Lp space

 ( Banach Spaces ) 

In mathematics, the spaces are defined using a natural generalization of the -norm for finite-dimensional . They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Nicolas Bourbaki group they were first introduced by Frigyes Riesz .

spaces form an important class of [[Banach space]]s in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, economics, finance, engineering, and other disciplines.
     

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Preliminaries

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The -norm in finite dimensions The Euclidean length of a vector $x\; =\; (x\_1,\; x\_2,\; \backslash dots,\; x\_n)$ in the $n$-dimensional real number vector space $\backslash Reals^n$ is given by the Euclidean norm: $$\backslash |x\backslash |\_2\; =\; \backslash left(\{x\_1\}^2\; +\; \{x\_2\}^2\; +\; \backslash dotsb\; +\; \{x\_n\}^2\backslash right)^\{1/2\}.$$

The Euclidean distance between two points $x$ and $y$ is the length $\backslash |x\; -\; y\backslash |\_2$ of the straight line between the two points. In many situations, the Euclidean distance is appropriate for capturing the actual distances in a given space. In contrast, consider taxi drivers in a grid street plan who should measure distance not in terms of the length of the straight line to their destination, but in terms of the taxicab geometry, which takes into account that streets are either orthogonal or parallel to each other. The class of $p$-norms generalizes these two examples and has an abundance of applications in many parts of mathematics, physics, and computer science.

For a real number $p\; \backslash geq\; 1,$ the $p$-norm or $L^p$-norm of $x$ is defined by $$\backslash |x\backslash |\_p\; =\; \backslash left(|x\_1|^p\; +\; |x\_2|^p\; +\; \backslash dotsb\; +\; |x\_n|^p\backslash right)^\{1/p\}.$$ The absolute value bars can be dropped when $p$ is a rational number with an even numerator in its reduced form, and $x$ is drawn from the set of real numbers, or one of its subsets.

The Euclidean norm from above falls into this class and is the $2$-norm, and the $1$-norm is the norm that corresponds to the taxicab geometry.

The $L^\backslash infty$-norm or maximum norm (or uniform norm) is the limit of the $L^p$-norms for $p\; \backslash to\; \backslash infty$, given by: $$\backslash |x\backslash |\_\backslash infty\; =\; \backslash max\; \backslash left\backslash$$

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For all $p\; \backslash geq\; 1,$ the $p$-norms and maximum norm satisfy the properties of a "length function" (or 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 (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 $\backslash Reals^n$ together with the $p$-norm is a normed vector space. Moreover, it turns out that this space is complete, thus making it a Banach space. Relations between -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: $$\backslash$$

_1 . This fact generalizes to $p$-norms in that the $p$-norm $\backslash$

_p of any given vector $x$ does not grow with $p$: For the opposite direction, the following relation between the $1$-norm and the $2$-norm is known: $$\backslash$$

_2 ~. This inequality depends on the dimension $n$ of the underlying vector space and follows directly from the Cauchy–Schwarz inequality. In general, for vectors in $\backslash Complex^n$ where $$\backslash$$

_p ~. This is a consequence of Hölder's inequality. When In $\backslash Reals^n$ for $n\; >\; 1,$ the formula $$\backslash$$

^p\right)^{1/p} defines an absolutely homogeneous function for however, the resulting function does not define a norm, because it is not subadditivity. On the other hand, the formula $$$$

^p defines a subadditive function at the cost of losing absolute homogeneity. It does define an F-space, though, which is homogeneous of degree $p.$ Hence, the function $$d\_p(x,\; y)\; =\; \backslash sum\_\{i=1\}^n$$

^p defines a metric space. The metric space $(\backslash Reals^n,\; d\_p)$ is denoted by $\backslash ell\_n^p.$ Although the $p$-unit ball $B\_n^p$ around the origin in this metric is "concave", the topology defined on $\backslash Reals^n$ by the metric $B\_p$ is the usual vector space topology of $\backslash Reals^n,$ hence $\backslash ell\_n^p$ is a locally convex topological vector space. Beyond this qualitative statement, a quantitative way to measure the lack of convexity of $\backslash ell\_n^p$ is to denote by $C\_p(n)$ the smallest constant $C$ such that the scalar multiple $C\; \backslash ,\; B\_n^p$ of the $p$-unit ball contains the convex hull of $B\_n^p,$ which is equal to $B\_n^1.$ The fact that for fixed we have $$C\_p(n)\; =\; n^\{\backslash tfrac\{1\}\{p\}\; -\; 1\}\; \backslash to\; \backslash infty,\; \backslash quad\; \backslash text\{as\; \}\; n\; \backslash to\; \backslash infty$$ shows that the infinite-dimensional sequence space $\backslash ell^p$ defined below, is no longer locally convex. When There is one $\backslash ell\_0$ norm and another function called the $\backslash ell\_0$ "norm" (with quotation marks). The mathematical definition of the $\backslash ell\_0$ norm was established by Stefan Banach's Theory of Linear Operations. The F-space of sequences has a complete metric topology provided by the F-space on the product metric: $$(x\_n)\; \backslash mapsto\; \backslash$$

{1 +|x_n|}. The $\backslash ell\_0$-normed space is studied in functional analysis, probability theory, and harmonic analysis.

Another function was called the $\backslash ell\_0$ "norm" by David Donoho—whose quotation marks warn that this function is not a proper norm—is the number of non-zero entries of the vector $x.$ Many authors abuse terminology by omitting the quotation marks. Defining $0^0\; =\; 0,$ the zero "norm" of $x$ is equal to $$|x\_1|^0\; +\; |x\_2|^0\; +\; \backslash cdots\; +\; |x\_n|^0\; .$$

This is not a norm because it is not homogeneous. For example, scaling the vector $x$ by a positive constant does not change the "norm". Despite these defects as a mathematical norm, the non-zero counting "norm" has uses in scientific computing, information theory, and statistics–notably in compressed sensing in signal processing and computational harmonic analysis. Despite not being a norm, the associated metric, known as Hamming distance, is a valid distance, since homogeneity is not required for distances.

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

• $\backslash ell^1,$ the space of sequences whose series are absolutely convergent,
• $\backslash ell^2,$ the space of square-summable sequences, which is a Hilbert space, and
• $\backslash ell^\backslash infty,$ the space of .

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 of real (or complex number) numbers are given by:

Define the $p$-norm: $$\backslash |x\backslash |\_p\; =\; \backslash left(|x\_1|^p\; +\; |x\_2|^p\; +\; \backslash cdots\; +|x\_n|^p\; +\; |x\_\{n+1\}|^p\; +\; \backslash cdots\backslash right)^\{1/p\}$$

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, $(1,\; 1,\; 1,\; \backslash ldots),$ will have an infinite $p$-norm for The space $\backslash ell^p$ is then defined as the set of all infinite sequences of real (or complex) numbers such that the $p$-norm is finite.

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

One also defines the $\backslash infty$-norm using the supremum: $$\backslash |x\backslash |\_\backslash infty\; =\; \backslash sup(|x\_1|,\; |x\_2|,\; \backslash dotsc,\; |x\_n|,|x\_\{n+1\}|,\; \backslash ldots)$$ and the corresponding space $\backslash ell^\backslash infty$ of all bounded sequences. It turns out that, page 16 $$\backslash |x\backslash |\_\backslash infty\; =\; \backslash lim\_\{p\; \backslash to\; \backslash infty\}\; \backslash |x\backslash |\_p$$ if the right-hand side is finite, or the left-hand side is infinite. Thus, we will consider $\backslash ell^p$ spaces for $1\; \backslash leq\; p\; \backslash leq\; \backslash infty.$

The $p$-norm thus defined on $\backslash ell^p$ is indeed a norm, and $\backslash ell^p$ together with this norm is a Banach space.

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General ℓ^p-space In complete analogy to the preceding definition one can define the space $\backslash ell^p(I)$ over a general index set $I$ (and ) as where convergence on the right requires that only countably many summands are nonzero (see also Absolute convergence over sets). With the norm $$\backslash |x\backslash |\_p\; =\; \backslash left(\backslash sum\_\{i\backslash in\; I\}\; |x\_i|^p\backslash right)^\{1/p\}$$ the space $\backslash ell^p(I)$ becomes a Banach space. In the case where $I$ is finite with $n$ elements, this construction yields $\backslash Reals^n$ with the $p$-norm defined above. If $I$ is countably infinite, this is exactly the sequence space $\backslash ell^p$ defined above. For uncountable sets $I$ this is a non-Separable space Banach space which can be seen as the locally convex direct limit of $\backslash ell^p$-sequence spaces.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

For $p\; =\; 2,$ the $\backslash |\backslash ,\backslash cdot\backslash ,\backslash |\_2$-norm is even induced by a canonical inner product $\backslash langle\; \backslash ,\backslash cdot,\backslash ,\backslash cdot\backslash rangle,$ called the , which means that $\backslash |\backslash mathbf\{x\}\backslash |\_2\; =\; \backslash sqrt\{\backslash langle\backslash mathbf\{x\},\; \backslash mathbf\{x\}\backslash rangle\}$ holds for all vectors $\backslash mathbf\{x\}.$ This inner product can expressed in terms of the norm by using the polarization identity. On $\backslash ell^2,$ it can be defined by $$\backslash langle\; \backslash left(x\_i\backslash right)\_\{i\},\; \backslash left(y\_n\backslash right)\_\{i\}\; \backslash rangle\_\{\backslash ell^2\}\; ~=~\; \backslash sum\_i\; x\_i\; \backslash overline\{y\_i\}.$$ Now consider the case $p\; =\; \backslash infty.$ Define where for all $x$

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

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L^p spaces and Lebesgue integrals An $L^p$ space may be defined as a space of measurable functions for which the $p$-th power of the absolute value is Lebesgue integrable, where functions which agree almost everywhere are identified. More generally, let $(S,\; \backslash Sigma,\; \backslash mu)$ be a measure space and $1\; \backslash leq\; p\; \backslash leq\; \backslash infty.$The definitions of $\backslash |\backslash cdot\backslash |\_p,$ $\backslash mathcal\{L\}^p(S,\backslash ,\; \backslash mu),$ and $L^p(S,\backslash ,\; \backslash mu)$ can be extended to all (rather than just $1\; \backslash leq\; p\; \backslash leq\; \backslash infty$), but it is only when $1\; \backslash leq\; p\; \backslash leq\; \backslash infty$ that $\backslash |\backslash cdot\backslash |\_p$ is guaranteed to be a norm (although $\backslash |\backslash cdot\backslash |\_p$ is a quasi-seminorm for all ). When $p\; \backslash neq\; \backslash infty$, consider the set $\backslash mathcal\{L\}^p(S,\backslash ,\; \backslash mu)$ of all measurable functions $f$ from $S$ to $\backslash Complex$ or $\backslash Reals$ whose absolute value raised to the $p$-th power has a finite integral, or in symbols:

To define the set for $p\; =\; \backslash infty,$ recall that two functions $f$ and $g$ defined on $S$ are said to be , written , if the set $\backslash \{s\; \backslash in\; S\; :\; f(s)\; \backslash neq\; g(s)\backslash \}$ is measurable and has measure zero. Similarly, a measurable function $f$ (and its absolute value) is (or ) by a real number $C,$ written , if the (necessarily) measurable set $\backslash \{s\; \backslash in\; S\; :\; |f(s)|\; >\; C\backslash \}$ has measure zero. The space $\backslash mathcal\{L\}^\backslash infty(S,\backslash mu)$ is the set of all measurable functions $f$ that are bounded almost everywhere (by some real $C$) and $\backslash |f\backslash |\_\backslash infty$ is defined as the infimum of these bounds: $$\backslash |f\backslash |\_\backslash infty\; ~\backslash stackrel\{\backslash scriptscriptstyle\backslash text\{def\}\}\{=\}~\; \backslash inf\; \backslash \{C\; \backslash in\; \backslash Reals\_\{\backslash geq\; 0\}\; :\; |f(s)|\; \backslash leq\; C\; \backslash text\{\; for\; almost\; every\; \}\; s\backslash \}.$$ When $\backslash mu(S)\; \backslash neq\; 0$ then this is the same as the essential supremum of the absolute value of $f$:

For example, if $f$ is a measurable function that is equal to $0$ almost everywhereFor example, if a non-empty measurable set $N\; \backslash neq\; \backslash varnothing$ of measure $\backslash mu(N)\; =\; 0$ exists then its indicator function $\backslash mathbf\{1\}\_N$ satisfies $\backslash |\backslash mathbf\{1\}\_N\backslash |\_p\; =\; 0$ although $\backslash mathbf\{1\}\_N\; \backslash neq\; 0.$ then $\backslash |f\backslash |\_p\; =\; 0$ for every $p$ and thus $f\; \backslash in\; \backslash mathcal\{L\}^p(S,\backslash ,\; \backslash mu)$ for all $p.$

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

Seminormed space of $p$-th power integrable functions

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

Absolute homogeneity, the triangle inequality, and non-negativity are the defining properties of a seminorm. Thus $\backslash |\backslash cdot\backslash |\_p$ is a seminorm and the set $\backslash mathcal\{L\}^p(S,\backslash ,\; \backslash mu)$ of $p$-th power integrable functions together with the function $\backslash |\backslash cdot\backslash |\_p$ defines a seminormed vector space. In general, the seminorm $\backslash |\backslash cdot\backslash |\_p$ is not a norm because there might exist measurable functions $f$ that satisfy $\backslash |f\backslash |\_p\; =\; 0$ but are not equal to $0$ ($\backslash |\backslash cdot\backslash |\_p$ is a norm if and only if no such $f$ exists).

Zero sets of $p$-seminorms

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

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

Quotient vector space

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

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

The $p$-norm on the quotient vector space

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

The Lebesgue $L^p$ space

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

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

The above definitions generalize to .

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

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

Similar to $\backslash ell^2$ spaces, $L^2$ is the only Hilbert space among $L^p$ spaces. In the complex case, the inner product on $L^2$ is defined by $$\backslash langle\; f,\; g\; \backslash rangle\; =\; \backslash int\_S\; f(x)\; \backslash overline\{g(x)\}\; \backslash ,\; \backslash mathrm\{d\}\backslash mu(x).$$ Functions in $L^2$ 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 .

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

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

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When If then $L^p(\backslash mu)$ can be defined as above, that is: In this case, however, the $p$-norm $\backslash |f\backslash |\_p\; =\; N\_p(f)^\{1/p\}$ does not satisfy the triangle inequality and defines only a quasi-norm. The inequality $(a\; +\; b)^p\; \backslash leq\; a^p\; +\; b^p,$ valid for $a,\; b\; \backslash geq\; 0,$ implies that $$N\_p(f\; +\; g)\; \backslash leq\; N\_p(f)\; +\; N\_p(g)$$ and so the function $$d\_p(f\; ,g)\; =\; N\_p(f\; -\; g)\; =\; \backslash |f\; -\; g\backslash |\_p^p$$ is a metric on $L^p(\backslash mu).$ The resulting metric space is complete.

In this setting $L^p$ satisfies a reverse Minkowski inequality, that is for $u,\; v\; \backslash in\; L^p$ $$\backslash Big\backslash ||u|\; +\; |v|\backslash Big\backslash |\_p\; \backslash geq\; \backslash |u\backslash |\_p\; +\; \backslash |v\backslash |\_p$$

This result may be used to prove Clarkson's inequalities, which are in turn used to establish the uniform convexity of the spaces $L^p$ for .

The space $L^p$ for 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 $\backslash ell^p$ or $L^p(0,),$ every open convex set containing the $0$ function is unbounded for the $p$-quasi-norm; therefore, the $0$ vector does not possess a fundamental system of convex neighborhoods. Specifically, this is true if the measure space $S$ contains an infinite family of disjoint measurable sets of finite positive measure.

The only nonempty convex open set in $L^p(0,)$ is the entire space. Consequently, there are no nonzero continuous linear functionals on $L^p(0,);$ the continuous dual space is the zero space. In the case of the counting measure on the natural numbers (i.e. $L^p(\backslash mu)\; =\; \backslash ell^p$), the bounded linear functionals on $\backslash ell^p$ are exactly those that are bounded on $\backslash ell^1$, i.e., those given by sequences in $\backslash ell^\backslash infty.$ Although $\backslash ell^p$ 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 $\backslash Reals^n,$ rather than work with $L^p$ for it is common to work with the Hardy space 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 for .

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Properties

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Hölder's inequality Suppose $p,\; q,\; r\; \backslash in\; 1,$ satisfy $\backslash tfrac\{1\}\{p\}\; +\; \backslash tfrac\{1\}\{q\}\; =\; \backslash tfrac\{1\}\{r\}$. If $f\; \backslash in\; L^p(S,\; \backslash mu)$ and $g\; \backslash in\; L^q(S,\; \backslash mu)$ then $f\; g\; \backslash in\; L^r(S,\; \backslash mu)$ and $$\backslash |f\; g\backslash |\_r\; ~\backslash leq~\; \backslash |f\backslash |\_p\; \backslash ,\; \backslash |g\backslash |\_q.$$

This inequality, called Hölder's inequality, is in some sense optimal since if $r\; =\; 1$ and $f$ is a measurable function such that where the supremum is taken over the closed unit ball of $L^q(S,\; \backslash mu),$ then $f\; \backslash in\; L^p(S,\; \backslash mu)$ and $$\backslash |f\backslash |\_p\; ~=~\; \backslash sup\_\{\backslash |g\backslash |\_q\; \backslash leq\; 1\}\; \backslash ,\; \backslash int\_S\; f\; g\; \backslash ,\; \backslash mathrm\{d\}\; \backslash mu.$$

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Generalized Minkowski inequality Minkowski inequality, which states that $\backslash |\backslash cdot\backslash |\_p$ satisfies the triangle inequality, can be generalized: If the measurable function $F\; :\; M\; \backslash times\; N\; \backslash to\; \backslash Reals$ is non-negative (where $(M,\; \backslash mu)$ and $(N,\; \backslash nu)$ are measure spaces) then for all $1\; \backslash leq\; p\; \backslash leq\; q\; \backslash leq\; \backslash infty,$ $$\backslash left\backslash |\backslash left\backslash |F(\backslash ,\backslash cdot,\; n)\backslash right\backslash |\_\{L^p(M,\; \backslash mu)\}\backslash right\backslash |\_\{L^q(N,\; \backslash nu)\}\; ~\backslash leq~\; \backslash left\backslash |\backslash left\backslash |F(m,\; \backslash cdot)\backslash right\backslash |\_\{L^q(N,\; \backslash nu)\}\backslash right\backslash |\_\{L^p(M,\; \backslash mu)\}\; \backslash \; .$$

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Atomic decomposition If then every non-negative $f\; \backslash in\; L^p(\backslash mu)$ has an , meaning that there exist a sequence $(r\_n)\_\{n\; \backslash in\; \backslash Z\}$ of non-negative real numbers and a sequence of non-negative functions $(f\_n)\_\{n\; \backslash in\; \backslash Z\},$ called , whose supports $\backslash left(\backslash operatorname\{supp\}\; f\_n\backslash right)\_\{n\; \backslash in\; \backslash Z\}$ are Disjoint sets of measure $\backslash mu\backslash left(\backslash operatorname\{supp\}\; f\_n\backslash right)\; \backslash leq\; 2^\{n+1\},$ such that $$f\; ~=~\; \backslash sum\_\{n\; \backslash in\; \backslash Z\}\; r\_n\; \backslash ,\; f\_n\; \backslash ,\; ,$$ and for every integer $n\; \backslash in\; \backslash Z,$ $$\backslash |f\_n\backslash |\_\backslash infty\; ~\backslash leq~\; 2^\{-\backslash tfrac\{n\}\{p\}\}\; \backslash ,\; ,$$ and $$\backslash tfrac\{1\}\{2\}\; \backslash |f\backslash |\_p^p\; ~\backslash leq~\; \backslash sum\_\{n\; \backslash in\; \backslash Z\}\; r\_n^p\; ~\backslash leq~\; 2\; \backslash |f\backslash |^p\_p\; \backslash ,\; ,$$ and where moreover, the sequence of functions $(r\_n\; f\_n)\_\{n\; \backslash in\backslash Z\}$ depends only on $f$ (it is independent of $p$). These inequalities guarantee that $\backslash |f\_n\backslash |\_p^p\; \backslash leq\; 2$ for all integers $n$ while the supports of $(f\_n)\_\{n\; \backslash in\; \backslash Z\}$ being pairwise disjoint implies $$\backslash |f\backslash |\_p^p\; ~=~\; \backslash sum\_\{n\; \backslash in\; \backslash Z\}\; r\_n^p\; \backslash ,\; \backslash |f\_n\backslash |^p\_p\; \backslash ,\; .$$

An atomic decomposition can be explicitly given by first defining for every integer $n\; \backslash in\; \backslash Z,$This infimum is attained by $t\_n;$ that is, holds. and then letting where $\backslash mu(f\; >\; t)\; =\; \backslash mu(\backslash \{s\; :\; f(s)\; >\; t\backslash \})$ denotes the measure of the set $(f\; >\; t)\; :=\; \backslash \{s\; \backslash in\; S\; :\; f(s)\; >\; t\backslash \}$ and denotes the indicator function of the set The sequence $(t\_n)\_\{n\; \backslash in\; \backslash Z\}$ is decreasing and converges to $0$ as $n\; \backslash to\; \backslash infty.$ Consequently, if $t\_n\; =\; 0$ then $t\_\{n+1\}\; =\; 0$ and so that is identically equal to $0$ (in particular, the division $\backslash tfrac\{1\}\{r\_n\}$ by $r\_n\; =\; 0$ causes no issues).

The complementary cumulative distribution function $t\; \backslash in\; \backslash Reals\; \backslash mapsto\; \backslash mu(|f|\; >\; t)$ of $|f|\; =\; f$ that was used to define the $t\_n$ also appears in the definition of the weak $L^p$-norm (given below) and can be used to express the $p$-norm $\backslash |\backslash cdot\backslash |\_p$ (for ) of $f\; \backslash in\; L^p(S,\; \backslash mu)$ as the integral $$\backslash |f\backslash |\_p^p\; ~=~\; p\; \backslash ,\; \backslash int\_0^\backslash infty\; t^\{p-1\}\; \backslash mu(|f|\; >\; t)\; \backslash ,\; \backslash mathrm\{d\}\; t\; \backslash ,\; ,$$ where the integration is with respect to the usual Lebesgue measure on $(0,\; \backslash infty).$

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Dual spaces The Continuous dual of $L^p(\backslash mu)$ for has a natural isomorphism with $L^q(\backslash mu),$ where $q$ is such that $\backslash tfrac\{1\}\{p\}\; +\; \backslash tfrac\{1\}\{q\}\; =\; 1$. This isomorphism associates $g\; \backslash in\; L^q(\backslash mu)$ with the functional $\backslash kappa\_p(g)\; \backslash in\; L^p(\backslash mu)^*$ defined by $$f\; \backslash mapsto\; \backslash kappa\_p(g)(f)\; =\; \backslash int\; f\; g\; \backslash ,\; \backslash mathrm\{d\}\backslash mu$$ for every $f\; \backslash in\; L^p(\backslash mu).$

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

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

$$j\_p\; :\; L^p(\backslash mu)\; \backslash mathrel\{\backslash overset\{\backslash kappa\_q\}\{\backslash longrightarrow\}\}\; L^q(\backslash mu)^*\; \backslash mathrel\{\backslash overset\{\backslash left(\backslash kappa\_p^\{-1\}\backslash right)^*\}\{\backslash longrightarrow\}\}\; L^p(\backslash mu)^\{**\}$$

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

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

The dual of $L^\backslash infty(\backslash mu)$ is subtler. Elements of $L^\backslash infty(\backslash mu)^*$ can be identified with bounded signed finitely additive measures on $S$ that are absolutely continuous with respect to $\backslash mu.$ See ba space for more details. If we assume the axiom of choice, this space is much bigger than $L^1(\backslash mu)$ 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 $\backslash ell^\backslash infty$ is $\backslash ell^1.$ See Sections 14.77 and 27.44–47

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

1. If finite measure).
2. If 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 $L^q$ to $L^p$ in the first case and $L^p$ to $L^q$ in the second. Indeed, if the domain $S$ has finite measure, one can make the following explicit calculation using Hölder's inequality $$\backslash \; \backslash |\backslash mathbf\{1\}f^p\backslash |\_1\; \backslash leq\; \backslash |\backslash mathbf\{1\}\backslash |\_\{q/(q-p)\}\; \backslash |f^p\backslash |\_\{q/p\}$$ leading to $$\backslash \; \backslash |f\backslash |\_p\; \backslash leq\; \backslash mu(S)^\{1/p\; -\; 1/q\}\; \backslash |f\backslash |\_q\; .$$

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

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

More can be said when $S$ is a Normal space topological space and $\backslash Sigma$ its Borel algebra.

Suppose $V\; \backslash subseteq\; S$ is an open set with Then for every Borel set $A\; \backslash in\; \backslash Sigma$ contained in $V$ there exist a closed set $F$ and an open set $U$ such that for every $\backslash varepsilon\; >\; 0$. Subsequently, there exists a Urysohn function $0\; \backslash leq\; \backslash varphi\; \backslash leq\; 1$ on $S$ that is $1$ on $F$ and $0$ on $S\; \backslash setminus\; U,$ with

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

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

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

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Applications

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Statistics In statistics, measures of central tendency and statistical dispersion, such as the mean, median, and standard deviation, can be defined in terms of $L^p$ 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 Taxicab geometry of a solution's vector of parameter values (i.e. the sum of its absolute values), or its squared $L^2$ norm (its Euclidean norm). 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 $L^1$ norm and the squared $L^2$ norm of the parameter vector.

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

By contrast, if $p\; >\; 2,$ the Fourier transform does not map into $L^q.$

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

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Generalizations and extensions

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Weak Let $(S,\; \backslash Sigma,\; \backslash mu)$ be a measure space, and $f$ a measurable function with real or complex values on $S.$ The distribution function of $f$ is defined for $t\; \backslash geq\; 0$ by $$\backslash lambda\_f(t)\; =\; \backslash mu\backslash \{x\; \backslash in\; S\; :\; |f(x)|\; >\; t\backslash \}.$$

If $f$ is in $L^p(S,\; \backslash mu)$ for some $p$ with then by Markov's inequality, $$\backslash lambda\_f(t)\; \backslash leq\; \backslash frac\{\backslash |f\backslash |\_p^p\}\{t^p\}$$

A function $f$ is said to be in the space weak $L^p(S,\; \backslash mu)$, or $L^\{p,w\}(S,\; \backslash mu),$ if there is a constant $C\; >\; 0$ such that, for all $t\; >\; 0,$ $$\backslash lambda\_f(t)\; \backslash leq\; \backslash frac\{C^p\}\{t^p\}$$

The best constant $C$ for this inequality is the $L^\{p,w\}$-norm of $f,$ and is denoted by $$\backslash |f\backslash |\_\{p,w\}\; =\; \backslash sup\_\{t\; >\; 0\}\; ~\; t\; \backslash lambda\_f^\{1/p\}(t).$$

The weak $L^p$ coincide with the $L^\{p,\backslash infty\},$ so this notation is also used to denote them.

The $L^\{p,w\}$-norm is not a true norm, since the triangle inequality fails to hold. Nevertheless, for $f$ in $L^p(S,\; \backslash mu),$ $$\backslash |f\backslash |\_\{p,w\}\; \backslash leq\; \backslash |f\backslash |\_p$$ and in particular $L^p(S,\; \backslash mu)\; \backslash subset\; L^\{p,w\}(S,\; \backslash mu).$

In fact, one has $$\backslash |f\backslash |^p\_\{L^p\}\; =\; \backslash int\; |f(x)|^p\; d\backslash mu(x)\; \backslash geq\; \backslash int\_\{\backslash \{|f(x)|\; >\; t\; \backslash \}\}\; t^p\; +\; \backslash int\_\{\backslash \{|f(x)|\; \backslash leq\; t\; \backslash \}\}\; |f|^p\; \backslash geq\; t^p\; \backslash mu(\backslash \{|f|\; >\; t\; \backslash \}),$$ and raising to power $1/p$ and taking the supremum in $t$ one has $$\backslash |f\backslash |\_\{L^p\}\; \backslash geq\; \backslash sup\_\{t\; >\; 0\}\; t\; \backslash ;\; \backslash mu(\backslash \{|f|\; >\; t\; \backslash \})^\{1/p\}\; =\; \backslash |f\backslash |\_\{L^\{p,w\}\}.$$

Under the convention that two functions are equal if they are equal $\backslash mu$ almost everywhere, then the spaces $L^\{p,w\}$ are complete .

For any the expression is comparable to the $L^\{p,w\}$-norm. Further in the case $p\; >\; 1,$ this expression defines a norm if $r\; =\; 1.$ Hence for $p\; >\; 1$ the weak $L^p$ spaces are .

A major result that uses the $L^\{p,w\}$-spaces is the Marcinkiewicz interpolation theorem, which has broad applications to harmonic analysis and the study of singular integrals.

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Weighted spaces As before, consider a measure space $(S,\; \backslash Sigma,\; \backslash mu).$ Let $w\; :\; S\; \backslash to\; [a,\; \backslash infty),\; a\; >\; 0$ be a measurable function. The $w$- weighted $L^p$ space is defined as $L^p(S,\; w\; \backslash ,\; \backslash mathrm\{d\}\; \backslash mu),$ where $w\; \backslash ,\; \backslash mathrm\{d\}\; \backslash mu$ means the measure $\backslash nu$ defined by $$\backslash nu(A)\; \backslash equiv\; \backslash int\_A\; w(x)\; \backslash ,\; \backslash mathrm\{d\}\; \backslash mu\; (x),\; \backslash qquad\; A\; \backslash in\; \backslash Sigma,$$

or, in terms of the Radon–Nikodym derivative, $w\; =\; \backslash tfrac\{\backslash mathrm\{d\}\; \backslash nu\}\{\backslash mathrm\{d\}\; \backslash mu\}$ the norm for $L^p(S,\; w\; \backslash ,\; \backslash mathrm\{d\}\; \backslash mu)$ is explicitly $$\backslash |u\backslash |\_\{L^p(S,\; w\; \backslash ,\; \backslash mathrm\{d\}\; \backslash mu)\}\; \backslash equiv\; \backslash left(\backslash int\_S\; w(x)\; |u(x)|^p\; \backslash ,\; \backslash mathrm\{d\}\; \backslash mu(x)\backslash right)^\{1/p\}$$

As $L^p$-spaces, the weighted spaces have nothing special, since $L^p(S,\; w\; \backslash ,\; \backslash mathrm\{d\}\; \backslash mu)$ is equal to $L^p(S,\; \backslash mathrm\{d\}\; \backslash nu).$ But they are the natural framework for several results in harmonic analysis ; they appear for example in the Muckenhoupt theorem: for the classical Hilbert transform is defined on $L^p(\backslash mathbf\{T\},\; \backslash lambda)$ where $\backslash mathbf\{T\}$ denotes the unit circle and $\backslash lambda$ the Lebesgue measure; the (nonlinear) Hardy–Littlewood maximal operator is bounded on $L^p(\backslash Reals^n,\; \backslash lambda).$ Muckenhoupt's theorem describes weights $w$ such that the Hilbert transform remains bounded on $L^p(\backslash mathbf\{T\},\; w\; \backslash ,\; \backslash mathrm\{d\}\; \backslash lambda)$ and the maximal operator on $L^p(\backslash Reals^n,\; w\; \backslash ,\; \backslash mathrm\{d\}\; \backslash lambda).$

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spaces on manifolds One may also define spaces $L^p(M)$ on a manifold, called the intrinsic $L^p$ spaces of the manifold, using densities.

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Vector-valued spaces Given a measure space $(\backslash Omega,\; \backslash Sigma,\; \backslash mu)$ and a locally convex space $E$ (here assumed to be complete), it is possible to define spaces of $p$-integrable $E$-valued functions on $\backslash Omega$ in a number of ways. One way is to define the spaces of Bochner integral and Pettis integral functions, and then endow them with locally convex Vector topology that are (each in their own way) a natural generalization of the usual $L^p$ topology. Another way involves topological tensor products of $L^p(\backslash Omega,\; \backslash Sigma,\; \backslash mu)$ with $E.$ Element of the vector space $L^p(\backslash Omega,\; \backslash Sigma,\; \backslash mu)\; \backslash otimes\; E$ are finite sums of simple tensors $f\_1\; \backslash otimes\; e\_1\; +\; \backslash cdots\; +\; f\_n\; \backslash otimes\; e\_n,$ where each simple tensor $f\; \backslash times\; e$ may be identified with the function $\backslash Omega\; \backslash to\; E$ that sends $x\; \backslash mapsto\; e\; f(x).$ This tensor product $L^p(\backslash Omega,\; \backslash Sigma,\; \backslash mu)\; \backslash otimes\; E$ 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 $L^p(\backslash Omega,\; \backslash Sigma,\; \backslash mu)\; \backslash otimes\_\backslash pi\; E,$ and the injective tensor product, denoted by $L^p(\backslash Omega,\; \backslash Sigma,\; \backslash mu)\; \backslash otimes\_\backslash varepsilon\; E.$ In general, neither of these space are complete so their completions are constructed, which are respectively denoted by $L^p(\backslash Omega,\; \backslash Sigma,\; \backslash mu)\; \backslash widehat\{\backslash otimes\}\_\backslash pi\; E$ and $L^p(\backslash Omega,\; \backslash Sigma,\; \backslash mu)\; \backslash widehat\{\backslash otimes\}\_\backslash varepsilon\; E$ (this is analogous to how the space of scalar-valued on $\backslash Omega,$ when seminormed by any $\backslash |\backslash cdot\backslash |\_p,$ is not complete so a completion is constructed which, after being quotiented by $\backslash ker\; \backslash |\backslash cdot\backslash |\_p,$ is isometrically isomorphic to the Banach space $L^p(\backslash Omega,\; \backslash mu)$). Alexander Grothendieck showed that when $E$ 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.

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space of measurable functions The vector space of (equivalence classes of) measurable functions on $(S,\; \backslash Sigma,\; \backslash mu)$ is denoted $L^0(S,\; \backslash Sigma,\; \backslash mu)$ . By definition, it contains all the $L^p,$ and is equipped with the topology of convergence in measure. When $\backslash mu$ is a probability measure (i.e., $\backslash mu(S)\; =\; 1$), this mode of convergence is named convergence in probability. The space $L^0$ is always a topological abelian group but is only a topological vector space if This is because scalar multiplication is continuous if and only if If $(S,\backslash Sigma,\backslash mu)$ is $\backslash sigma$-finite then the weaker topology of local convergence in measure is an F-space, i.e. a completely metrizable topological vector space. Moreover, this topology is isometric to global convergence in measure $(S,\backslash Sigma,\backslash nu)$ for a suitable choice of probability measure $\backslash nu.$

The description is easier when $\backslash mu$ is finite. If $\backslash mu$ is a finite measure on $(S,\; \backslash Sigma),$ the $0$ function admits for the convergence in measure the following fundamental system of neighborhoods

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

For the infinite Lebesgue measure $\backslash lambda$ on $\backslash Reals^n,$ the definition of the fundamental system of neighborhoods could be modified as follows

The resulting space $L^0(\backslash Reals^n,\; \backslash lambda)$, with the topology of local convergence in measure, is isomorphic to the space $L^0(\backslash Reals^n,\; g\; \backslash ,\; \backslash lambda),$ for any positive $\backslash lambda$–integrable density $g.$

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See also

• $\backslash left(\; L^1\_\{\backslash text\{loc\}\}\backslash right)$

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Notes

• .
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• Elias M. Stein (2025). 9781400840557, Princeton University Press. ISBN 9781400840557

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External links

• Proof that L ^p spaces are complete

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