystyle 1\le p\le \mathrm{\infty }}$ (the triangle inequality does not hold for). That${\displaystyle {\mathcal{L}}^p(S,\mu )}$ is closed under scalar multiplication is due to${\displaystyle \Vert \cdot {\Vert }_p}$ beingabsolutely homogeneous, which means that${\displaystyle \Vert sf{\Vert }_p=|s|\Vert f{\Vert }_p}$ for every scalar${\displaystyle s}$ and every function${\displaystyle f.}$

Absolute homogeneity, thetriangle inequality, and non-negativity are the defining properties of aseminorm. Thus${\displaystyle \Vert \cdot {\Vert }_p}$ is a seminorm and the set${\displaystyle {\mathcal{L}}^p(S,\mu )}$ of${\displaystyle p}$-th power integrable functions together with the function${\displaystyle \Vert \cdot {\Vert }_p}$ defines aseminormed vector space. In general, theseminorm${\displaystyle \Vert \cdot {\Vert }_p}$ is not anorm because there might exist measurable functions${\displaystyle f}$ that satisfy${\displaystyle \Vert f{\Vert }_p=0}$ but are notidentically equal to${\displaystyle 0}$^{[note 5]} (${\displaystyle \Vert \cdot {\Vert }_p}$ is a norm if and only if no such${\displaystyle f}$ exists).

Zero sets of${\displaystyle p}$-seminorms

If${\displaystyle f}$ is measurable and equals${\displaystyle 0}$ a.e. then${\displaystyle \Vert f{\Vert }_p=0}$ for all positive${\displaystyle p\le \mathrm{\infty }.}$On the other hand, if${\displaystyle f}$ is a measurable function for which there exists some such that${\displaystyle \Vert f{\Vert }_p=0}$ then${\displaystyle f=0}$ almost everywhere. When${\displaystyle p}$ is finite then this follows from the${\displaystyle p=1}$ case and the formula${\displaystyle \Vert f{\Vert }_p^p=\Vert |f{|}^p{\Vert }_1}$ mentioned above.

Thus if${\displaystyle p\le \mathrm{\infty }}$ is positive and${\displaystyle f}$ is any measurable function, then${\displaystyle \Vert f{\Vert }_p=0}$ if and only if${\displaystyle f=0}$almost everywhere. Since the right hand side (${\displaystyle f=0}$ a.e.) does not mention${\displaystyle p,}$ it follows that all${\displaystyle \Vert \cdot {\Vert }_p}$ have the samezero set (it does not depend on${\displaystyle p}$). So denote this common set by$${\displaystyle \mathcal{N}\stackrel{\text{def}}{=}\{f:f=0\mu \text{-almost everywhere}\}=\{f\in {\mathcal{L}}^p(S,\mu ):\Vert f{\Vert }_p=0\}\mathrm{\forall }p.}$$This set is a vector subspace of${\displaystyle {\mathcal{L}}^p(S,\mu )}$ for every positive${\displaystyle p\le \mathrm{\infty }.}$

Quotient vector space

Like everyseminorm, the seminorm${\displaystyle \Vert \cdot {\Vert }_p}$ induces anorm (defined shortly) on the canonicalquotient vector space of${\displaystyle {\mathcal{L}}^p(S,\mu )}$ by its vector subspace$\mathcal{N}=\{f\in {\mathcal{L}}^p(S,\mu ):\Vert f{\Vert }_p=0\}.$This normed quotient space is calledLebesgue space and it is the subject of this article. We begin by defining the quotient vector space.

Given any${\displaystyle f\in {\mathcal{L}}^p(S,\mu ),}$ thecoset${\displaystyle f+\mathcal{N}\stackrel{\text{def}}{=}\{f+h:h\in \mathcal{N}\}}$ consists of all measurable functions${\displaystyle g}$ that are equal to${\displaystyle f}$almost everywhere. The set of all cosets, typically denoted by$${\displaystyle {\mathcal{L}}^p(S,\mu )/\mathcal{N}\stackrel{\text{def}}{=}\{f+\mathcal{N}:f\in {\mathcal{L}}^p(S,\mu )\},}$$forms a vector space with origin${\displaystyle 0+\mathcal{N}=\mathcal{N}}$ when vector addition and scalar multiplication are defined by${\displaystyle (f+\mathcal{N})+(g+\mathcal{N})\stackrel{\text{def}}{=}(f+g)+\mathcal{N}}$ and${\displaystyle s(f+\mathcal{N})\stackrel{\text{def}}{=}(sf)+\mathcal{N}.}$ This particular quotient vector space will be denoted by$${\displaystyle L^p(S,\mu )\stackrel{\text{def}}{=}{\mathcal{L}}^p(S,\mu )/\mathcal{N}.}$$Two cosets are equal${\displaystyle f+\mathcal{N}=g+\mathcal{N}}$ if and only if${\displaystyle g\in f+\mathcal{N}}$ (or equivalently,${\displaystyle f-g\in \mathcal{N}}$), which happens if and only if${\displaystyle f=g}$ almost everywhere; if this is the case then${\displaystyle f}$ and${\displaystyle g}$ are identified in the quotient space. Hence, strictly speaking${\displaystyle L^p(S,\mu )}$ consists ofequivalence classes of functions.^[5]

The${\displaystyle p}$-norm on the quotient vector space

Given any${\displaystyle f\in {\mathcal{L}}^p(S,\mu ),}$ the value of the seminorm${\displaystyle \Vert \cdot {\Vert }_p}$ on thecoset${\displaystyle f+\mathcal{N}=\{f+h:h\in \mathcal{N}\}}$ is constant and equal to${\displaystyle \Vert f{\Vert }_p;}$ denote this unique value by${\displaystyle \Vert f+\mathcal{N}{\Vert }_p,}$ so that:$${\displaystyle \Vert f+\mathcal{N}{\Vert }_p\stackrel{\text{def}}{=}\Vert f{\Vert }_p.}$$ This assignment${\displaystyle f+\mathcal{N}\mapsto \Vert f+\mathcal{N}{\Vert }_p}$ defines a map, which will also be denoted by${\displaystyle \Vert \cdot {\Vert }_p,}$ on thequotient vector space$${\displaystyle L^p(S,\mu )\stackrel{\text{def}}{=}{\mathcal{L}}^p(S,\mu )/\mathcal{N}=\{f+\mathcal{N}:f\in {\mathcal{L}}^p(S,\mu )\}.}$$This map is anorm on${\displaystyle L^p(S,\mu )}$ called the${\displaystyle p}$-norm. The value${\displaystyle \Vert f+\mathcal{N}{\Vert }_p}$ of a coset${\displaystyle f+\mathcal{N}}$ is independent of the particular function${\displaystyle f}$ that was chosen to represent the coset, meaning that if${\displaystyle \mathcal{C}\in L^p(S,\mu )}$ is any coset then${\displaystyle \Vert \mathcal{C}{\Vert }_p=\Vert f{\Vert }_p}$ for every${\displaystyle f\in \mathcal{C}}$ (since${\displaystyle \mathcal{C}=f+\mathcal{N}}$ for every${\displaystyle f\in \mathcal{C}}$).

The Lebesgue${\displaystyle L^p}$ space

Thenormed vector space${\displaystyle \left(L^p(S,\mu ),\Vert \cdot {\Vert }_p\right)}$ is called${\displaystyle L^p}$ space or theLebesgue space of${\displaystyle p}$-th power integrable functions and it is aBanach space for every${\displaystyle 1\le p\le \mathrm{\infty }}$ (meaning that it is acomplete metric space, a result that is sometimes called theRiesz–Fischer theorem). When the underlying measure space${\displaystyle S}$ is understood then${\displaystyle L^p(S,\mu )}$ is often abbreviated${\displaystyle L^p(\mu ),}$ or even just${\displaystyle L^p.}$ Depending on the author, the subscript notation${\displaystyle L_p}$ might denote either${\displaystyle L^p(S,\mu )}$ or${\displaystyle L^{1/p}(S,\mu ).}$

If the seminorm${\displaystyle \Vert \cdot {\Vert }_p}$ on${\displaystyle {\mathcal{L}}^p(S,\mu )}$ happens to be a norm (which happens if and only if${\displaystyle \mathcal{N}=\{0\}}$) then the normed space${\displaystyle \left({\mathcal{L}}^p(S,\mu ),\Vert \cdot {\Vert }_p\right)}$ will belinearlyisometrically isomorphic to the normed quotient space${\displaystyle \left(L^p(S,\mu ),\Vert \cdot {\Vert }_p\right)}$ via the canonical map${\displaystyle g\in {\mathcal{L}}^p(S,\mu )\mapsto \{g\}}$ (since${\displaystyle g+\mathcal{N}=\{g\}}$); in other words, they will be,up to alinear isometry, the same normed space and so they may both be called "${\displaystyle L^p}$ space".

The above definitions generalize toBochner spaces.

In general, this process cannot be reversed: there is no consistent way to define a "canonical" representative of each coset of${\displaystyle \mathcal{N}}$ in${\displaystyle L^p.}$ For${\displaystyle L^{\mathrm{\infty }},}$ however, there is atheory of lifts enabling such recovery.

For${\displaystyle 1\le p\le \mathrm{\infty }}$ the${\displaystyle {\ell }^p}$ spaces are a special case of${\displaystyle L^p}$ spaces; when${\displaystyle S}$ are thenatural numbers${\displaystyle \mathbb{N}}$ and${\displaystyle \mu }$ is thecounting measure. More generally, if one considers any set${\displaystyle S}$ with the counting measure, the resulting${\displaystyle L^p}$ space is denoted${\displaystyle {\ell }^p(S).}$ For example,${\displaystyle {\ell }^p(\mathbb{Z})}$ is the space of all sequences indexed by the integers, and when defining the${\displaystyle p}$-norm on such a space, one sums over all the integers. The space${\displaystyle {\ell }^p(n),}$ where${\displaystyle n}$ is the set with${\displaystyle n}$ elements, is${\displaystyle {\mathbb{R}}^n}$ with its${\displaystyle p}$-norm as defined above.

Similar to${\displaystyle {\ell }^2}$ spaces,${\displaystyle L^2}$ is the onlyHilbert space among${\displaystyle L^p}$ spaces. In the complex case, the inner product on${\displaystyle L^2}$ is defined by$${\displaystyle ⟨f,g⟩={\int }_{S}f(x)\overline{g(x)}\mathrm{d}\mu (x).}$$Functions in${\displaystyle L^2}$ are sometimes calledsquare-integrable functions,quadratically integrable functions orsquare-summable functions, but sometimes these terms are reserved for functions that are square-integrable in some other sense, such as in the sense of aRiemann integral (Titchmarsh 1976).

As any Hilbert space, every space${\displaystyle L^2}$ is linearly isometric to a suitable${\displaystyle {\ell }^2(I),}$ where the cardinality of the set${\displaystyle I}$ is the cardinality of an arbitrary basis for this particular${\displaystyle L^2.}$

If we use complex-valued functions, the space${\displaystyle L^{\mathrm{\infty }}}$ is acommutativeC*-algebra with pointwise multiplication and conjugation. For many measure spaces, including all sigma-finite ones, it is in fact a commutativevon Neumann algebra. An element of${\displaystyle L^{\mathrm{\infty }}}$ defines abounded operator on any${\displaystyle L^p}$ space bymultiplication.

If then${\displaystyle L^p(\mu )}$ can be defined as above, that is:In this case, however, the${\displaystyle p}$-norm${\displaystyle \Vert f{\Vert }_p=N_p(f{)}^{1/p}}$ does not satisfy the triangle inequality and defines only aquasi-norm. The inequality${\displaystyle (a+b{)}^p\le a^p+b^p,}$ valid for${\displaystyle a,b\ge 0,}$ implies that$${\displaystyle N_p(f+g)\le N_p(f)+N_p(g)}$$and so the function$${\displaystyle d_p(f,g)=N_p(f-g)=\Vert f-g{\Vert }_p^p}$$is a metric on${\displaystyle L^p(\mu ).}$ The resulting metric space iscomplete.^[6]

In this setting${\displaystyle L^p}$ satisfies areverse Minkowski inequality, that is for${\displaystyle u,v\in L^p}$$${\displaystyle \Vert |u|+|v|{\Vert }_p\ge \Vert u{\Vert }_p+\Vert v{\Vert }_p}$$

This result may be used to proveClarkson's inequalities, which are in turn used to establish theuniform convexity of the spaces${\displaystyle L^p}$ for (Adams & Fournier 2003).

The space${\displaystyle L^p}$ for is anF-space: it admits a complete translation-invariant metric with respect to which the vector space operations are continuous. It is the prototypical example of anF-space that, for most reasonable measure spaces, is notlocally convex: in${\displaystyle {\ell }^p}$ or${\displaystyle L^p([0,1]),}$ every open convex set containing the${\displaystyle 0}$ function is unbounded for the${\displaystyle p}$-quasi-norm; therefore, the${\displaystyle 0}$ vector does not possess a fundamental system of convex neighborhoods. Specifically, this is true if the measure space${\displaystyle S}$ contains an infinite family of disjoint measurable sets of finite positive measure.

The only nonempty convex open set in${\displaystyle L^p([0,1])}$ is the entire space. Consequently, there are no nonzero continuous linear functionals on${\displaystyle L^p([0,1]);}$ thecontinuous dual space is the zero space. In the case of thecounting measure on the natural numbers (i.e.${\displaystyle L^p(\mu )={\ell }^p}$), the bounded linear functionals on${\displaystyle {\ell }^p}$ are exactly those that are bounded on${\displaystyle {\ell }^1}$, i.e., those given by sequences in${\displaystyle {\ell }^{\mathrm{\infty }}.}$ Although${\displaystyle {\ell }^p}$ does contain non-trivial convex open sets, it fails to have enough of them to give a base for the topology.

Having no linear functionals is highly undesirable for the purposes of doing analysis. In case of the Lebesgue measure on${\displaystyle {\mathbb{R}}^n,}$ rather than work with${\displaystyle L^p}$ for it is common to work with theHardy spaceH^p whenever possible, as this has quite a few linear functionals: enough to distinguish points from one another. However, theHahn–Banach theorem still fails inH^p for (Duren 1970, §7.5).

Suppose${\displaystyle p,q,r\in [1,\mathrm{\infty }]}$ satisfy${\displaystyle \frac{1}{p}+\frac{1}{q}=\frac{1}{r}}$. If${\displaystyle f\in L^p(S,\mu )}$ and${\displaystyle g\in L^q(S,\mu )}$ then${\displaystyle fg\in L^r(S,\mu )}$ and^[7]$${\displaystyle \Vert fg{\Vert }_r\le \Vert f{\Vert }_p\Vert g{\Vert }_q.}$$

This inequality, calledHölder's inequality, is in some sense optimal since if${\displaystyle r=1}$ and${\displaystyle f}$ is a measurable function such that where thesupremum is taken over the closed unit ball of${\displaystyle L^q(S,\mu ),}$ then${\displaystyle f\in L^p(S,\mu )}$ and$${\displaystyle \Vert f{\Vert }_p=\underset{\Vert g{\Vert }_q\le 1}{sup}{\int }_{S}fg\mathrm{d}\mu .}$$

Minkowski inequality, which states that${\displaystyle \Vert \cdot {\Vert }_p}$ satisfies thetriangle inequality, can be generalized: If the measurable function${\displaystyle F:M\times N\to \mathbb{R}}$ is non-negative (where${\displaystyle (M,\mu )}$ and${\displaystyle (N,\nu )}$ are measure spaces) then for all${\displaystyle 1\le p\le q\le \mathrm{\infty },}$^[8]$${\displaystyle {\Vert {\Vert F(\cdot ,n)\Vert }_{L^p(M,\mu )}\Vert }_{L^q(N,\nu )}\le {\Vert {\Vert F(m,\cdot )\Vert }_{L^q(N,\nu )}\Vert }_{L^p(M,\mu )}.}$$

If then every non-negative${\displaystyle f\in L^p(\mu )}$ has anatomic decomposition,^[9] meaning that there exist a sequence${\displaystyle (r_n{)}_{n\in \mathbb{Z}}}$ of non-negative real numbers and a sequence of non-negative functions${\displaystyle (f_n{)}_{n\in \mathbb{Z}},}$ calledthe atoms, whose supports${\displaystyle {\left(\mathrm{supp}{f}_n\right)}_{n\in \mathbb{Z}}}$ arepairwise disjoint sets of measure${\displaystyle \mu \left(\mathrm{supp}{f}_n\right)\le 2^{n+1},}$ such that$${\displaystyle f=\sum _{n\in \mathbb{Z}}{r}_n{f}_n,}$$and for every integer${\displaystyle n\in \mathbb{Z},}$$${\displaystyle \Vert f_n{\Vert }_{\mathrm{\infty }}\le 2^{-\frac{n}{p}},}$$ and$${\displaystyle \frac{1}{2}\Vert f{\Vert }_p^p\le \sum _{n\in \mathbb{Z}}{r}_n^p\le 2\Vert f{\Vert }_p^p,}$$and where moreover, the sequence of functions${\displaystyle (r_n{f}_n{)}_{n\in \mathbb{Z}}}$ depends only on${\displaystyle f}$ (it is independent of${\displaystyle p}$).^[9] These inequalities guarantee that${\displaystyle \Vert f_n{\Vert }_p^p\le 2}$ for all integers${\displaystyle n}$ while the supports of${\displaystyle (f_n{)}_{n\in \mathbb{Z}}}$ being pairwise disjoint implies^[9]$${\displaystyle \Vert f{\Vert }_p^p=\sum _{n\in \mathbb{Z}}{r}_n^p\Vert f_n{\Vert }_p^p.}$$

An atomic decomposition can be explicitly given by first defining for every integer${\displaystyle n\in \mathbb{Z},}$^{[9][note 7]}and then lettingwhere${\displaystyle \mu (f>t)=\mu (\{s:f(s)>t\})}$ denotes the measure of the set${\displaystyle (f>t):=\{s\in S:f(s)>t\}}$ and denotes theindicator function of the set The sequence${\displaystyle (t_n{)}_{n\in \mathbb{Z}}}$ is decreasing and converges to${\displaystyle 0}$ as${\displaystyle n\to \mathrm{\infty }.}$^[9] Consequently, if${\displaystyle t_n=0}$ then${\displaystyle t_{n+1}=0}$ and so that is identically equal to${\displaystyle 0}$ (in particular, the division${\displaystyle \frac{1}{r_n}}$ by${\displaystyle r_n=0}$ causes no issues).