Lp space

ℓp spaces and sequence spaces

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 are absolutely convergent,
• ${\displaystyle {\ell }^2,}$ the space of square-summable sequences, which is a Hilbert space, and
• ${\displaystyle {\ell }^{\mathrm{\infty }},}$ the space of bounded 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 infinite sequences of real (or complex) numbers are given by: $${\displaystyle \begin{array}{cc}& (x_1,x_2,\dots ,x_n,x_{n+1},\dots )+(y_1,y_2,\dots ,y_n,y_{n+1},\dots )\\ =& (x_1+y_1,x_2+y_2,\dots ,x_n+y_n,x_{n+1}+y_{n+1},\dots ),\\ & \lambda \cdot \left(x_1,x_2,\dots ,x_n,x_{n+1},\dots \right)\\ =& (\lambda x_1,\lambda x_2,\dots ,\lambda x_n,\lambda x_{n+1},\dots ).\end{array}}$$

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 the series 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 ${\displaystyle 1\le p<\mathrm{\infty }.}$ 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 ${\displaystyle p=1}$ (the harmonic series), but is convergent for ${\displaystyle p>1.}$

One also defines the ${\displaystyle \mathrm{\infty }}$-norm using the supremum: $${\displaystyle \Vert x{\Vert }_{\mathrm{\infty }}=sup(|x_1|,|x_2|,\dots ,|x_n|,|x_{n+1}|,\dots )}$$ and the corresponding space ${\displaystyle {\ell }^{\mathrm{\infty }}}$ of all bounded sequences. It turns out that $${\displaystyle \Vert x{\Vert }_{\mathrm{\infty }}=\underset{p\to \mathrm{\infty }}{lim}\Vert x{\Vert }_p}$$ 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 a Banach space.

General ℓp-space

In complete analogy to the preceding definition one can define the space ${\displaystyle {\ell }^p(I)}$ over a general index set ${\displaystyle I}$ (and ${\displaystyle 1\le p<\mathrm{\infty }}$) as $${\displaystyle {\ell }^p(I)=\left\{(x_i{)}_{i\in I}\in {\mathbb{K}}^I:\sum _{i\in I}|x_i{|}^p<+\mathrm{\infty }\right\},}$$ where convergence on the right requires that only countably many summands are nonzero (see also Absolute 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 the locally convex direct limit of ${\displaystyle {\ell }^p}$-sequence spaces.

For ${\displaystyle p=2,}$ the ${\displaystyle \Vert \cdot {\Vert }_2}$-norm is even induced by a canonical inner product ${\displaystyle ⟨\cdot ,\cdot ⟩,}$ called the Euclidean 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 the polarization 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 $${\displaystyle {\ell }^{\mathrm{\infty }}(I)=\{x\in {\mathbb{K}}^I:sup\mathrm{range}|x|<+\mathrm{\infty }\},}$$ where for all ${\displaystyle x}$ $${\displaystyle \Vert x{\Vert }_{\mathrm{\infty }}\equiv inf\{C\in {\mathbb{R}}_{\ge 0}:|x_i|\le C\text{ for all }i\in I\}=\{\begin{array}{ll}sup\mathrm{range}|x|& \text{if }X\ne \varnothing ,\\ 0& \text{if }X=\varnothing .\end{array}}$$

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

Lp spaces and Lebesgue integrals

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 ${\displaystyle (S,\mathrm{\Sigma },\mu )}$ be a measure space and ${\displaystyle 1\le p\le \mathrm{\infty }.}$ When ${\displaystyle p\ne \mathrm{\infty }}$, 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: $${\displaystyle \Vert f{\Vert }_p\stackrel{\text{def}}{=}{\left({\int }_S|f{|}^p\mathrm{d}\mu \right)}^{1/p}<\mathrm{\infty }.}$$

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 be equal almost 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 its absolute value) is bounded (or dominated) 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 the infimum 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 the essential supremum of the absolute value of ${\displaystyle f}$: $${\displaystyle \Vert f{\Vert }_{\mathrm{\infty }}=\{\begin{array}{ll}\mathrm{esssup}|f|& \text{if }\mu (S)>0,\\ 0& \text{if }\mu (S)=0.\end{array}}$$

For example, if ${\displaystyle f}$ is a measurable function that is equal to ${\displaystyle 0}$ almost everywhere 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 ${\displaystyle 0<p\le \mathrm{\infty }}$ (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 a vector space when addition and scalar multiplication are defined pointwise. 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 $\Vert f+g{\Vert }_p^p\le 2^{p-1}\left(\Vert f{\Vert }_p^p+\Vert g{\Vert }_p^p\right),$ although it is also a consequence of Minkowski'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 the triangle inequality for ${\displaystyle 1\le p\le \mathrm{\infty }}$ (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 ${\displaystyle \Vert \cdot {\Vert }_p}$ being absolutely 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, the triangle inequality, and non-negativity are the defining properties of a seminorm. 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 a seminormed vector space. In general, the seminorm ${\displaystyle \Vert \cdot {\Vert }_p}$ is not a norm because there might exist measurable functions ${\displaystyle f}$ that satisfy ${\displaystyle \Vert f{\Vert }_p=0}$ but are not identically equal to ${\displaystyle 0}$ (${\displaystyle \Vert \cdot {\Vert }_p}$ 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 ${\displaystyle 0<p\le \mathrm{\infty }}$ 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 same zero 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 every seminorm, the seminorm ${\displaystyle \Vert \cdot {\Vert }_p}$ induces a norm (defined shortly) on the canonical quotient 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 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 ${\displaystyle f+\mathcal{N}\stackrel{\text{def}}{=}\{f+h:h\in \mathcal{N}\}}$ 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 of equivalence classes of functions.

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 the coset ${\displaystyle f+\mathcal{N}=\{f+h:h\in \mathcal{N}\}}$ 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 the quotient 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 a norm 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

The normed vector space ${\displaystyle \left(L^p(S,\mu ),\Vert \cdot {\Vert }_p\right)}$ 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 ${\displaystyle 1\le p\le \mathrm{\infty }}$ (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 ${\displaystyle \Vert \cdot {\Vert }_p}$ 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 be linearly isometrically 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 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 ${\displaystyle L^{\mathrm{\infty }},}$ however, there is a theory of lifts enabling such recovery.

Special cases

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 the natural numbers ${\displaystyle \mathbb{N}}$ and ${\displaystyle \mu }$ is the counting 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 only Hilbert 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 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).

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 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 ${\displaystyle L^{\mathrm{\infty }}}$ defines a bounded operator on any ${\displaystyle L^p}$ space by multiplication.

When 0 < p < 1

If ${\displaystyle 0<p<1,}$ then ${\displaystyle L^p(\mu )}$ can be defined as above, that is: $${\displaystyle N_p(f)={\int }_S|f{|}^{p}d\mu <\mathrm{\infty }.}$$ 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 a quasi-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 is complete, but not locally convex.

In this setting ${\displaystyle L^p}$ satisfies a reverse 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 prove Clarkson's inequalities, which are in turn used to establish the uniform convexity of the spaces ${\displaystyle L^p}$ for ${\displaystyle 1<p<\mathrm{\infty }}$ (Adams & Fournier 2003).

The space ${\displaystyle L^p}$ for ${\displaystyle 0<p<1}$ is an F-space: it admits a complete translation-invariant metric with respect to which the vector space operations are continuous. It is the prototypical example of an F-space that, for most reasonable measure spaces, is not locally 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 ${\displaystyle L^p([0,1]);}$ the continuous dual space is the zero space. In the case of the counting 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 ${\displaystyle 0<p<1,}$ it is common to work with the Hardy space H^p 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^p for ${\displaystyle p<1}$ (Duren 1970, §7.5).

When p = 0

For a finite measure space ${\displaystyle (X,\mu )}$, ${\displaystyle L^0(\mu )}$ is defined as the space of all ${\displaystyle \mu }$-measurable functions, with functions identified if they are equal almost everywhere. The F-norm is defined by $${\displaystyle \Vert f{\Vert }_0={\int }_X\frac{|f(x)|}{1+|f(x)|}d\mu (x).}$$ This is not a true norm. As with ${\displaystyle 0<p<1}$, the topology it defines is not locally convex.

The space ${\displaystyle {\ell }_0}$ sometimes denotes the space of all sequences, with the F-norm $${\displaystyle \Vert f{\Vert }_0=\sum _n{2}^{-n}\frac{|f_n|}{1+|f_n|}.}$$ This is a special case of ${\displaystyle L^0}$ with a weighted measure on the positive integers.

Properties

Hölder's inequality

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 $${\displaystyle \Vert fg{\Vert }_r\le \Vert f{\Vert }_p\Vert g{\Vert }_q.}$$

This inequality, called Hölder's inequality, is in some sense optimal since if ${\displaystyle r=1}$ and ${\displaystyle f}$ is a measurable function such that $${\displaystyle \underset{\Vert g{\Vert }_q\le 1}{sup}{\int }_S|fg|\mathrm{d}\mu <\mathrm{\infty }}$$ 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 $${\displaystyle \Vert f{\Vert }_p=\underset{\Vert g{\Vert }_q\le 1}{sup}{\int }_{S}fg\mathrm{d}\mu .}$$

Generalized Minkowski inequality

Minkowski inequality, which states that ${\displaystyle \Vert \cdot {\Vert }_p}$ satisfies the triangle 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 ${\displaystyle 1\le p\le q\le \mathrm{\infty },}$ $${\displaystyle {\Vert {\Vert F(\cdot ,n)\Vert }_{L^p(M,\mu )}\Vert }_{L^q(N,\nu )}\le {\Vert {\Vert F(m,\cdot )\Vert }_{L^q(N,\nu )}\Vert }_{L^p(M,\mu )}.}$$

Atomic decomposition

If ${\displaystyle 1\le p<\mathrm{\infty }}$ then every non-negative ${\displaystyle f\in L^p(\mu )}$ has an atomic decomposition, 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 ${\displaystyle {\left(\mathrm{supp}{f}_n\right)}_{n\in \mathbb{Z}}}$ are pairwise 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 ${\displaystyle p}$). 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 $${\displaystyle \Vert f{\Vert }_p^p=\sum _{n\in \mathbb{Z}}{r}_n^p\Vert f_n{\Vert }_p^p.}$$

An atomic decomposition can be explicitly given by first defining for every integer ${\displaystyle n\in \mathbb{Z},}$ $${\displaystyle t_n=inf\{t\in \mathbb{R}:\mu (f>t)<2^n\}}$$ and then letting $${\displaystyle r_n=2^{n/p}{t}_n\text{ and }{f}_n=\frac{f}{r_n}{\mathbf{1}}_{(t_{n+1}<f\le t_n)}}$$ 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 ${\displaystyle {\mathbf{1}}_{(t_{n+1}<f\le t_n)}}$ denotes the indicator function of the set ${\displaystyle (t_{n+1}<f\le t_n):=\{s\in S:t_{n+1}<f(s)\le t_n\}.}$ The sequence ${\displaystyle (t_n{)}_{n\in \mathbb{Z}}}$ is decreasing and converges to ${\displaystyle 0}$ as ${\displaystyle n\to \mathrm{\infty }.}$ Consequently, if ${\displaystyle t_n=0}$ then ${\displaystyle t_{n+1}=0}$ and ${\displaystyle (t_{n+1}<f\le t_n)=\varnothing }$ so that ${\displaystyle f_n=\frac{1}{r_n}f{\mathbf{1}}_{(t_{n+1}<f\le t_n)}}$ is identically equal to ${\displaystyle 0}$ (in particular, the division ${\displaystyle \frac{1}{r_n}}$ 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 ${\displaystyle \Vert \cdot {\Vert }_p}$ (for ${\displaystyle 1\le p<\mathrm{\infty }}$) of ${\displaystyle f\in L^p(S,\mu )}$ as the integral $${\displaystyle \Vert f{\Vert }_p^p=p{\int }_0^{\mathrm{\infty }}{t}^{p-1}\mu (|f|>t)\mathrm{d}t,}$$ where the integration is with respect to the usual Lebesgue measure on ${\displaystyle (0,\mathrm{\infty }).}$

Dual spaces

The dual space of ${\displaystyle L^p(\mu )}$ for ${\displaystyle 1<p<\mathrm{\infty }}$ has a natural isomorphism with ${\displaystyle L^q(\mu ),}$ where ${\displaystyle q}$ is such that ${\displaystyle \frac{1}{p}+\frac{1}{q}=1}$. This isomorphism associates ${\displaystyle g\in L^q(\mu )}$ with the functional ${\displaystyle {\kappa }_p(g)\in L^p(\mu {)}^{\ast }}$ defined by $${\displaystyle f\mapsto {\kappa }_p(g)(f)=\int fg\mathrm{d}\mu }$$ for every ${\displaystyle f\in L^p(\mu ).}$

${\displaystyle {\kappa }_p:L^q(\mu )\to L^p(\mu {)}^{\ast }}$ is a well defined continuous linear mapping which is an isometry by the extremal case of Hölder's inequality. If ${\displaystyle (S,\mathrm{\Sigma },\mu )}$ is a ${\displaystyle \sigma }$-finite measure space one can use the Radon–Nikodym theorem to show that any ${\displaystyle G\in L^p(\mu {)}^{\ast }}$ can be expressed this way, i.e., ${\displaystyle {\kappa }_p}$ is an isometric isomorphism of Banach spaces. Hence, it is usual to say simply that ${\displaystyle L^q(\mu )}$ is the continuous dual space of ${\displaystyle L^p(\mu ).}$

For ${\displaystyle 1<p<\mathrm{\infty },}$ the space ${\displaystyle L^p(\mu )}$ is reflexive. Let ${\displaystyle {\kappa }_p}$ be as above and let ${\displaystyle {\kappa }_q:L^p(\mu )\to L^q(\mu {)}^{\ast }}$ be the corresponding linear isometry. Consider the map from ${\displaystyle L^p(\mu )}$ to ${\displaystyle L^p(\mu {)}^{\ast \ast },}$ obtained by composing ${\displaystyle {\kappa }_q}$ with the transpose (or adjoint) of the inverse of ${\displaystyle {\kappa }_p:}$

$${\displaystyle j_p:L^p(\mu )\stackrel{{\kappa }_q}{⟶}{L}^q(\mu {)}^{\ast }\stackrel{{\left({\kappa }_p^{-1}\right)}^{\ast }}{⟶}{L}^p(\mu {)}^{\ast \ast }}$$

This map coincides with the canonical embedding ${\displaystyle J}$ of ${\displaystyle L^p(\mu )}$ into its bidual. Moreover, the map ${\displaystyle j_p}$ is onto, as composition of two onto isometries, and this proves reflexivity.

If the measure ${\displaystyle \mu }$ on ${\displaystyle S}$ is sigma-finite, then the dual of ${\displaystyle L^1(\mu )}$ is isometrically isomorphic to ${\displaystyle L^{\mathrm{\infty }}(\mu )}$ (more precisely, the map ${\displaystyle {\kappa }_1}$ corresponding to ${\displaystyle p=1}$ is an isometry from ${\displaystyle L^{\mathrm{\infty }}(\mu )}$ onto ${\displaystyle L^1(\mu {)}^{\ast }.}$

The dual of ${\displaystyle L^{\mathrm{\infty }}(\mu )}$ is subtler. Elements of ${\displaystyle L^{\mathrm{\infty }}(\mu {)}^{\ast }}$ can be identified with bounded signed finitely additive measures on ${\displaystyle S}$ that are absolutely continuous with respect to ${\displaystyle \mu .}$ See ba space for more details. If we assume the axiom of choice, this space is much bigger than ${\displaystyle L^1(\mu )}$ except in some trivial cases. However, Saharon Shelah proved that there are relatively consistent extensions of Zermelo–Fraenkel set theory (ZF + DC + "Every subset of the real numbers has the Baire property") in which the dual of ${\displaystyle {\ell }^{\mathrm{\infty }}}$ is ${\displaystyle {\ell }^1.}$

Embeddings

Colloquially, if ${\displaystyle 1\le p<q\le \mathrm{\infty },}$ then ${\displaystyle L^p(S,\mu )}$ contains functions that are more locally singular, while elements of ${\displaystyle L^q(S,\mu )}$ can be more spread out. Consider the Lebesgue measure on the half line ${\displaystyle (0,\mathrm{\infty }).}$ A continuous function in ${\displaystyle L^1}$ might blow up near ${\displaystyle 0}$ but must decay sufficiently fast toward infinity. On the other hand, continuous functions in ${\displaystyle L^{\mathrm{\infty }}}$ need not decay at all but no blow-up is allowed. More formally:

1. If ${\displaystyle 0<p<q<\mathrm{\infty }}$: ${\displaystyle L^q(S,\mu )\subseteq L^p(S,\mu )}$ if and only if ${\displaystyle S}$ does not contain sets of finite but arbitrarily large measure (e.g. any finite measure).
2. If ${\displaystyle 0<p<q\le \mathrm{\infty }}$: ${\displaystyle L^p(S,\mu )\subseteq L^q(S,\mu )}$ if and only if ${\displaystyle S}$ does not contain sets of non-zero but arbitrarily small measure (e.g. the counting measure).

Neither condition holds for the Lebesgue measure on the real line while both conditions holds for the counting measure on any finite set. As a consequence of the closed graph theorem, the embedding is continuous, i.e., the identity operator is a bounded linear map from ${\displaystyle L^q}$ to ${\displaystyle L^p}$ in the first case and ${\displaystyle L^p}$ to ${\displaystyle L^q}$ in the second. Indeed, if the domain ${\displaystyle S}$ has finite measure, one can make the following explicit calculation using Hölder's inequality $${\displaystyle \Vert \mathbf{1}{f}^p{\Vert }_1\le \Vert \mathbf{1}{\Vert }_{q/(q-p)}\Vert f^p{\Vert }_{q/p}}$$ leading to $${\displaystyle \Vert f{\Vert }_p\le \mu (S{)}^{1/p-1/q}\Vert f{\Vert }_q.}$$

The constant appearing in the above inequality is optimal, in the sense that the operator norm of the identity ${\displaystyle I:L^q(S,\mu )\to L^p(S,\mu )}$ is precisely $${\displaystyle \Vert I{\Vert }_{q,p}=\mu (S{)}^{1/p-1/q}}$$ the case of equality being achieved exactly when ${\displaystyle f=1}$ ${\displaystyle \mu }$-almost-everywhere.

Dense subspaces

Let ${\displaystyle 1\le p<\mathrm{\infty }}$ and ${\displaystyle (S,\mathrm{\Sigma },\mu )}$ be a measure space and consider an integrable simple function ${\displaystyle f}$ on ${\displaystyle S}$ given by $${\displaystyle f=\sum _{j=1}^n{a}_j{\mathbf{1}}_{A_j},}$$ where ${\displaystyle a_j}$ are scalars, ${\displaystyle A_j\in \mathrm{\Sigma }}$ has finite measure and ${\displaystyle {\mathbf{1}}_{A_j}}$ is the indicator function of the set ${\displaystyle A_j,}$ for ${\displaystyle j=1,\dots ,n.}$ By construction of the integral, the vector space of integrable simple functions is dense in ${\displaystyle L^p(S,\mathrm{\Sigma },\mu ).}$

More can be said when ${\displaystyle S}$ is a normal topological space and ${\displaystyle \mathrm{\Sigma }}$ its Borel 𝜎–algebra.

Suppose ${\displaystyle V\subseteq S}$ is an open set with ${\displaystyle \mu (V)<\mathrm{\infty }.}$ Then for every Borel set ${\displaystyle A\in \mathrm{\Sigma }}$ contained in ${\displaystyle V}$ there exist a closed set ${\displaystyle F}$ and an open set ${\displaystyle U}$ such that $${\displaystyle F\subseteq A\subseteq U\subseteq V\text{and}\mu (U\setminus F)=\mu (U)-\mu (F)<\epsilon ,}$$ for every ${\displaystyle \epsilon >0}$. Subsequently, there exists a Urysohn function ${\displaystyle 0\le \phi \le 1}$ on ${\displaystyle S}$ that is ${\displaystyle 1}$ on ${\displaystyle F}$ and ${\displaystyle 0}$ on ${\displaystyle S\setminus U,}$ with $${\displaystyle {\int }_S|{\mathbf{1}}_A-\phi |\mathrm{d}\mu <\epsilon .}$$

If ${\displaystyle S}$ can be covered by an increasing sequence ${\displaystyle (V_n)}$ of open sets that have finite measure, then the space of ${\displaystyle p}$–integrable continuous functions is dense in ${\displaystyle L^p(S,\mathrm{\Sigma },\mu ).}$ More precisely, one can use bounded continuous functions that vanish outside one of the open sets ${\displaystyle V_n.}$

This applies in particular when ${\displaystyle S={\mathbb{R}}^d}$ and when ${\displaystyle \mu }$ is the Lebesgue measure. For example, the space of continuous and compactly supported functions as well as the space of integrable step functions are dense in ${\displaystyle L^p({\mathbb{R}}^d)}$.

Closed subspaces

If ${\displaystyle 0<p<\mathrm{\infty }}$ is any positive real number, ${\displaystyle \mu }$ is a probability measure on a measurable space ${\displaystyle (S,\mathrm{\Sigma })}$ (so that ${\displaystyle L^{\mathrm{\infty }}(\mu )\subseteq L^p(\mu )}$), and ${\displaystyle V\subseteq L^{\mathrm{\infty }}(\mu )}$ is a vector subspace, then ${\displaystyle V}$ is a closed subspace of ${\displaystyle L^p(\mu )}$ if and only if ${\displaystyle V}$ is finite-dimensional (${\displaystyle V}$ was chosen independent of ${\displaystyle p}$). In this theorem, which is due to Alexander Grothendieck, it is crucial that the vector space ${\displaystyle V}$ be a subset of ${\displaystyle L^{\mathrm{\infty }}}$ since it is possible to construct an infinite-dimensional closed vector subspace of ${\displaystyle L^1\left(S^1,\frac{1}{2\pi }\lambda \right)}$ (which is even a subset of ${\displaystyle L^4}$), where ${\displaystyle \lambda }$ is Lebesgue measure on the unit circle ${\displaystyle S^1}$ and ${\displaystyle \frac{1}{2\pi }\lambda }$ is the probability measure that results from dividing it by its mass ${\displaystyle \lambda (S^1)=2\pi .}$

Applications

Statistics

In statistics, measures of central tendency and statistical dispersion, such as the mean, median, and standard deviation, can be defined in terms of ${\displaystyle L^p}$ metrics, and measures of central tendency can be characterized as solutions to variational problems.

In penalized regression, "L1 penalty" and "L2 penalty" refer to penalizing either the ${\displaystyle L^1}$ norm of a solution's vector of parameter values (i.e. the sum of its absolute values), or its squared ${\displaystyle L^2}$ norm (its Euclidean length). Techniques which use an L1 penalty, like LASSO, encourage sparse solutions (where the many parameters are zero). Elastic net regularization uses a penalty term that is a combination of the ${\displaystyle L^1}$ norm and the squared ${\displaystyle L^2}$ norm of the parameter vector.

Hausdorff–Young inequality

The Fourier transform for the real line (or, for periodic functions, see Fourier series), maps ${\displaystyle L^p(\mathbb{R})}$ to ${\displaystyle L^q(\mathbb{R})}$ (or ${\displaystyle L^p(\mathbf{T})}$ to ${\displaystyle {\ell }^q}$) respectively, where ${\displaystyle 1\le p\le 2}$ and ${\displaystyle \frac{1}{p}+\frac{1}{q}=1.}$ This is a consequence of the Riesz–Thorin interpolation theorem, and is made precise with the Hausdorff–Young inequality.