Editing Lp space

{{DISPLAYTITLE:''L''<sup>''p''</sup> space}}
{{Short description|Function spaces generalizing finite-dimensional p norm spaces}}
In [[mathematics]], the '''{{math|''L''<sup>''p''</sup>}} spaces''' are [[function space]]s defined using a natural generalization of the [[Norm (mathematics)#p-norm|{{math|''p''}}-norm]] for finite-dimensional [[vector space]]s. They are sometimes called '''Lebesgue spaces''', named after [[Henri Lebesgue]] {{harv|Dunford|Schwartz|1958|loc=III.3}}, although according to the [[Nicolas Bourbaki|Bourbaki]] group {{harv|Bourbaki|1987}} they were first introduced by [[Frigyes Riesz]] {{harv|Riesz|1910}}. 

{{math|''L''<sup>''p''</sup>}} spaces form an important class of [[Banach space]]s in [[functional analysis]], and of [[topological vector space]]s. 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.
==Preliminaries==
===The {{math|''p''}}-norm in finite dimensions===
[[Image:Vector-p-Norms qtl1.svg|thumb|right|Illustrations of [[unit circle]]s (see also [[superellipse]]) in <math>\Reals^2</math> based on different <math>p</math>-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 <math>p</math>).]]

The Euclidean length of a vector <math>x = (x_1, x_2, \dots, x_n)</math> in the <math>n</math>-dimensional [[real number|real]] [[vector space]] <math>\Reals^n</math> is given by the [[Euclidean norm]]:
<math display="block">\|x\|_2 = \left({x_1}^2 + {x_2}^2 + \dotsb + {x_n}^2\right)^{1/2}.</math>

The Euclidean distance between two points <math>x</math> and <math>y</math> is the length <math>\|x - y\|_2</math> 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 the [[taxicab geometry|rectilinear distance]], which takes into account that streets are either orthogonal or parallel to each other. The class of <math>p</math>-norms generalizes these two examples and has an abundance of applications in many parts of [[mathematics]], [[physics]], and [[computer science]].

For a [[real number]] <math>p \geq 1,</math> the '''<math>p</math>-norm''' or '''<math>L^p</math>-norm''' of <math>x</math> is defined by
<math display="block">\|x\|_p = \left(|x_1|^p + |x_2|^p + \dotsb + |x_n|^p\right)^{1/p}.</math>
The absolute value bars can be dropped when <math>p</math> is a rational number with an even numerator in its reduced form, and <math>x</math> 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 <math>2</math>-norm, and the <math>1</math>-norm is the norm that corresponds to the [[taxicab geometry|rectilinear distance]].

The '''<math>L^\infty</math>-norm''' or [[Chebyshev distance|maximum norm]] (or uniform norm) is the limit of the <math>L^p</math>-norms for <math>p \to \infty</math>, given by:
<math display="block">\|x\|_\infty = \max \left\{|x_1|, |x_2|, \dotsc, |x_n|\right\}</math>

For all <math>p \geq 1,</math> the <math>p</math>-norms and maximum norm satisfy the properties of a "length function" (or [[norm (mathematics)|norm]]), that is:
*only the zero vector has zero length,
*the length of the vector is positive homogeneous with respect to multiplication by a scalar ([[Euler's homogeneous function theorem|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 <math>\Reals^n</math> together with the <math>p</math>-norm is a [[normed vector space]]. Moreover, it turns out that this space is [[Complete_metric_space|complete]], thus making it a [[Banach space]].

====Relations between {{math|''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:
<math display="block">\|x\|_2 \leq \|x\|_1 .</math>

This fact generalizes to <math>p</math>-norms in that the <math>p</math>-norm <math>\|x\|_p</math> of any given vector <math>x</math> does not grow with <math>p</math>:
{{block indent | em = 1.5 | text = <math>\|x\|_{p+a} \leq \|x\|_p</math> for any vector <math>x</math> and real numbers <math>p \geq 1</math> and <math>a \geq 0.</math> (In fact this remains true for <math>0 < p < 1</math> and <math>a \geq 0</math> .)}}

For the opposite direction, the following relation between the <math>1</math>-norm and the <math>2</math>-norm is known:
<math display="block">\|x\|_1 \leq \sqrt{n} \|x\|_2 ~.</math>

This inequality depends on the dimension <math>n</math> of the underlying vector space and follows directly from the [[Cauchy–Schwarz inequality]].

In general, for vectors in <math>\Complex^n</math> where <math>0 < r < p:</math>
<math display="block">\|x\|_p \leq \|x\|_r \leq n^{\frac{1}{r} - \frac{1}{p}} \|x\|_p ~.</math>

This is a consequence of [[Hölder's inequality]].

====When {{math|0 < ''p'' < 1}}====

[[Image:Astroid.svg|thumb|right|[[Astroid]], unit circle in <math>p = \tfrac{2}{3}</math> metric]]
In <math>\Reals^n</math> for <math>n > 1,</math> the formula
<math display="block">\|x\|_p = \left(|x_1|^p + |x_2| ^p + \cdots + |x_n|^p\right)^{1/p}</math>
defines an absolutely [[homogeneous function]] for <math>0 < p < 1;</math> however, the resulting function does not define a norm, because it is not [[subadditivity|subadditive]]. On the other hand, the formula
<math display="block">|x_1|^p + |x_2|^p + \dotsb + |x_n|^p</math>
defines a subadditive function at the cost of losing absolute homogeneity. It does define an [[F-space|F-norm]], though, which is homogeneous of degree <math>p.</math>

Hence, the function
<math display="block">d_p(x, y) = \sum_{i=1}^n |x_i - y_i|^p</math>
defines a [[metric space|metric]]. The [[metric space]] <math>(\Reals^n, d_p)</math> is denoted by <math>\ell_n^p.</math>

Although the <math>p</math>-unit ball <math>B_n^p</math> around the origin in this metric is "concave", the topology defined on <math>\Reals^n</math> by the metric <math>B_p</math> is the usual vector space topology of <math>\Reals^n,</math> hence <math>\ell_n^p</math> is a [[locally convex]] topological vector space. Beyond this qualitative statement, a quantitative way to measure the lack of convexity of <math>\ell_n^p</math> is to denote by <math>C_p(n)</math> the smallest constant <math>C</math> such that the scalar multiple <math>C \, B_n^p</math> of the <math>p</math>-unit ball contains the convex hull of <math>B_n^p,</math> which is equal to <math>B_n^1.</math> The fact that for fixed <math>p < 1</math> we have
<math display="block">C_p(n) = n^{\tfrac{1}{p} - 1} \to \infty, \quad \text{as } n \to \infty</math>
shows that the infinite-dimensional sequence space <math>\ell^p</math> defined below, is no longer locally convex.{{citation needed|date=November 2015}}

====When {{math|1=''p'' = 0}}====

There is one <math>\ell_0</math> norm and another function called the <math>\ell_0</math> "norm" (with quotation marks).

The mathematical definition of the <math>\ell_0</math> norm was established by [[Stefan Banach|Banach]]'s ''[[Theory of Linear Operations]]''. The [[F-space|space]] of sequences has a complete metric topology provided by the [[F-space|F-norm]] on the [[Metric_space#Product_metric_spaces|product metric]]:{{Citation needed|date=December 2024}}
<math display="block">(x_n) \mapsto \|x\|:=d(0,x)=\sum_n 2^{-n} \frac{|x_n|}{1 +|x_n|}.</math> 
The <math>\ell_0</math>-normed space is studied in functional analysis, probability theory, and harmonic analysis.

Another function was called the <math>\ell_0</math> "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 <math>x.</math>{{Citation needed|date=September 2022}} Many authors [[abuse of terminology|abuse terminology]] by omitting the quotation marks. Defining [[zero to the power of zero|<math>0^0 = 0,</math>]] the zero "norm" of <math>x</math> is equal to
<math display="block">|x_1|^0 + |x_2|^0 + \cdots + |x_n|^0 .</math>

[[File:Lp space animation.gif|alt=An animated gif of unit circles in p-norms 0.1 through 2 with a step of 0.05.|thumb|An animated gif of p-norms 0.1 through 2 with a step of 0.05.]]
This is not a [[norm (mathematics)|norm]] because it is not [[Homogeneous function|homogeneous]]. For example, scaling the vector <math>x</math> 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.

=={{math|''ℓ''{{i sup|''p''}}}} spaces and sequence spaces==
{{Details|Sequence space}}
The <math>p</math>-norm can be extended to vectors that have an infinite number of components ([[sequence]]s), which yields the space <math>\ell^p.</math> This contains as special cases:
* <math>\ell^1,</math> the space of sequences whose series are [[absolute convergence|absolutely convergent]],
* <math>\ell^2,</math> the space of '''square-summable''' sequences, which is a [[Hilbert space]], and
* <math>\ell^\infty,</math> the space of [[bounded sequence]]s.

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 [[sequence]]s of real (or [[complex number|complex]]) numbers are given by:
<math display="block">\begin{align}
& (x_1, x_2, \ldots, x_n, x_{n+1},\ldots)+(y_1, y_2, \ldots, y_n, y_{n+1},\ldots) \\
= {} & (x_1+y_1, x_2+y_2, \ldots, x_n+y_n, x_{n+1}+y_{n+1},\ldots), \\[6pt]
& \lambda \cdot \left (x_1, x_2, \ldots, x_n, x_{n+1},\ldots \right) \\
= {} & (\lambda x_1, \lambda x_2, \ldots, \lambda x_n, \lambda x_{n+1},\ldots).
\end{align}</math>

Define the <math>p</math>-norm:
<math display="block">\|x\|_p = \left(|x_1|^p + |x_2|^p + \cdots +|x_n|^p + |x_{n+1}|^p + \cdots\right)^{1/p}</math>

Here, a complication arises, namely that the [[series (mathematics)|series]] on the right is not always convergent, so for example, the sequence made up of only ones, <math>(1, 1, 1, \ldots),</math> will have an infinite <math>p</math>-norm for <math>1 \leq p < \infty.</math> The space <math>\ell^p</math> is then defined as the set of all infinite sequences of real (or complex) numbers such that the <math>p</math>-norm is finite.

One can check that as <math>p</math> increases, the set <math>\ell^p</math> grows larger. For example, the sequence
<math display="block">\left(1, \frac{1}{2}, \ldots, \frac{1}{n}, \frac{1}{n+1}, \ldots\right)</math>
is not in <math>\ell^1,</math> but it is in <math>\ell^p</math> for <math>p > 1,</math> as the series
<math display="block">1^p + \frac{1}{2^p} + \cdots + \frac{1}{n^p} + \frac{1}{(n+1)^p} + \cdots,</math>
diverges for <math>p = 1</math> (the [[harmonic series (mathematics)|harmonic series]]), but is convergent for <math>p > 1.</math>

One also defines the <math>\infty</math>-norm using the [[supremum]]:
<math display="block">\|x\|_\infty = \sup(|x_1|, |x_2|, \dotsc, |x_n|,|x_{n+1}|, \ldots)</math>
and the corresponding space <math>\ell^\infty</math> of all bounded sequences. It turns out that<ref>{{Citation| last1=Maddox | first1=I. J. | title=Elements of Functional Analysis | publisher=CUP | location=Cambridge | edition=2nd | year=1988}}, page 16</ref>
<math display="block">\|x\|_\infty = \lim_{p \to \infty} \|x\|_p</math>
if the right-hand side is finite, or the left-hand side is infinite. Thus, we will consider <math>\ell^p</math> spaces for <math>1 \leq p \leq \infty.</math>

The <math>p</math>-norm thus defined on <math>\ell^p</math> is indeed a norm, and <math>\ell^p</math> together with this norm is a [[Banach space]].

===General ℓ<sup>''p''</sup>-space===

In complete analogy to the preceding definition one can define the space <math>\ell^p(I)</math> over a general [[index set]] <math>I</math> (and <math>1 \leq p < \infty</math>) as
<math display="block">\ell^p(I) = \left\{(x_i)_{i\in I} \in \mathbb{K}^I : \sum_{i \in I} |x_i|^p < +\infty\right\},</math>
where convergence on the right means that only countably many summands are nonzero (see also [[Unconditional convergence]]).
With the norm
<math display="block">\|x\|_p = \left(\sum_{i\in I} |x_i|^p\right)^{1/p}</math>
the space <math>\ell^p(I)</math> becomes a Banach space.
In the case where <math>I</math> is finite with <math> n</math> elements, this construction yields <math>\Reals^n</math> with the <math>p</math>-norm defined above.
If <math>I</math> is countably infinite, this is exactly the sequence space <math>\ell^p</math> defined above.
For uncountable sets <math>I</math> this is a non-[[Separable space|separable]] Banach space which can be seen as the [[Locally convex topological vector space|locally convex]] [[direct limit]] of <math>\ell^p</math>-sequence spaces.<ref>Rafael Dahmen, Gábor Lukács: ''Long colimits of topological groups I: Continuous maps and homeomorphisms.'' in: ''Topology and its Applications'' Nr. 270, 2020. Example 2.14 </ref>

For <math>p = 2,</math> the <math>\|\,\cdot\,\|_2</math>-norm is even induced by a canonical [[inner product]] <math>\langle \,\cdot,\,\cdot\rangle,</math> called the ''{{visible anchor|Euclidean inner product}}'', which means that <math>\|\mathbf{x}\|_2 = \sqrt{\langle\mathbf{x}, \mathbf{x}\rangle}</math> holds for all vectors <math>\mathbf{x}.</math> This inner product can expressed in terms of the norm by using the [[polarization identity]]. 
On <math>\ell^2,</math> it can be defined by
<math display="block">\langle \left(x_i\right)_{i}, \left(y_n\right)_{i} \rangle_{\ell^2} ~=~ \sum_i x_i \overline{y_i}.</math>
Now consider the case <math>p = \infty.</math> Define{{refn|group=note|The condition <math>\sup\operatorname{range} |x| < + \infty.</math> is not equivalent to <math>\sup\operatorname{range} |x|</math> being finite, unless <math>X \neq \varnothing.</math>}}
<math display="block">\ell^\infty(I)=\{x\in \mathbb K^I : \sup\operatorname{range}|x|<+\infty\},</math>
where for all <math>x</math><ref>{{cite book|last1=Garling|first1=D. J. H.|title=Inequalities: A Journey into Linear Analysis|date=2007|publisher=Cambridge University Press|isbn=978-0-521-87624-7|page=54}}</ref>{{refn|group=note|If <math>X = \varnothing</math> then <math>\sup\operatorname{range} |x| = - \infty.</math>}}
<math display="block">\|x\|_\infty\equiv\inf\{C \in \Reals_{\geq 0}:|x_i| \leq C\text{ for all } i \in I\} = \begin{cases}\sup\operatorname{range}|x|&\text{if } X\neq\varnothing,\\0&\text{if } X=\varnothing.\end{cases}</math>

The index set <math>I</math> can be turned into a [[measure space]] by giving it the [[Σ-algebra#Simple set-based examples|discrete σ-algebra]] and the [[counting measure]]. Then the space <math>\ell^p(I)</math> is just a special case of the more general <math>L^p</math>-space (defined below).

==''L<sup>p</sup>'' spaces and Lebesgue integrals==

An <math>L^p</math> space may be defined as a space of measurable functions for which the <math>p</math>-th power of the [[absolute value]] is [[Lebesgue integrable]], where functions which agree almost everywhere are identified. More generally, let <math>(S, \Sigma, \mu)</math> be a [[measure space]] and <math>1 \leq p \leq \infty.</math><ref group=note>The definitions of <math>\|\cdot\|_p,</math> <math>\mathcal{L}^p(S,\, \mu),</math> and <math>L^p(S,\, \mu)</math> can be extended to all <math>0 < p \leq \infty</math> (rather than just <math>1 \leq p \leq \infty</math>), but it is only when <math>1 \leq p \leq \infty</math> that <math>\|\cdot\|_p</math> is guaranteed to be a norm (although <math>\|\cdot\|_p</math> is a [[quasi-seminorm]] for all <math>0 < p \leq \infty,</math>).</ref> 
When <math>p \neq \infty</math>, consider the set <math>\mathcal{L}^p(S,\, \mu)</math> of all [[measurable function]]s <math>f</math> from <math>S</math> to <math>\Complex</math> or <math>\Reals</math> whose [[absolute value]] raised to the <math>p</math>-th power has a finite integral, or in symbols:{{sfn|Rudin|1987|p=65}}
<math display="block">\|f\|_p ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \left(\int_S |f|^p\;\mathrm{d}\mu\right)^{1/p} < \infty.</math>

To define the set for <math>p = \infty,</math> recall that two functions <math>f</math> and <math>g</math> defined on <math>S</math> are said to be {{em|equal [[almost everywhere]]}}, written {{em|<math>f = g</math> a.e.}}, if the set <math>\{s \in S : f(s) \neq g(s)\}</math> is measurable and has measure zero. 
Similarly, a measurable function <math>f</math> (and its [[absolute value]]) is {{em|bounded}} (or {{em|dominated}}) {{em|almost everywhere}} by a real number <math>C,</math> written {{em|<math>|f| \leq C</math> a.e.}}, if the (necessarily) measurable set <math>\{s \in S : |f(s)| > C\}</math> has measure zero. 
The space <math>\mathcal{L}^\infty(S,\mu)</math> is the set of all measurable functions <math>f</math> that are bounded almost everywhere (by some real <math>C</math>) and <math>\|f\|_\infty</math> is defined as the [[infimum]] of these bounds:
<math display="block">\|f\|_\infty ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \inf \{C \in \Reals_{\geq 0} : |f(s)| \leq C \text{ for almost every } s\}.</math> 
When <math>\mu(S) \neq 0</math> then this is the same as the [[essential supremum]] of the absolute value of <math>f</math>:{{refn|group=note|If <math>\mu(S) = 0</math> then <math>\operatorname{esssup}|f| = -\infty.</math>}}
<math display="block">\|f\|_\infty ~=~ \begin{cases}\operatorname{esssup}|f| & \text{if } \mu(S) > 0,\\ 0 & \text{if } \mu(S) = 0.\end{cases}</math> 

For example, if <math>f</math> is a measurable function that is equal to <math>0</math> almost everywhere<ref group=note name=Non0Value0Example>For example, if a non-empty measurable set <math>N \neq \varnothing</math> of measure <math>\mu(N) = 0</math> exists then its [[indicator function]] <math>\mathbf{1}_N</math> satisfies <math>\|\mathbf{1}_N\|_p = 0</math> although <math>\mathbf{1}_N \neq 0.</math></ref> then <math>\|f\|_p = 0</math> for every <math>p</math> and thus <math>f \in \mathcal{L}^p(S,\, \mu)</math> for all <math>p.</math> 

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

'''Seminormed space of <math>p</math>-th power integrable functions'''

Each set of functions <math>\mathcal{L}^p(S,\, \mu)</math> forms a [[vector space]] when addition and scalar multiplication are defined pointwise.<ref group=note>Explicitly, the vector space operations are defined by:
<math display="block">\begin{align}
(f+g)(x) &= f(x)+g(x), \\
(s f)(x) &= s f(x)
\end{align}</math>
for all <math>f, g \in \mathcal{L}^p(S,\, \mu)</math> and all scalars <math>s.</math> These operations make <math>\mathcal{L}^p(S,\, \mu)</math> into a vector space because if <math>s</math> is any scalar and <math>f, g \in \mathcal{L}^p(S,\, \mu)</math> then both <math>s f</math> and <math>f + g</math> also belong to <math>\mathcal{L}^p(S,\, \mu).</math></ref> 
That the sum of two <math>p</math>-th power integrable functions <math>f</math> and <math>g</math> is again <math>p</math>-th power integrable follows from <math display=inline>\|f + g\|_p^p \leq 2^{p-1} \left(\|f\|_p^p + \|g\|_p^p\right),</math><ref group=proof name=UpperBoundForNormOfSum>When <math>1 \leq p < \infty,</math> the inequality <math>\|f + g\|_p^p \leq 2^{p-1} \left(\|f\|_p^p + \|g\|_p^p\right)</math> can be deduced from the fact that the function <math>F : [0, \infty) \to \Reals</math> defined by <math>F(t) = t^p</math> is [[Convex function|convex]], which by definition means that <math>F(t x + (1 - t) y) \leq t F(x) + (1 - t) F(y)</math> for all <math>0 \leq t \leq 1</math> and all <math>x, y</math> in the domain of <math>F.</math> Substituting <math>|f|, |g|,</math> and <math>\tfrac{1}{2}</math> in for <math>x, y,</math> and <math>t</math> gives <math>\left(\tfrac{1}{2}|f| + \tfrac{1}{2}|g|\right)^p \leq \tfrac{1}{2} |f|^p + \tfrac{1}{2} |g|^p,</math> which proves that <math>(|f| + |g|)^p \leq 2^{p-1} (|f|^p + |g|^p).</math> The triangle inequality <math>|f + g| \leq |f| + |g|</math> now implies <math>|f + g|^p \leq 2^{p-1} (|f|^p + |g|^p).</math> The desired inequality follows by integrating both sides. <math>\blacksquare</math></ref> 
although it is also a consequence of ''[[Minkowski inequality|Minkowski's inequality]]'' 
<math display="block">\|f + g\|_p \leq \|f\|_p + \|g\|_p</math> 
which establishes that <math>\|\cdot\|_p</math> satisfies the [[triangle inequality]] for <math>1 \leq p \leq \infty</math> (the triangle inequality does not hold for <math>0 < p < 1</math>). 
That <math>\mathcal{L}^p(S,\, \mu)</math> is closed under scalar multiplication is due to <math>\|\cdot\|_p</math> being [[Absolute homogeneity|absolutely homogeneous]], which means that <math>\|s f\|_p = |s| \|f\|_p</math> for every scalar <math>s</math> and every function <math>f.</math> 

[[Absolute homogeneity]], the [[triangle inequality]], and non-negativity are the defining properties of a [[seminorm]]. 
Thus <math>\|\cdot\|_p</math> is a seminorm and the set <math>\mathcal{L}^p(S,\, \mu)</math> of <math>p</math>-th power integrable functions together with the function <math>\|\cdot\|_p</math> defines a [[seminormed vector space]]. In general, the [[seminorm]] <math>\|\cdot\|_p</math> is not a [[Norm (mathematics)|norm]] because there might exist measurable functions <math>f</math> that satisfy <math>\|f\|_p = 0</math> but are not {{em|identically}} equal to <math>0</math><ref group=note name=Non0Value0Example /> (<math>\|\cdot\|_p</math> is a norm if and only if no such <math>f</math> exists). 

'''Zero sets of <math>p</math>-seminorms'''

{{anchor|kernel}}
If <math>f</math> is measurable and equals <math>0</math> a.e. then <math>\|f\|_p = 0</math> for all positive <math>p \leq \infty.</math>
On the other hand, if <math>f</math> is a measurable function for which there exists some <math>0 < p \leq \infty</math> such that <math>\|f\|_p = 0</math> then <math>f = 0</math> almost everywhere. When <math>p</math> is finite then this follows from the <math>p = 1</math> case and the formula <math>\|f\|_p^p = \||f|^p\|_1</math> mentioned above. <!--(this formula itself follows from <math>\|f\|_p^r = \|f^r\|_{p/r},</math> which holds whenever <math>f \geq 0</math> is measurable, <math>r > 0</math> is real, and <math>0 < p \leq \infty</math> (where <math>\infty / r \;\stackrel{\scriptscriptstyle\text{def}}{=}\; \infty</math> when <math>p = \infty</math>)). -->

Thus if <math>p \leq \infty</math> is positive and <math>f</math> is any measurable function, then <math>\|f\|_p = 0</math> if and only if <math>f = 0</math> [[almost everywhere]]. Since the right hand side (<math>f = 0</math> a.e.) does not mention <math>p,</math> it follows that all <math>\|\cdot\|_p</math> have the same [[zero set]] (it does not depend on <math>p</math>). So denote this common set by
<math display="block">\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.</math>
This set is a vector subspace of <math>\mathcal{L}^p(S,\, \mu)</math> for every positive <math>p \leq \infty.</math> 

'''Quotient vector space'''

Like every [[seminorm]], the seminorm <math>\|\cdot\|_p</math> induces a [[Norm (mathematics)|norm]] (defined shortly) on the canonical [[Quotient space (linear algebra)|quotient vector space]] of <math>\mathcal{L}^p(S,\, \mu)</math> by its vector subspace 
<math display="inline">\mathcal{N} = \{f \in \mathcal{L}^p(S,\, \mu) : \|f\|_p = 0\}.</math>
This normed quotient space is called {{em|Lebesgue space}} and it is the subject of this article. We begin by defining the quotient vector space.

Given any <math>f \in \mathcal{L}^p(S,\, \mu),</math> the [[coset]] <math>f + \mathcal{N} \;\stackrel{\scriptscriptstyle\text{def}}{=}\; \{f + h : h \in \mathcal{N}\}</math> consists of all measurable functions <math>g</math> that are equal to <math>f</math> [[almost everywhere]]. 
The set of all cosets, typically denoted by 
<math display="block">\mathcal{L}^p(S, \mu) / \mathcal{N} ~~\stackrel{\scriptscriptstyle\text{def}}{=}~~ \{f + \mathcal{N} : f \in \mathcal{L}^p(S, \mu)\},</math>
forms a vector space with origin <math>0 + \mathcal{N} = \mathcal{N}</math> when vector addition and scalar multiplication are defined by <math>(f + \mathcal{N}) + (g + \mathcal{N}) \;\stackrel{\scriptscriptstyle\text{def}}{=}\; (f + g) + \mathcal{N}</math> and <math>s (f + \mathcal{N}) \;\stackrel{\scriptscriptstyle\text{def}}{=}\; (s f) + \mathcal{N}.</math> 
This particular quotient vector space will be denoted by
<math display="block">L^p(S,\, \mu) ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \mathcal{L}^p(S, \mu) / \mathcal{N}.</math>
Two cosets are equal <math>f + \mathcal{N} = g + \mathcal{N}</math> if and only if <math>g \in f + \mathcal{N}</math> (or equivalently, <math>f - g \in \mathcal{N}</math>), which happens if and only if <math>f = g</math> almost everywhere; if this is the case then <math>f</math> and <math>g</math> are identified in the quotient space. Hence, strictly speaking <math>L^p(S,\, \mu) </math> consists of [[equivalence class]]es of functions.{{sfn|Stein|Shakarchi|2012|p=2}}

'''The <math>p</math>-norm on the quotient vector space'''

Given any <math>f \in \mathcal{L}^p(S,\, \mu),</math> the value of the seminorm <math>\|\cdot\|_p</math> on the [[coset]] <math>f + \mathcal{N} = \{f + h : h \in \mathcal{N}\}</math> is constant and equal to <math>\|f\|_p;</math> denote this unique value by <math>\|f + \mathcal{N}\|_p,</math> so that:
<math display=block>\|f + \mathcal{N}\|_p \;\stackrel{\scriptscriptstyle\text{def}}{=}\; \|f\|_p.</math> 
This assignment <math>f + \mathcal{N} \mapsto \|f + \mathcal{N}\|_p</math> defines a map, which will also be denoted by <math>\|\cdot\|_p,</math> on the [[Quotient space (linear algebra)|quotient vector space]] 
<math display="block">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)\}.</math>
This map is a [[Norm (mathematics)|norm]] on <math>L^p(S, \mu)</math> called the {{em|{{visible anchor|p-norm|text=<math>p</math>-norm}}}}. 
The value <math>\|f + \mathcal{N}\|_p</math> of a coset <math>f + \mathcal{N}</math> is independent of the particular function <math>f</math> that was chosen to represent the coset, meaning that if <math>\mathcal{C} \in L^p(S, \mu)</math> is any coset then <math>\|\mathcal{C}\|_p = \|f\|_p</math> for every <math>f \in \mathcal{C}</math> (since <math>\mathcal{C} = f + \mathcal{N}</math> for every <math>f \in \mathcal{C}</math>). 

'''The Lebesgue <math>L^p</math> space'''

The [[normed vector space]] <math>\left(L^p(S, \mu), \|\cdot\|_p\right)</math> is called {{em|<math>L^p</math> space}} or the {{em|Lebesgue space}} of <math>p</math>-th power integrable functions and it is a [[Banach space]] for every <math>1 \leq p \leq \infty</math> (meaning that it is a [[complete metric space]], a result that is sometimes called the [[Riesz–Fischer theorem#Completeness of Lp, 0 < p ≤ ∞|Riesz–Fischer theorem]]). 
When the underlying measure space <math>S</math> is understood then <math>L^p(S, \mu)</math> is often abbreviated <math>L^p(\mu),</math> or even just <math>L^p.</math> 
Depending on the author, the subscript notation <math>L_p</math> might denote either <math>L^p(S, \mu)</math> or <math>L^{1/p}(S, \mu).</math> 

If the seminorm <math>\|\cdot\|_p</math> on <math>\mathcal{L}^p(S,\, \mu)</math> happens to be a norm (which happens if and only if <math>\mathcal{N} = \{0\}</math>) then the normed space <math>\left(\mathcal{L}^p(S,\, \mu), \|\cdot\|_p\right)</math> will be [[Linear map|linearly]] [[isometrically isomorphic]] to the normed quotient space <math>\left(L^p(S, \mu), \|\cdot\|_p\right)</math> via the canonical map <math>g \in \mathcal{L}^p(S,\, \mu) \mapsto \{g\}</math> (since <math>g + \mathcal{N} = \{g\}</math>); in other words, they will be, [[up to]] a [[linear isometry]], the same normed space and so they may both be called "<math>L^p</math> space".

The above definitions generalize to [[Bochner space]]s. 

In general, this process cannot be reversed: there is no consistent way to define a "canonical" representative of each coset of <math>\mathcal{N}</math> in <math>L^p.</math> For <math>L^\infty,</math> however, there is a [[Lifting theory|theory of lifts]] enabling such recovery.

===Special cases===
For <math>1 \leq p \leq \infty</math> the <math>\ell^p</math> spaces are a special case of <math>L^p</math> spaces; when <math>S</math> are the [[natural number]]s <math>\mathbb{N}</math> and <math>\mu</math> is the [[counting measure]]. More generally, if one considers any set <math>S</math> with the counting measure, the resulting <math>L^p</math> space is denoted <math>\ell^p(S).</math> For example, <math>\ell^p(\mathbb{Z})</math> is the space of all sequences indexed by the integers, and when defining the <math>p</math>-norm on such a space, one sums over all the integers. The space <math>\ell^p(n),</math> where <math>n</math> is the set with <math>n</math> elements, is <math>\Reals^n</math> with its <math>p</math>-norm as defined above. 

Similar to <math>\ell^2</math> spaces, <math>L^2</math> is the only [[Hilbert space]] among <math>L^p</math> spaces. In the complex case, the inner product on <math>L^2</math> is defined by
<math display="block">\langle f, g \rangle = \int_S f(x) \overline{g(x)} \, \mathrm{d}\mu(x).</math>
Functions in <math>L^2</math> are sometimes called '''[[square-integrable function]]s''', '''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]] {{harv|Titchmarsh|1976}}.

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

If we use complex-valued functions, the space <math>L^\infty</math> 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 <math>L^\infty</math> defines a [[bounded operator]] on any <math>L^p</math> space by [[multiplication operator|multiplication]].

===When {{math|(0 < ''p'' < 1)}}===

If <math>0 < p < 1,</math> then <math>L^p(\mu)</math> can be defined as above, that is:
<math display="block">N_p(f) = \int_S |f|^p\, d\mu < \infty.</math>
In this case, however, the <math>p</math>-norm <math>\|f\|_p = N_p(f)^{1/p}</math> does not satisfy the triangle inequality and defines only a [[quasi-norm]]. The inequality <math>(a + b)^p \leq a^p + b^p,</math> valid for <math>a, b \geq 0,</math> implies that
<math display="block">N_p(f + g) \leq N_p(f) + N_p(g)</math>
and so the function
<math display="block">d_p(f ,g) = N_p(f - g) = \|f - g\|_p^p</math>
is a metric on <math>L^p(\mu).</math> The resulting metric space is [[Complete metric space|complete]].{{sfn|Rudin|1991|p=37}}

In this setting <math>L^p</math> satisfies a ''reverse Minkowski inequality'', that is for <math>u, v \in L^p</math>
<math display="block">\Big\||u| + |v|\Big\|_p \geq \|u\|_p + \|v\|_p</math>

This result may be used to prove [[Clarkson's inequalities]], which are in turn used to establish the [[uniformly convex space|uniform convexity]] of the spaces <math>L^p</math> for <math>1 < p < \infty</math> {{harv|Adams|Fournier|2003}}.

The space <math>L^p</math> for <math>0 < p < 1</math> 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 topological vector space|locally convex]]: in <math>\ell^p</math> or <math>L^p([0, 1]),</math> every open convex set containing the <math>0</math> function is unbounded for the <math>p</math>-quasi-norm; therefore, the <math>0</math> vector does not possess a fundamental system of convex neighborhoods. Specifically, this is true if the measure space <math>S</math> contains an infinite family of disjoint measurable sets of finite positive measure.

The only nonempty convex open set in <math>L^p([0, 1])</math> is the entire space. Consequently, there are no nonzero continuous linear functionals on <math>L^p([0, 1]);</math> the [[continuous dual space]] is the zero space. In the case of the [[counting measure]] on the natural numbers (i.e. <math>L^p(\mu) = \ell^p</math>), the bounded linear functionals on <math>\ell^p</math> are exactly those that are bounded on <math>\ell^1</math>, i.e., those given by sequences in <math>\ell^\infty.</math> Although <math>\ell^p</math> 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 <math>\Reals^n,</math> rather than work with <math>L^p</math> for <math>0 < p < 1,</math> it is common to work with the [[Hardy space]] {{math|''H{{i sup|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 {{math|''H{{i sup|p}}''}} for <math>p < 1</math> {{harv|Duren|1970|loc=§7.5}}.

==Properties==

===Hölder's inequality===

Suppose <math>p, q, r \in [1, \infty]</math> satisfy <math>\tfrac{1}{p} + \tfrac{1}{q} = \tfrac{1}{r}</math>. If <math>f \in L^p(S, \mu)</math> and <math>g \in L^q(S, \mu)</math> then <math>f g \in L^r(S, \mu)</math> and{{sfn|Bahouri|Chemin|Danchin|2011|pp=1–4}} 
<math display=block>\|f g\|_r ~\leq~ \|f\|_p \, \|g\|_q.</math>

This inequality, called [[Hölder's inequality]], is in some sense optimal since if <math>r = 1</math> and <math>f</math> is a measurable function such that
<math display=block>\sup_{\|g\|_q \leq 1} \, \int_S |f g| \, \mathrm{d} \mu ~<~ \infty</math> 
where the [[supremum]] is taken over the closed unit ball of <math>L^q(S, \mu),</math> then <math>f \in L^p(S, \mu)</math> and
<math display=block>\|f\|_p ~=~ \sup_{\|g\|_q \leq 1} \, \int_S f g \, \mathrm{d} \mu.</math>

===Generalized Minkowski inequality===

[[Minkowski inequality]], which states that <math>\|\cdot\|_p</math> satisfies the [[triangle inequality]], can be generalized: 
If the measurable function <math>F : M \times N \to \Reals</math> is non-negative (where <math>(M, \mu)</math> and <math>(N, \nu)</math> are measure spaces) then for all <math>1 \leq p \leq q \leq \infty,</math>{{sfn|Bahouri|Chemin|Danchin|2011|p=4}}
<math display=block>\left\|\left\|F(\,\cdot, n)\right\|_{L^p(M, \mu)}\right\|_{L^q(N, \nu)}
~\leq~ \left\|\left\|F(m, \cdot)\right\|_{L^q(N, \nu)}\right\|_{L^p(M, \mu)} \ .</math>

===Atomic decomposition===


If <math>1 \leq p < \infty</math> then every non-negative <math>f \in L^p(\mu)</math> has an {{em|atomic decomposition}},{{sfn|Bahouri|Chemin|Danchin|2011|pp=7–8}} meaning that there exist a sequence <math>(r_n)_{n \in \Z}</math> of non-negative real numbers and a sequence of non-negative functions <math>(f_n)_{n \in \Z},</math> called {{em|the atoms}}, whose supports <math>\left(\operatorname{supp} f_n\right)_{n \in \Z}</math> are [[Disjoint sets|pairwise disjoint sets]] of measure <math>\mu\left(\operatorname{supp} f_n\right) \leq 2^{n+1},</math> such that
<math display=block>f ~=~ \sum_{n \in \Z} r_n \, f_n \, ,</math>
and for every integer <math>n \in \Z,</math> 
<math display=block>\|f_n\|_\infty ~\leq~ 2^{-\tfrac{n}{p}} \, ,</math> 
and
<math display=block>\tfrac{1}{2} \|f\|_p^p ~\leq~ \sum_{n \in \Z} r_n^p ~\leq~ 2 \|f\|^p_p \, ,</math>
and where moreover, the sequence of functions <math>(r_n f_n)_{n \in\Z}</math> depends only on <math>f</math> (it is independent of <math>p</math>).{{sfn|Bahouri|Chemin|Danchin|2011|pp=7–8}} 
These inequalities guarantee that <math>\|f_n\|_p^p \leq 2</math> for all integers <math>n</math> while the supports of <math>(f_n)_{n \in \Z}</math> being pairwise disjoint implies{{sfn|Bahouri|Chemin|Danchin|2011|pp=7–8}} 
<math display=block>\|f\|_p^p ~=~ \sum_{n \in \Z} r_n^p \, \|f_n\|^p_p \, .</math>

An atomic decomposition can be explicitly given by first defining for every integer <math>n \in \Z,</math>{{sfn|Bahouri|Chemin|Danchin|2011|pp=7–8}}<ref group=note>This [[infimum]] is attained by <math>t_n;</math> that is, <math>\mu(f > t_n) < 2^n</math> holds.</ref> 
<math display=block>t_n = \inf \{t \in \Reals : \mu(f > t) < 2^n\}</math>
and then letting
<math display=block>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 )}</math>
where <math>\mu(f > t) = \mu(\{s : f(s) > t\})</math> denotes the measure of the set <math>(f > t) := \{s \in S : f(s) > t\}</math> and <math>\mathbf{1}_{(t_{n+1} < f \leq t_n)}</math> denotes the [[indicator function]] of the set <math>(t_{n+1} < f \leq t_n) := \{s \in S : t_{n+1} < f(s) \leq t_n\}.</math> 
The sequence <math>(t_n)_{n \in \Z}</math> is decreasing and converges to <math>0</math> as <math>n \to \infty.</math>{{sfn|Bahouri|Chemin|Danchin|2011|pp=7–8}} Consequently, if <math>t_n = 0</math> then <math>t_{n+1} = 0</math> and <math>(t_{n+1} < f \leq t_n) = \varnothing</math> so that <math>f_n = \frac{1}{r_n} \, f \,\mathbf{1}_{(t_{n+1} < f \leq t_n)}</math> is identically equal to <math>0</math> (in particular, the division <math>\tfrac{1}{r_n}</math> by <math>r_n = 0</math> causes no issues). 

The [[complementary cumulative distribution function]] <math>t \in \Reals \mapsto \mu(|f| > t)</math> of <math>|f| = f</math> that was used to define the <math>t_n</math> also appears in the definition of the weak <math>L^p</math>-norm (given below) and can be used to express the <math>p</math>-norm <math>\|\cdot\|_p</math> (for <math>1 \leq p < \infty</math>) of <math>f \in L^p(S, \mu)</math> as the integral{{sfn|Bahouri|Chemin|Danchin|2011|pp=7–8}}
<math display=block>\|f\|_p^p ~=~ p \, \int_0^\infty t^{p-1} \mu(|f| > t) \, \mathrm{d} t \, ,</math>
where the integration is with respect to the usual Lebesgue measure on <math>(0, \infty).</math>

===Dual spaces===

The [[Continuous dual|dual space]] of <math>L^p(\mu)</math> for <math>1 < p < \infty</math> has a natural isomorphism with <math>L^q(\mu),</math> where <math>q</math> is such that <math>\tfrac{1}{p} + \tfrac{1}{q} = 1</math>. This isomorphism associates <math>g \in L^q(\mu)</math> with the functional <math>\kappa_p(g) \in L^p(\mu)^*</math> defined by
<math display="block">f \mapsto \kappa_p(g)(f) = \int f g \, \mathrm{d}\mu</math> 
for every <math>f \in L^p(\mu).</math>

<math>\kappa_p : L^q(\mu) \to L^p(\mu)^*</math> is a well defined continuous linear mapping which is an [[isometry]] by the [[Hölder's inequality#Extremal equality|extremal case]] of Hölder's inequality. If <math>(S,\Sigma,\mu)</math> is a [[Measure_space#Important_classes_of_measure_spaces|<math>\sigma</math>-finite measure space]] one can use the [[Radon–Nikodym theorem]] to show that any <math>G \in L^p(\mu)^*</math> can be expressed this way, i.e., <math>\kappa_p</math> is an [[Isometry#Definition|isometric isomorphism]] of [[Banach space]]s.{{sfn|Rudin|1987|loc=Theorem 6.16}} Hence, it is usual to say simply that <math>L^q(\mu)</math> is the [[continuous dual space]] of <math>L^p(\mu).</math>

For <math>1 < p < \infty,</math> the space <math>L^p(\mu)</math> is [[reflexive space|reflexive]]. Let <math>\kappa_p</math> be as above and let <math>\kappa_q : L^p(\mu) \to L^q(\mu)^*</math> be the corresponding linear isometry. Consider the map from <math>L^p(\mu)</math> to <math>L^p(\mu)^{**},</math> obtained by composing <math>\kappa_q</math> with the [[dual space#Transpose of a continuous linear map|transpose]] (or adjoint) of the inverse of <math>\kappa_p:</math>

<math display="block">j_p : L^p(\mu) \mathrel{\overset{\kappa_q}{\longrightarrow}} L^q(\mu)^* \mathrel{\overset{\left(\kappa_p^{-1}\right)^*}{\longrightarrow}} L^p(\mu)^{**}</math>

This map coincides with the [[Reflexive space#Definitions|canonical embedding]] <math>J</math> of <math>L^p(\mu)</math> into its bidual. Moreover, the map <math>j_p</math> is onto, as composition of two onto isometries, and this proves reflexivity.

If the measure <math>\mu</math> on <math>S</math> is [[sigma-finite]], then the dual of <math>L^1(\mu)</math> is isometrically isomorphic to <math>L^\infty(\mu)</math> (more precisely, the map <math>\kappa_1</math> corresponding to <math>p = 1</math> is an isometry from <math>L^\infty(\mu)</math> onto <math>L^1(\mu)^*.</math>

The dual of <math>L^\infty(\mu)</math> is subtler. Elements of <math>L^\infty(\mu)^*</math> can be identified with bounded signed ''finitely'' additive measures on <math>S</math> that are [[absolutely continuous]] with respect to <math>\mu.</math> See [[ba space]] for more details. If we assume the axiom of choice, this space is much bigger than <math>L^1(\mu)</math> except in some trivial cases. However, [[Saharon Shelah]] proved that there are relatively consistent extensions of [[Zermelo–Fraenkel set theory]] (ZF + [[Axiom of dependent choice|DC]] + "Every subset of the real numbers has the [[Baire property]]") in which the dual of <math>\ell^\infty</math> is <math>\ell^1.</math><ref>{{Citation|title=Handbook of Analysis and its Foundations|last=Schechter |first=Eric|year=1997| publisher=Academic Press Inc.|location=London}} See Sections 14.77 and 27.44–47</ref>

===Embeddings===

Colloquially, if <math>1 \leq p < q \leq \infty,</math> then <math>L^p(S, \mu)</math> contains functions that are more locally singular, while elements of <math>L^q(S, \mu)</math> can be more spread out. Consider the [[Lebesgue measure]] on the half line <math>(0, \infty).</math> A continuous function in <math>L^1</math> might blow up near <math>0</math> but must decay sufficiently fast toward infinity. On the other hand, continuous functions in <math>L^\infty</math> need not decay at all but no blow-up is allowed. More formally:<ref name="VillaniEmbeddings">{{Citation|title=Another note on the inclusion {{math|''L<sup>p</sup>''(''μ'') ⊂ ''L<sup>q</sup>''(''μ'')}}|last=Villani|first=Alfonso|year=1985|journal=Amer. Math. Monthly|volume=92|number=7|pages=485–487|doi=10.2307/2322503|mr=801221|jstor=2322503}}</ref>

#If <math>0<p<q<\infty</math>: <math>L^q(S, \mu) \subseteq L^p(S, \mu)</math> if and only if <math>S</math> does not contain sets of finite but arbitrarily large measure (e.g. any [[finite measure]]). 
#If <math>0<p<q\le\infty</math>: <math>L^p(S, \mu) \subseteq L^q(S, \mu)</math> if and only if <math>S</math> 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 <math>L^q</math> to <math>L^p</math> in the first case and <math>L^p</math> to <math>L^q</math> in the second. Indeed, if the domain <math>S</math> has finite measure, one can make the following explicit calculation using [[Hölder's inequality]]
<math display="block">\ \|\mathbf{1}f^p\|_1 \leq \|\mathbf{1}\|_{q/(q-p)} \|f^p\|_{q/p}</math>
leading to
<math display="block">\ \|f\|_p \leq \mu(S)^{1/p - 1/q} \|f\|_q .</math>

The constant appearing in the above inequality is optimal, in the sense that the [[operator norm]] of the identity <math>I : L^q(S, \mu) \to L^p(S, \mu)</math> is precisely
<math display="block">\|I\|_{q,p} = \mu(S)^{1/p - 1/q}</math>
the case of equality being achieved exactly when <math>f = 1</math> <math>\mu</math>-almost-everywhere.

===Dense subspaces===

Let <math>1 \leq p < \infty</math> and <math>(S, \Sigma, \mu)</math> be a measure space and consider an integrable [[simple function]] <math>f</math> on <math>S</math> given by
<math display="block">f = \sum_{j=1}^n a_j \mathbf{1}_{A_j},</math>
where <math>a_j</math> are scalars, <math>A_j \in \Sigma</math> has finite measure and <math>{\mathbf 1}_{A_j}</math> is the [[indicator function]] of the set <math>A_j,</math> for <math>j = 1, \dots, n.</math> By construction of the [[Lebesgue integration|integral]], the vector space of integrable simple functions is [[dense_set|dense]] in <math>L^p(S, \Sigma, \mu).</math>

More can be said when <math>S</math> is a [[Normal space|normal]] [[topological space]] and <math>\Sigma</math> its [[Borel algebra|Borel {{sigma}}&ndash;algebra]]. 

Suppose <math>V \subseteq S</math> is an open set with <math>\mu(V) < \infty.</math> Then for every Borel set <math>A \in \Sigma</math> contained in <math>V</math> there exist a closed set <math>F</math> and an open set <math>U</math> such that
<math display="block">F \subseteq A \subseteq U \subseteq V \quad \text{and} \quad \mu(U \setminus F)= \mu(U) - \mu(F)  < \varepsilon,</math>
for every <math>\varepsilon > 0</math>. Subsequently, there exists a [[Urysohn function]] <math>0 \leq \varphi \leq 1</math> on <math>S</math> that is <math>1</math> on <math>F</math> and <math>0</math> on <math>S \setminus U,</math> with
<math display="block">\int_S |\mathbf{1}_A - \varphi| \, \mathrm{d}\mu < \varepsilon \, .</math>

If <math>S</math> can be covered by an increasing sequence <math>(V_n)</math> of open sets that have finite measure, then the space of <math>p</math>&ndash;integrable continuous functions is dense in <math>L^p(S, \Sigma, \mu).</math> More precisely, one can use bounded continuous functions that vanish outside one of the open sets <math>V_n.</math> 

This applies in particular when <math>S = \Reals^d</math> and when <math>\mu</math> is the Lebesgue measure. For example, the space of continuous and compactly supported functions as well as the  space of integrable [[step function]]s are dense in <math>L^p(\Reals^d)</math>.

===Closed subspaces===

If <math>0 < p < \infty</math> is any positive real number, <math>\mu</math> is a [[probability measure]] on a measurable space <math>(S, \Sigma)</math> (so that <math>L^\infty(\mu) \subseteq L^p(\mu)</math>), and <math>V \subseteq L^\infty(\mu)</math> is a vector subspace, then <math>V</math> is a closed subspace of <math>L^p(\mu)</math> if and only if <math>V</math> is finite-dimensional{{sfn|Rudin|1991|pp=117–119}} (<math>V</math> was chosen independent of <math>p</math>). 
In this theorem, which is due to [[Alexander Grothendieck]],{{sfn|Rudin|1991|pp=117–119}} it is crucial that the vector space <math>V</math> be a subset of <math>L^\infty</math> since it is possible to construct an infinite-dimensional closed vector subspace of <math>L^1\left(S^1, \tfrac{1}{2\pi}\lambda\right)</math> (which is even a subset of <math>L^4</math>), where <math>\lambda</math> is [[Lebesgue measure]] on the [[unit circle]] <math>S^1</math> and <math>\tfrac{1}{2\pi} \lambda</math> is the probability measure that results from dividing it by its mass <math>\lambda(S^1) = 2 \pi.</math>{{sfn|Rudin|1991|pp=117–119}}

==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 <math>L^p</math> metrics, and measures of central tendency can be characterized as [[Central tendency#Solutions to variational problems|solutions to variational problems]].

In [[penalized regression]], "L1 penalty" and "L2 penalty" refer to penalizing either the [[Taxicab geometry|<math>L^1</math> norm]] of a solution's vector of parameter values (i.e. the sum of its absolute values), or its squared <math>L^2</math> norm (its [[Euclidean norm|Euclidean length]]). Techniques which use an L1 penalty, like [[LASSO]], encourage sparse solutions (where the many parameters are zero).<ref>{{cite book |last=Hastie |first=T. J. |authorlink=Trevor Hastie |last2=Tibshirani |first2=R. |author2link=Robert Tibshirani |last3=Wainwright |first3=M. J. |year=2015 |title=Statistical Learning with Sparsity: The Lasso and Generalizations |location= |publisher=CRC Press |isbn=978-1-4987-1216-3 }}</ref> [[Elastic net regularization]] uses a penalty term that is a combination of the <math>L^1</math> norm and the squared <math>L^2</math> norm of the parameter vector.

===Hausdorff–Young inequality===

The [[Fourier transform]] for the real line (or, for [[periodic functions]], see [[Fourier series]]), maps <math>L^p(\Reals)</math> to <math>L^q(\Reals)</math> (or <math>L^p(\mathbf{T})</math> to <math>\ell^q</math>) respectively, where <math>1 \leq p \leq 2</math> and <math>\tfrac{1}{p} + \tfrac{1}{q} = 1.</math>  This is a consequence of the [[Riesz–Thorin_theorem#Hausdorff–Young_inequality|Riesz–Thorin interpolation theorem]], and is made precise with the [[Hausdorff–Young inequality]].

By contrast, if <math>p > 2,</math> the Fourier transform does not map into <math>L^q.</math>

===Hilbert spaces===
[[Hilbert space]]s are central to many applications, from [[quantum mechanics]] to [[stochastic calculus]]. The spaces <math>L^2</math> and <math>\ell^2</math> are both Hilbert spaces. In fact, by choosing a Hilbert basis <math>E,</math> i.e., a maximal orthonormal subset of <math>L^2</math> or any Hilbert space, one sees that every Hilbert space is isometrically isomorphic to <math>\ell^2(E)</math> (same <math>E</math> as above), i.e., a Hilbert space of type <math>\ell^2.</math>

==Generalizations and extensions==

===Weak {{math|''L<sup>p</sup>''}}===

Let <math>(S, \Sigma, \mu)</math> be a measure space, and <math>f</math> a [[measurable function]] with real or complex values on <math>S.</math> The [[cumulative distribution function|distribution function]] of <math>f</math> is defined for <math>t \geq 0</math> by
<math display="block">\lambda_f(t) = \mu\{x \in S : |f(x)| > t\}.</math>

If <math>f</math> is in <math>L^p(S, \mu)</math> for some <math>p</math> with <math>1 \leq p < \infty,</math> then by [[Markov's inequality]],
<math display="block">\lambda_f(t) \leq \frac{\|f\|_p^p}{t^p}</math>

A function <math>f</math> is said to be in the space '''weak <math>L^p(S, \mu)</math>''', or <math>L^{p,w}(S, \mu),</math> if there is a constant <math>C > 0</math> such that, for all <math>t > 0,</math>
<math display="block">\lambda_f(t) \leq \frac{C^p}{t^p}</math>

The best constant <math>C</math> for this inequality is the <math>L^{p,w}</math>-norm of <math>f,</math> and is denoted by
<math display="block">\|f\|_{p,w} = \sup_{t > 0} ~ t \lambda_f^{1/p}(t).</math>

The weak <math>L^p</math> coincide with the [[Lorentz space]]s <math>L^{p,\infty},</math> so this notation is also used to denote them.

The <math>L^{p,w}</math>-norm is not a true norm, since the [[triangle inequality]] fails to hold. Nevertheless, for <math>f</math> in <math>L^p(S, \mu),</math>
<math display="block">\|f\|_{p,w} \leq \|f\|_p</math>
and in particular <math>L^p(S, \mu) \subset L^{p,w}(S, \mu).</math> 

In fact, one has
<math display="block">\|f\|^p_{L^p} = \int |f(x)|^p d\mu(x) \geq \int_{\{|f(x)| > t \}} t^p + \int_{\{|f(x)| \leq t \}} |f|^p \geq t^p \mu(\{|f| > t \}),</math>
and raising to power <math>1/p</math> and taking the supremum in <math>t</math> one has
<math display="block">\|f\|_{L^p} \geq \sup_{t > 0} t \; \mu(\{|f| > t \})^{1/p} = \|f\|_{L^{p,w}}.</math>

Under the convention that two functions are equal if they are equal <math>\mu</math> almost everywhere, then the spaces <math>L^{p,w}</math> are complete {{harv|Grafakos|2004}}.

For any <math>0 < r < p</math> the expression
<math display="block">\|| f |\|_{L^{p,\infty}} = \sup_{0<\mu(E)<\infty} \mu(E)^{-1/r + 1/p} \left(\int_E |f|^r\, d\mu\right)^{1/r}</math>
is comparable to the <math>L^{p,w}</math>-norm. Further in the case <math>p > 1,</math> this expression defines a norm if <math>r = 1.</math> Hence for <math>p > 1</math> the weak <math>L^p</math> spaces are [[Banach space]]s {{harv|Grafakos|2004}}.

A major result that uses the <math>L^{p,w}</math>-spaces is the [[Marcinkiewicz interpolation|Marcinkiewicz interpolation theorem]], which has broad applications to [[harmonic analysis]] and the study of [[singular integrals]].

===Weighted {{math|''L<sup>p</sup>''}} spaces===

As before, consider a [[measure space]] <math>(S, \Sigma, \mu).</math> Let <math>w : S \to [a, \infty), a > 0</math> be a measurable function. The <math>w</math>-'''weighted <math>L^p</math> space''' is defined as <math>L^p(S, w \, \mathrm{d} \mu),</math> where <math>w \, \mathrm{d} \mu</math> means the measure <math>\nu</math> defined by
<math display="block">\nu(A) \equiv \int_A w(x) \, \mathrm{d} \mu (x), \qquad A \in \Sigma,</math>

or, in terms of the [[Radon–Nikodym theorem|Radon–Nikodym derivative]], <math>w = \tfrac{\mathrm{d} \nu}{\mathrm{d} \mu}</math> the [[Norm (mathematics)|norm]] for <math>L^p(S, w \, \mathrm{d} \mu)</math> is explicitly
<math display="block">\|u\|_{L^p(S, w \, \mathrm{d} \mu)} \equiv \left(\int_S w(x) |u(x)|^p \, \mathrm{d} \mu(x)\right)^{1/p}</math>

As <math>L^p</math>-spaces, the weighted spaces have nothing special, since <math>L^p(S, w \, \mathrm{d} \mu)</math> is equal to <math>L^p(S, \mathrm{d} \nu).</math> But they are the natural framework for several results in harmonic analysis {{harv|Grafakos|2004}}<!--Please check this reference. Appears in Grafakos "Modern Fourier analysis", Chapter 9.-->; they appear for example in the [[Muckenhoupt weights|Muckenhoupt theorem]]: for <math>1 < p < \infty,</math> the classical [[Hilbert transform]] is defined on <math>L^p(\mathbf{T}, \lambda)</math> where <math>\mathbf{T}</math> denotes the [[unit circle]] and <math>\lambda</math> the Lebesgue measure; the (nonlinear) [[Hardy–Littlewood maximal operator]] is bounded on <math>L^p(\Reals^n, \lambda).</math> Muckenhoupt's theorem describes weights <math>w</math> such that the Hilbert transform remains bounded on <math>L^p(\mathbf{T}, w \, \mathrm{d} \lambda)</math> and the maximal operator on <math>L^p(\Reals^n, w \, \mathrm{d} \lambda).</math> 

==={{math|''L<sup>p</sup>''}} spaces on manifolds===

One may also define spaces <math>L^p(M)</math> on a manifold, called the '''intrinsic <math>L^p</math> spaces''' of the manifold, using [[Density on a manifold|densities]].

===Vector-valued {{math|''L<sup>p</sup>''}} spaces===

Given a measure space <math>(\Omega, \Sigma, \mu)</math> and a [[Locally convex topological vector space|locally convex space]] <math>E</math> (here assumed to be [[Complete topological vector space|complete]]), it is possible to define spaces of <math>p</math>-integrable <math>E</math>-valued functions on <math>\Omega</math> in a number of ways. One way is to define the spaces of [[Bochner integral|Bochner integrable]] and [[Pettis integral|Pettis integrable]] functions, and then endow them with [[Locally convex topological vector space|locally convex]] [[Vector topology|TVS-topologies]] that are (each in their own way) a natural generalization of the usual <math>L^p</math> topology. Another way involves [[topological tensor product]]s of <math>L^p(\Omega, \Sigma, \mu)</math> with <math>E.</math> Element of the vector space <math>L^p(\Omega, \Sigma, \mu) \otimes E</math> are finite sums of simple tensors <math>f_1 \otimes e_1 + \cdots + f_n \otimes e_n,</math> where each simple tensor <math>f \times e</math> may be identified with the function <math>\Omega \to E</math> that sends <math>x \mapsto e f(x).</math> This [[tensor product]] <math>L^p(\Omega, \Sigma, \mu) \otimes E</math> is then endowed with a locally convex topology that turns it into a [[topological tensor product]], the most common of which are the [[projective tensor product]], denoted by <math>L^p(\Omega, \Sigma, \mu) \otimes_\pi E,</math> and the [[injective tensor product]], denoted by <math>L^p(\Omega, \Sigma, \mu) \otimes_\varepsilon E.</math> In general, neither of these space are complete so their [[Complete topological vector space|completions]] are constructed, which are respectively denoted by <math>L^p(\Omega, \Sigma, \mu) \widehat{\otimes}_\pi E</math> and <math>L^p(\Omega, \Sigma, \mu) \widehat{\otimes}_\varepsilon E</math> (this is analogous to how the space of scalar-valued [[simple function]]s on <math>\Omega,</math> when seminormed by any <math>\|\cdot\|_p,</math> is not complete so a completion is constructed which, after being quotiented by <math>\ker \|\cdot\|_p,</math> is isometrically isomorphic to the Banach space <math>L^p(\Omega, \mu)</math>). [[Alexander Grothendieck]] showed that when <math>E</math> is a [[nuclear space]] (a concept he introduced), then these two constructions are, respectively, canonically TVS-isomorphic with the spaces of Bochner and Pettis integral functions mentioned earlier; in short, they are indistinguishable.

==={{math|''L''<sup>0</sup>}} space of measurable functions===

The vector space of ([[equivalence class]]es of) measurable functions on <math>(S, \Sigma, \mu)</math> is denoted <math>L^0(S, \Sigma, \mu)</math> {{harv|Kalton|Peck|Roberts|1984}}. By definition, it contains all the <math>L^p,</math> and is equipped with the topology of ''[[convergence in measure]]''. When <math>\mu</math> is a probability measure (i.e., <math>\mu(S) = 1</math>), this mode of convergence is named ''[[convergence in probability]]''. The space <math>L^0</math> is always a [[topological abelian group]] but  is only a [[topological vector space]] if <math>\mu(S)<\infty.</math> This is because scalar multiplication is continuous if and only if <math>\mu(S)<\infty.</math> If <math>(S,\Sigma,\mu)</math> is <math>\sigma</math>-finite then the [[weaker topology]] of [[local convergence in measure]] is an [[F-space]], i.e. a [[Complete topological vector space|completely]] [[metrizable topological vector space]]. Moreover, this topology is isometric to global convergence in measure <math>(S,\Sigma,\nu)</math> for a suitable choice of [[probability measure]] <math>\nu.</math>

The description is easier when <math>\mu</math> is finite. If <math>\mu</math> is a [[finite measure]] on <math>(S, \Sigma),</math> the <math>0</math> function admits for the convergence in measure the following [[fundamental system of neighborhoods]]
<math display="block">V_\varepsilon = \Bigl\{f : \mu \bigl(\{x : |f(x)| > \varepsilon\} \bigr) < \varepsilon \Bigr\}, \qquad \varepsilon > 0.</math>

The topology can be defined by any metric <math>d</math> of the form
<math display="block">d(f, g) = \int_S \varphi \bigl(|f(x) - g(x)|\bigr)\, \mathrm{d}\mu(x)</math>
where <math>\varphi</math> is bounded continuous concave and non-decreasing on <math>[0, \infty),</math> with <math>\varphi(0) = 0</math> and <math>\varphi(t) > 0</math> when <math>t > 0</math> (for example, <math>\varphi(t) = \min(t, 1).</math> Such a metric is called [[Paul Lévy (mathematician)|Lévy]]-metric for <math>L^0.</math> Under this metric the space <math>L^0</math> is complete. However, as mentioned above, scalar multiplication is continuous with respect to this metric only if <math>\mu(S)<\infty</math>. To see this, consider the Lebesgue measurable function <math>f:\mathbb R\rightarrow \mathbb R</math> defined by <math>f(x)=x</math>. Then clearly <math>\lim_{c\rightarrow 0}d(cf,0)=\infty</math>. The space <math>L^0</math> is in general not locally bounded, and not locally convex.

For the infinite Lebesgue measure <math>\lambda</math> on <math>\Reals^n,</math> the definition of the fundamental system of neighborhoods could be modified as follows
<math display="block">W_\varepsilon = \left\{f : \lambda \left(\left\{x : |f(x)| > \varepsilon \text{ and } |x| < \tfrac{1}{\varepsilon}\right\}\right) < \varepsilon\right\}</math>

The resulting space <math>L^0(\Reals^n, \lambda)</math>, with the topology of local convergence in measure, is isomorphic to the space <math>L^0(\Reals^n, g \, \lambda),</math> for any positive <math>\lambda</math>&ndash;integrable density <math>g.</math>

==See also==
{{Div col|colwidth=30em}}
* {{annotated link|Absolutely integrable function}}
* {{annotated link|Bochner space}}
* {{annotated link|Orlicz space}}
* {{annotated link|Hardy space}}
* {{annotated link|Riesz–Thorin theorem}}
* {{annotated link|Hölder mean}}
* {{annotated link|Hölder space}}
* {{annotated link|Root mean square}}
* {{annotated link|Least absolute deviations}}
* {{annotated link|Locally integrable function}} <math>\left( L^1_{\text{loc}}\right)</math>
* {{annotated link|Pontryagin duality#Haar measure|<math> L^p(G)</math> spaces over a locally compact group <math>G</math>}}
* {{annotated link|Least-squares spectral analysis}}
* {{annotated link|List of Banach spaces}}
* {{annotated link|Minkowski distance}}
* {{annotated link|L-infinity}}
* {{annotated link|Lp sum|''L<sup>p</sup>'' sum}}
* {{annotated link|Cm sum|''C<sup>m</sup>'' space}}

{{div col end}}

==Notes==

{{reflist}}
{{reflist|group=note}}

{{reflist|group=proof}}

==References==
{{sfn whitelist|CITEREFBahouriCheminDanchin2011}}
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* {{Bahouri Chemin Danchin Fourier Analysis and Nonlinear Partial Differential Equations 2011}} <!--{{sfn|Bahouri|Chemin|Danchin|2011|p=}}-->
* {{citation | last=Bourbaki | first=Nicolas | author-link=Nicolas Bourbaki | title=Topological vector spaces|series=Elements of mathematics|publisher= Springer-Verlag | location=Berlin | year=1987 | isbn=978-3-540-13627-9}}.
* {{citation | last=DiBenedetto | first=Emmanuele | title=Real analysis | publisher=Birkhäuser | year=2002 | isbn=3-7643-4231-5}}.
* {{citation | last1=Dunford | first1=Nelson | last2=Schwartz | first2=Jacob T. | title=Linear operators, volume I | publisher=Wiley-Interscience | year=1958}}.
* {{citation | last= Duren | first=P. | title=Theory of H<sup>p</sup>-Spaces | year=1970 | publisher= Academic Press | location= New York}}
* {{citation | last=Grafakos | first=Loukas | authorlink=Loukas Grafakos | title=Classical and Modern Fourier Analysis | publisher=Pearson Education, Inc. | pages=253&ndash;257 | year=2004 | isbn=0-13-035399-X}}.
* {{citation | last1=Hewitt | first1=Edwin | last2=Stromberg | first2=Karl | title=Real and abstract analysis | publisher=Springer-Verlag | year=1965}}.
* {{citation | last1=Kalton | first1=Nigel J. | author-link=Nigel Kalton | last2=Peck | first2=N. Tenney | last3=Roberts | first3=James W. | title = An F-space sampler | series = London Mathematical Society Lecture Note Series | volume=89 | publisher = Cambridge University Press| location = Cambridge | year = 1984 | isbn = 0-521-27585-7 | mr=808777 | doi=10.1017/CBO9780511662447}}
* {{citation | last=Riesz | first=Frigyes | author-link=Frigyes Riesz | title=Untersuchungen über Systeme integrierbarer Funktionen | journal=Mathematische Annalen | volume=69 | year=1910 | pages=449–497 | doi=10.1007/BF01457637 | issue=4| s2cid=120242933 | url=https://zenodo.org/record/2456593 }}
* {{Rudin Walter Functional Analysis|edition=2}} <!--{{sfn|Rudin|1991|p=}}-->
* {{Citation | last1=Rudin | first1=Walter | title=Real and complex analysis | publisher=[[McGraw-Hill]] | location=New York | edition=3rd | isbn=978-0-07-054234-1 | mr=924157 | year=1987}}
* {{cite book | last=Stein | first=Elias M. | author-link1=Elias Stein | last2=Shakarchi | first2=Rami | title=Functional Analysis: Introduction to Further Topics in Analysis | publisher=Princeton University Press | date=2012 | isbn=978-1-4008-4055-7 | doi=10.1515/9781400840557}}
* {{citation | last=Titchmarsh | first=EC | author-link=Edward Charles Titchmarsh | title=The theory of functions | publisher=Oxford University Press | year=1976 | isbn=978-0-19-853349-8}}

==External links==

* {{springer|title=Lebesgue space|id=p/l057910}}
* [http://planetmath.org/ProofThatLpSpacesAreComplete Proof that ''L''<sup>''p''</sup> spaces are complete]

{{Lp spaces}}
{{Measure theory}}
{{Banach spaces}}
{{Functional analysis}}

{{DEFAULTSORT:Lp Space}}

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