L^p space

Illustrations of unit circles (see also superellipse) in ${\displaystyle \mathbb {R} ^{2}}$ based on different ${\displaystyle p}$-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 ${\displaystyle p}$).

The length of a vector ${\displaystyle x=(x_{1},x_{2},\dots ,x_{n})}$ in the ${\displaystyle n}$-dimensional real vector space ${\displaystyle \mathbb {R} ^{n}}$ is usually given by the Euclidean norm:

$${\displaystyle \|x\|_{2}=\left({x_{1}}^{2}+{x_{2}}^{2}+\dotsb +{x_{n}}^{2}\right)^{1/2}.}$$

The Euclidean distance between two points ${\displaystyle x}$ and ${\displaystyle y}$ is the length ${\displaystyle \|x-y\|_{2}}$ 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. An analogy to this is suggested by 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, 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 of mathematics, physics, and computer science.

Definition

For a real number ${\displaystyle p\geq 1,}$ the ${\displaystyle p}$-norm or ${\displaystyle L^{p}}$-norm of ${\displaystyle x}$ is defined by

$${\displaystyle \|x\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\dotsb +|x_{n}|^{p}\right)^{1/p}.}$$

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 the rectilinear distance.

The ${\displaystyle L^{\infty }}$-norm or maximum norm (or uniform norm) is the limit of the ${\displaystyle L^{p}}$-norms for ${\displaystyle p\to \infty .}$ It turns out that this limit is equivalent to the following definition:

$${\displaystyle \|x\|_{\infty }=\max \left\{|x_{1}|,|x_{2}|,\dotsc ,|x_{n}|\right\}}$$

See L-infinity.

For all ${\displaystyle p\geq 1,}$ the ${\displaystyle p}$-norms and maximum norm as defined above indeed satisfy the properties of a "length function" (or norm), which are that:

• 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 a normed vector space. Moreover, it turns out that this space is complete, thus making it a Banach space. This Banach space is the ${\displaystyle L^{p}}$-space over ${\displaystyle \{1,2,\ldots ,n\}.}$

Relations between p-norms

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:

$${\displaystyle \|x\|_{2}\leq \|x\|_{1}.}$$

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

${\displaystyle \|x\|_{p+a}\leq \|x\|_{p}}$ for any vector ${\displaystyle x}$ and real numbers ${\displaystyle p\geq 1}$ and ${\displaystyle a\geq 0.}$ (In fact this remains true for ${\displaystyle 0<p<1}$ and ${\displaystyle a\geq 0}$ .)

For the opposite direction, the following relation between the ${\displaystyle 1}$-norm and the ${\displaystyle 2}$-norm is known:

$${\displaystyle \|x\|_{1}\leq {\sqrt {n}}\|x\|_{2}~.}$$

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

In general, for vectors in ${\displaystyle \mathbb {C} ^{n}}$ where ${\displaystyle 0<r<p:}$

$${\displaystyle \|x\|_{p}\leq \|x\|_{r}\leq n^{{\frac {1}{r}}-{\frac {1}{p}}}\|x\|_{p}~.}$$

This is a consequence of Hölder's inequality.

When 0 < p < 1

Astroid, unit circle in ${\displaystyle p={\tfrac {2}{3}}}$ metric

In ${\displaystyle \mathbb {R} ^{n}}$ for ${\displaystyle n>1,}$ the formula

$${\displaystyle \|x\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\cdots +|x_{n}|^{p}\right)^{1/p}}$$

defines an absolutely homogeneous function for ${\displaystyle 0<p<1;}$ however, the resulting function does not define a norm, because it is not subadditive. On the other hand, the formula

$${\displaystyle |x_{1}|^{p}+|x_{2}|^{p}+\dotsb +|x_{n}|^{p}}$$

defines a subadditive function at the cost of losing absolute homogeneity. It does define an F-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 a metric. The metric 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 a locally 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 ${\displaystyle p<1}$ we have

$${\displaystyle C_{p}(n)=n^{{\tfrac {1}{p}}-1}\to \infty ,\quad {\text{as }}n\to \infty }$$

shows that the infinite-dimensional sequence space ${\displaystyle \ell ^{p}}$ defined below, is no longer locally convex.

When p = 0

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 by Banach's Theory of Linear Operations. The space of sequences has a complete metric topology provided by the F-norm

$${\displaystyle (x_{n})\mapsto \sum _{n}2^{-n}{\frac {|x_{n}|}{1+|x_{n}|}},}$$

which is discussed by Stefan Rolewicz in Metric Linear Spaces.[1] 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" 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 ${\displaystyle x.}$ Many authors abuse terminology by omitting the quotation marks. Defining ${\displaystyle 0^{0}=0,}$ the zero "norm" of ${\displaystyle x}$ is equal to

$${\displaystyle |x_{1}|^{0}+|x_{2}|^{0}+\cdots +|x_{n}|^{0}.}$$

An animated gif of p-norms 0.1 through 2 with a step of 0.05.

This is not a norm because it is not homogeneous. 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 in scientific computing, information theory, and statistics–notably in compressed sensing in signal processing and computational harmonic analysis. Despite not being a norm, the associated metric, known as Hamming distance, is a valid distance, since homogeneity is not required for distances.

An ${\displaystyle L^{p}}$ space may be defined as a space of measurable functions for which the ${\displaystyle p}$-th power of the absolute value is Lebesgue integrable, where functions which agree almost everywhere are identified. More generally, let ${\displaystyle (S,\Sigma ,\mu )}$ be a measure space and ${\displaystyle 1\leq p\leq \infty .}$[note 3] When ${\displaystyle p}$ is real (that is, ${\displaystyle p\neq \infty }$), consider the set ${\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}$ of all measurable functions ${\displaystyle f}$ from ${\displaystyle S}$ to ${\displaystyle \mathbb {C} }$ or ${\displaystyle \mathbb {R} }$ whose absolute value raised to the ${\displaystyle p}$-th power has a finite integral, or in symbols:

$${\displaystyle \|f\|_{p}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left(\int _{S}|f|^{p}\;\mathrm {d} \mu \right)^{1/p}<\infty .}$$

To define the set for ${\displaystyle p=\infty ,}$ recall that two functions ${\displaystyle f}$ and ${\displaystyle g}$ defined on ${\displaystyle S}$ are said to be equal almost everywhere, written ${\displaystyle f=g}$ a.e., if the set ${\displaystyle \{s\in S:f(s)\neq g(s)\}}$ is measurable and has measure zero. Similarly, a measurable function ${\displaystyle f}$ (and its absolute value) is bounded (or dominated) almost everywhere by a real number ${\displaystyle C,}$ written ${\displaystyle |f|\leq C}$ a.e., if the (necessarily) measurable set ${\displaystyle \{s\in S:|f(s)|>C\}}$ has measure zero. The space ${\displaystyle {\mathcal {L}}^{\infty }(S,\mu )}$ is the set of all measurable functions ${\displaystyle f}$ that are bounded almost everywhere (by some real ${\displaystyle C}$) and ${\displaystyle \|f\|_{\infty }}$ is defined as the infimum of these bounds:

$${\displaystyle \|f\|_{\infty }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\inf\{C\in \mathbb {R} _{\geq 0}:|f(s)|\leq C{\text{ for almost every }}s\}.}$$

When ${\displaystyle \mu (S)\neq 0}$ then this is the same as the essential supremum of the absolute value of ${\displaystyle f}$:[note 4]

$${\displaystyle \|f\|_{\infty }~=~{\begin{cases}\operatorname {esssup} |f|&{\text{if }}\mu (S)>0,\\0&{\text{if }}\mu (S)=0.\end{cases}}}$$

For example, if ${\displaystyle f}$ is a measurable function that is equal to ${\displaystyle 0}$ almost everywhere[note 5] then ${\displaystyle \|f\|_{p}=0}$ 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 ${\displaystyle \|\,\cdot \,\|_{p}}$ of a measurable function ${\displaystyle f}$ and its absolute value ${\displaystyle |f|:S\to [0,\infty ]}$ are always the same (that is, ${\displaystyle \|f\|_{p}=\||f|\|_{p}}$ 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 ${\displaystyle \|f\|_{p}^{r}=\|f^{r}\|_{p/r},}$ which holds whenever ${\displaystyle f\geq 0}$ is measurable, ${\displaystyle r>0}$ is real, and ${\displaystyle 0<p\leq \infty }$ (here ${\displaystyle \infty /r\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;\infty }$ when ${\displaystyle p=\infty }$). The non-negativity requirement ${\displaystyle f\geq 0}$ can be removed by substituting ${\displaystyle |f|}$ in for ${\displaystyle f,}$ which gives ${\displaystyle \|\,|f|\,\|_{p}^{r}=\|\,|f|^{r}\,\|_{p/r}.}$ Note in particular that when ${\displaystyle p=r}$ is finite then the formula ${\displaystyle \|f\|_{p}^{p}=\||f|^{p}\|_{1}}$ 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 a vector 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 ${\textstyle \|f+g\|_{p}^{p}\leq 2^{p-1}\left(\|f\|_{p}^{p}+\|g\|_{p}^{p}\right),}$[proof 1] although it is also a consequence of Minkowski's inequality

$${\displaystyle \|f+g\|_{p}\leq \|f\|_{p}+\|g\|_{p}}$$

which establishes that ${\displaystyle \|\cdot \|_{p}}$ satisfies the triangle inequality for ${\displaystyle 1\leq p\leq \infty }$ (the triangle inequality does not hold for ${\displaystyle 0<p<1}$). That ${\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}$ is closed under scalar multiplication is due to ${\displaystyle \|\cdot \|_{p}}$ being absolutely homogeneous, which means that ${\displaystyle \|sf\|_{p}=|s|\|f\|_{p}}$ for every scalar ${\displaystyle s}$ and every function ${\displaystyle f.}$

Absolute homogeneity, the triangle inequality, and non-negativity are the defining properties of a seminorm. Thus ${\displaystyle \|\cdot \|_{p}}$ is a seminorm and the set ${\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}$ of ${\displaystyle p}$-th power integrable functions together with the function ${\displaystyle \|\cdot \|_{p}}$ defines a seminormed vector space. In general, the seminorm ${\displaystyle \|\cdot \|_{p}}$ is not a norm because there might exist measurable functions ${\displaystyle f}$ that satisfy ${\displaystyle \|f\|_{p}=0}$ but are not identically equal to ${\displaystyle 0}$[note 5] (${\displaystyle \|\cdot \|_{p}}$ 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 ${\displaystyle \|f\|_{p}=0}$ for all positive ${\displaystyle p\leq \infty .}$ On the other hand, if ${\displaystyle f}$ is a measurable function for which there exists some ${\displaystyle 0<p\leq \infty }$ such that ${\displaystyle \|f\|_{p}=0}$ then ${\displaystyle f=0}$ almost everywhere. When ${\displaystyle p}$ is finite then this follows from the ${\displaystyle p=1}$ case and the formula ${\displaystyle \|f\|_{p}^{p}=\||f|^{p}\|_{1}}$ mentioned above.

Thus if ${\displaystyle p\leq \infty }$ is positive and ${\displaystyle f}$ is any measurable function, then ${\displaystyle \|f\|_{p}=0}$ 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 ${\displaystyle \|\cdot \|_{p}}$ have the same zero set (it does not depend on ${\displaystyle p}$). So denote this common set by

$${\displaystyle {\mathcal {N}}\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;\{f:f=0\ \mu {\text{-almost everywhere}}\}=\{f\in {\mathcal {L}}^{p}(S,\,\mu ):\|f\|_{p}=0\}\qquad \forall \ p.}$$

This set is a vector subspace of ${\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}$ for every positive ${\displaystyle p\leq \infty .}$

Quotient vector space

Like every seminorm, the seminorm ${\displaystyle \|\cdot \|_{p}}$ induces a norm (defined shortly) on the canonical quotient vector space of ${\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}$ by its vector subspace ${\textstyle {\mathcal {N}}=\{f\in {\mathcal {L}}^{p}(S,\,\mu ):\|f\|_{p}=0\}.}$ This normed quotient space is called Lebesgue space and it is the subject of this article. We begin by defining the quotient vector space.

Given any ${\displaystyle f\in {\mathcal {L}}^{p}(S,\,\mu ),}$ the coset ${\displaystyle f+{\mathcal {N}}\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;\{f+h:h\in {\mathcal {N}}\}}$ consists of all measurable functions ${\displaystyle g}$ that are equal to ${\displaystyle f}$ almost everywhere. The set of all cosets, typically denoted by

$${\displaystyle {\mathcal {L}}^{p}(S,\mu )/{\mathcal {N}}~~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~~\{f+{\mathcal {N}}:f\in {\mathcal {L}}^{p}(S,\mu )\},}$$

forms a vector space with origin ${\displaystyle 0+{\mathcal {N}}={\mathcal {N}}}$ when vector addition and scalar multiplication are defined by ${\displaystyle (f+{\mathcal {N}})+(g+{\mathcal {N}})\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;(f+g)+{\mathcal {N}}}$ and ${\displaystyle s(f+{\mathcal {N}})\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;(sf)+{\mathcal {N}}.}$ This particular quotient vector space will be denoted by ${\displaystyle L^{p}(S,\,\mu )~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\mathcal {L}}^{p}(S,\mu )/{\mathcal {N}}.}$

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.

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

Given any ${\displaystyle f\in {\mathcal {L}}^{p}(S,\,\mu ),}$ the value of the seminorm ${\displaystyle \|\cdot \|_{p}}$ on the coset ${\displaystyle f+{\mathcal {N}}=\{f+h:h\in {\mathcal {N}}\}}$ is constant and equal to ${\displaystyle \|f\|_{p};}$ denote this unique value by ${\displaystyle \|f+{\mathcal {N}}\|_{p},}$ so that:

$${\displaystyle \|f+{\mathcal {N}}\|_{p}\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;\|f\|_{p}.}$$

This assignment ${\displaystyle f+{\mathcal {N}}\mapsto \|f+{\mathcal {N}}\|_{p}}$ defines a map, which will also be denoted by ${\displaystyle \|\cdot \|_{p},}$ on the quotient vector space

$${\displaystyle L^{p}(S,\mu )~~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~~{\mathcal {L}}^{p}(S,\mu )/{\mathcal {N}}~=~\{f+{\mathcal {N}}:f\in {\mathcal {L}}^{p}(S,\mu )\}.}$$

This map is a norm on ${\displaystyle L^{p}(S,\mu )}$ called the ${\displaystyle p}$-norm. The value ${\displaystyle \|f+{\mathcal {N}}\|_{p}}$ 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 ${\displaystyle \|{\mathcal {C}}\|_{p}=\|f\|_{p}}$ 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

The normed vector space ${\displaystyle \left(L^{p}(S,\mu ),\|\cdot \|_{p}\right)}$ is called ${\displaystyle L^{p}}$ space or the Lebesgue space of ${\displaystyle p}$-th power integrable functions and it is a Banach space for every ${\displaystyle 1\leq p\leq \infty }$ (meaning that it is a complete metric space, a result that is sometimes called the Riesz–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 ${\displaystyle \|\cdot \|_{p}}$ 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 ${\displaystyle \left({\mathcal {L}}^{p}(S,\,\mu ),\|\cdot \|_{p}\right)}$ will be linearly isometrically isomorphic to the normed quotient space ${\displaystyle \left(L^{p}(S,\mu ),\|\cdot \|_{p}\right)}$ 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 a linear isometry, the same normed space and so they may both be called "${\displaystyle L^{p}}$ space".

The above definitions generalize to Bochner 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 ${\displaystyle L^{\infty },}$ however, there is a theory of lifts enabling such recovery.

Special cases

Similar to the ${\displaystyle \ell ^{p}}$ spaces, ${\displaystyle L^{2}}$ is the only Hilbert space among ${\displaystyle L^{p}}$ spaces. In the complex case, the inner product on ${\displaystyle L^{2}}$ is defined by

$${\displaystyle \langle f,g\rangle =\int _{S}f(x){\overline {g(x)}}\,\mathrm {d} \mu (x)}$$

The additional inner product structure allows for a richer theory, with applications to, for instance, Fourier series and quantum mechanics. Functions in ${\displaystyle L^{2}}$ are sometimes called square-integrable functions, 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 (Titchmarsh 1976).

If we use complex-valued functions, the space ${\displaystyle L^{\infty }}$ 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 ${\displaystyle L^{\infty }}$ defines a bounded operator on any ${\displaystyle L^{p}}$ space by multiplication.

For ${\displaystyle 1\leq p\leq \infty }$ the ${\displaystyle \ell ^{p}}$ spaces are a special case of ${\displaystyle L^{p}}$ spaces, when ${\displaystyle S=\mathbf {N} )}$ consists of the natural numbers and ${\displaystyle \mu }$ is the counting measure on ${\displaystyle \mathbf {N} .}$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, the space ${\displaystyle \ell ^{p}(\mathbf {Z} )}$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. 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 Hilbertian basis for this particular ${\displaystyle L^{2}.}$

As in the discrete case, if there exists ${\displaystyle q<\infty }$ such that ${\displaystyle f\in L^{\infty }(S,\mu )\cap L^{q}(S,\mu ),}$ then

$${\displaystyle \|f\|_{\infty }=\lim _{p\to \infty }\|f\|_{p}.}$$

Hölder's inequality

Suppose ${\displaystyle p,q,r\in [1,\infty ]}$ satisfy ${\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}={\tfrac {1}{r}}}$ (where ${\displaystyle {\tfrac {1}{\infty }}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~0}$). 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[5]

$${\displaystyle \|fg\|_{r}~\leq ~\|f\|_{p}\,\|g\|_{q}.}$$

This inequality, called Hölder's inequality, is in some sense optimal[5] since if ${\displaystyle r=1}$ (so ${\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}$) and ${\displaystyle f}$ is a measurable function such that

$${\displaystyle \sup _{\|g\|_{q}\leq 1}\,\int _{S}|fg|\,\mathrm {d} \mu ~<~\infty }$$

where the supremum is taken over the closed unit ball of ${\displaystyle L^{q}(S,\mu ),}$ then ${\displaystyle f\in L^{p}(S,\mu )}$ and

$${\displaystyle \|f\|_{p}~=~\sup _{\|g\|_{q}\leq 1}\,\int _{S}fg\,\mathrm {d} \mu .}$$

Minkowski inequality

Minkowski inequality, which states that ${\displaystyle \|\cdot \|_{p}}$ satisfies the triangle inequality, can be generalized: If the measurable function ${\displaystyle F:S_{1}\times S_{2}\to \mathbb {R} }$ is non-negative then for all ${\displaystyle 1\leq p\leq q\leq \infty ,}$[6]

$${\displaystyle \left\|\left\|F(\,\cdot ,s_{2})\right\|_{L^{p}(S_{1},\mu _{1})}\right\|_{L^{q}(S_{2},\mu _{2})}~\leq ~\left\|\left\|F(s_{1},\cdot )\right\|_{L^{q}(S_{2},\mu _{2})}\right\|_{L^{p}(S_{1},\mu _{1})}\ .}$$

Atomic decomposition

If ${\displaystyle 1\leq p<\infty }$ then every non-negative ${\displaystyle f\in L^{p}(\mu )}$ has an atomic decomposition,[7] 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 ${\displaystyle (f_{n})_{n\in \mathbb {Z} },}$ called the atoms, whose supports ${\displaystyle \left(\operatorname {supp} f_{n}\right)_{n\in \mathbb {Z} }}$ are pairwise disjoint sets of measure ${\displaystyle \mu \left(\operatorname {supp} f_{n}\right)\leq 2^{n+1},}$ such that

$${\displaystyle f~=~\sum _{n\in \mathbb {Z} }r_{n}\,f_{n}\,,}$$

and for every integer ${\displaystyle n\in \mathbb {Z} ,}$

$${\displaystyle \|f_{n}\|_{\infty }~\leq ~2^{-{\tfrac {n}{p}}}\,,}$$

and

$${\displaystyle {\tfrac {1}{2}}\|f\|_{p}^{p}~\leq ~\sum _{n\in \mathbb {Z} }r_{n}^{p}~\leq ~2\|f\|_{p}^{p}\,,}$$

and where moreover, the sequence of functions ${\displaystyle (r_{n}f_{n})_{n\in \mathbb {Z} }}$ depends only on ${\displaystyle f}$ (it is independent of ${\displaystyle p}$).[7] These inequalities guarantee that ${\displaystyle \|f_{n}\|_{p}^{p}\leq 2}$ for all integers ${\displaystyle n}$ while the supports of ${\displaystyle (f_{n})_{n\in \mathbb {Z} }}$ being pairwise disjoint implies[7]

$${\displaystyle \|f\|_{p}^{p}~=~\sum _{n\in \mathbb {Z} }r_{n}^{p}\,\|f_{n}\|_{p}^{p}\,.}$$

An atomic decomposition can be explicitly given by first defining for every integer ${\displaystyle n\in \mathbb {Z} ,}$[7]

$${\displaystyle t_{n}=\inf\{t\in \mathbb {R}$$ :\mu (f>t)<2^{n}\}}

(this infimum is attained by ${\displaystyle t_{n};}$ that is, ${\displaystyle \mu (f>t_{n})<2^{n}}$ holds) and then letting

$${\displaystyle r_{n}~=~2^{n/p}\,t_{n}~{\text{ and }}\quad f_{n}~=~{\frac {f}{r_{n}}}\,\mathbf {1} _{(t_{n+1}<f\leq t_{n})}}$$

where ${\displaystyle \mu (f>t)=\mu (\{s:f(s)>t\})}$ denotes the measure of the set ${\displaystyle (f>t):=\{s\in S:f(s)>t\}}$ and ${\displaystyle \mathbf {1} _{(t_{n+1}<f\leq t_{n})}}$ denotes the indicator function of the set ${\displaystyle (t_{n+1}<f\leq t_{n}):=\{s\in S:t_{n+1}<f(s)\leq t_{n}\}.}$ The sequence ${\displaystyle (t_{n})_{n\in \mathbb {Z} }}$ is decreasing and converges to ${\displaystyle 0}$ as ${\displaystyle n\to \infty .}$[7] Consequently, if ${\displaystyle t_{n}=0}$ then ${\displaystyle t_{n+1}=0}$ and ${\displaystyle (t_{n+1}<f\leq t_{n})=\varnothing }$ so that ${\displaystyle f_{n}={\frac {1}{r_{n}}}\,f\,\mathbf {1} _{(t_{n+1}<f\leq t_{n})}}$ is identically equal to ${\displaystyle 0}$ (in particular, the division ${\displaystyle {\tfrac {1}{r_{n}}}}$ by ${\displaystyle r_{n}=0}$ causes no issues).

The complementary cumulative distribution function ${\displaystyle t\in \mathbb {R} \mapsto \mu (|f|>t)}$ of ${\displaystyle |f|=f}$ that was used to define the ${\displaystyle t_{n}}$ also appears in the definition of the weak ${\displaystyle L^{p}}$-norm (given below) and can be used to express the ${\displaystyle p}$-norm ${\displaystyle \|\cdot \|_{p}}$ (for ${\displaystyle 1\leq p<\infty }$) of ${\displaystyle f\in L^{p}(S,\mu )}$ as the integral[7]

$${\displaystyle \|f\|_{p}^{p}~=~p\,\int _{0}^{\infty }t^{p-1}\mu (|f|>t)\,\mathrm {d} t\,,}$$

where the integration is with respect to the usual Lebesgue measure on ${\displaystyle (0,\infty ).}$