# Lp space

An \( L^p \) space is a space of measurable functions whose \( p \)-th power of absolute value is integrable, equipped with a norm built from that integral; the spaces are also called Lebesgue spaces and are central objects of functional analysis.<sup>[1](https://encyclopediaofmath.org/wiki/Lp_spaces)</sup> For a measure space \( (S, F, \mu) \) and \( 1 \le p < \infty \), the space \( L^p(S, F, \mu) \) contains the equivalence classes of complex measurable functions \( f \) with \( \int_S |f(s)|^p \, d\mu(s) < \infty \), where two functions are identified when they agree almost everywhere (equal except on a set of measure zero) with respect to \( \mu \).<sup>[1](https://encyclopediaofmath.org/wiki/Lp_spaces)</sup>

| Key fact | Detail |
|---|---|
| Definition | \( L^p(S,F,\mu) \) for \( 1 \le p < \infty \): equivalence classes of measurable \( f \) with \( \int_S \|f\|^p \, d\mu < \infty \), modulo equality almost everywhere<sup>[1](https://encyclopediaofmath.org/wiki/Lp_spaces)</sup> |
| Norm | \( \|f\|_p = \left( \int_S \|f(s)\|^p \, d\mu(s) \right)^{1/p} \); for \( p = \infty \), the essential supremum<sup>[1](https://encyclopediaofmath.org/wiki/Lp_spaces)</sup><sup> • </sup><sup>[5](https://sites.duke.edu/dums/files/2024/03/Notes_for_PDE__Weak_Derivatives_and_Sobolev_Spaces-1-1.pdf)</sup> |
| Completeness | \( L^p(X,\mu) \) is a Banach space for every measure space and \( 1 \le p \le \infty \) (Riesz–Fischer theorem)<sup>[2](https://faculty.etsu.edu/gardnerr/5210/notes/19-1.pdf)</sup><sup> • </sup><sup>[10](https://qft.org/mathematical-methods/analysis-measure-fourier/lp-spaces-inequalities-weak-convergence/)</sup> |
| Core inequalities | Hölder: \( \|fg\|_1 \le \|f\|_p \|g\|_q \) when \( 1/p + 1/q = 1 \); Minkowski: \( \|f+g\|_p \le \|f\|_p + \|g\|_p \)<sup>[2](https://faculty.etsu.edu/gardnerr/5210/notes/19-1.pdf)</sup> |
| Duality | \( (L^p)^* \cong L^q \) when \( 1/p + 1/q = 1 \)<sup>[5](https://sites.duke.edu/dums/files/2024/03/Notes_for_PDE__Weak_Derivatives_and_Sobolev_Spaces-1-1.pdf)</sup> |
| Special case | \( L^2 \) is a Hilbert space with \( \langle f, g \rangle = \int_\Omega f\overline{g} \) for complex-valued functions; for \( p \ne 2 \) the norm is not induced by an inner product<sup>[5](https://sites.duke.edu/dums/files/2024/03/Notes_for_PDE__Weak_Derivatives_and_Sobolev_Spaces-1-1.pdf)</sup><sup> • </sup><sup>[14](https://www.math.toronto.edu/ivrii/PDE-textbook/Chapter5/S5.2.html)</sup> |
| Excluded range | For \( 0 < p < 1 \) the quantity \( \|f\|_p \) fails the triangle inequality and is not a norm<sup>[1](https://encyclopediaofmath.org/wiki/Lp_spaces)</sup> |

## Definition and norm

The defining condition is \( p \)-power summability: a measurable function \( f \) belongs to \( L^p \) when \( \int_\Omega \|f\|^p \, d\mu < \infty \), and its norm is

\[ \|f\|_p = \left( \int_\Omega \|f\|^p \, d\mu \right)^{1/p}. \]

Because individual functions of norm zero differ only on a set of measure zero, the elements are formally equivalence classes under the relation \( f = g \) almost everywhere; in practice one speaks simply of functions.<sup>[1](https://encyclopediaofmath.org/wiki/Lp_spaces)</sup><sup> • </sup><sup>[2](https://faculty.etsu.edu/gardnerr/5210/notes/19-1.pdf)</sup> The case \( p = \infty \) is defined separately: \( L^\infty \) consists of equivalence classes of measurable functions bounded almost everywhere by some finite constant \( M \), with the essential supremum as norm, that is \( \inf\{x : m\{\|f\| > x\} = 0\} \).<sup>[1](https://encyclopediaofmath.org/wiki/Lp_spaces)</sup><sup> • </sup><sup>[5](https://sites.duke.edu/dums/files/2024/03/Notes_for_PDE__Weak_Derivatives_and_Sobolev_Spaces-1-1.pdf)</sup>

The restriction to \( p \ge 1 \) is not cosmetic. For \( 0 < p < 1 \) the quantity \( \|f\|_p \) does not satisfy the triangle inequality, so it fails to be a norm, although the spaces can still be defined as quasi-normed vector spaces.<sup>[1](https://encyclopediaofmath.org/wiki/Lp_spaces)</sup><sup> • </sup><sup>[3](https://www.math.ucdavis.edu/~krc/analysis_notes.pdf)</sup>

## The basic inequalities

Three named inequalities carry the theory. [Hölder's inequality](https://www.edgepedia.org/holders-inequality) states that if \( 1 \le p \le \infty \) and \( q \) is the conjugate exponent with \( 1/p + 1/q = 1 \), then for \( f \in L^p \) and \( g \in L^q \) the product \( fg \) lies in \( L^1 \) and \( \|fg\|_1 \le \|f\|_p \|g\|_q \).<sup>[2](https://faculty.etsu.edu/gardnerr/5210/notes/19-1.pdf)</sup> Taking \( p = q = 2 \) gives the [Cauchy–Schwarz inequality](https://www.edgepedia.org/cauchy-schwarz-inequality).<sup>[2](https://faculty.etsu.edu/gardnerr/5210/notes/19-1.pdf)</sup> Minkowski's inequality, \( \|f + g\|_p \le \|f\|_p + \|g\|_p \) for \( 1 \le p \le \infty \), is exactly the triangle inequality and makes \( L^p \) a normed linear space.<sup>[2](https://faculty.etsu.edu/gardnerr/5210/notes/19-1.pdf)</sup> Young's inequality for convolutions extends the scale: if \( u \in L^p(\mathbb{R}^N) \) and \( v \in L^1(\mathbb{R}^N) \), then \( u * v \in L^p \) with \( \|u * v\|_{L^p} \le \|u\|_{L^p} \|v\|_{L^1} \), and in general form \( \|u * v\|_{L^r} \le \|u\|_{L^p} \|v\|_{L^q} \) when \( 1/p + 1/q = 1 + 1/r \).<sup>[11](https://people.cas.uab.edu/~mswyatt/Sobolev_Notes_June_6.pdf)</sup>

## Completeness and density

The Riesz–Fischer theorem states that for every measure space \( (X, M, \mu) \) and \( 1 \le p \le \infty \), the space \( L^p(X, \mu) \) is a [Banach space](https://www.edgepedia.org/banach-space), that is, every [Cauchy sequence](https://www.edgepedia.org/cauchy-sequence) converges within the space; moreover, if a sequence converges to \( f \) in \( L^p \) norm, a subsequence converges pointwise almost everywhere to \( f \).<sup>[2](https://faculty.etsu.edu/gardnerr/5210/notes/19-1.pdf)</sup><sup> • </sup><sup>[10](https://qft.org/mathematical-methods/analysis-measure-fourier/lp-spaces-inequalities-weak-convergence/)</sup> Completeness is what makes norm limits stay inside the same space and underlies most applications.<sup>[10](https://qft.org/mathematical-methods/analysis-measure-fourier/lp-spaces-inequalities-weak-convergence/)</sup>

Simple functions anchor the space from below: for \( 1 \le p < \infty \), the subspace of simple functions on \( X \) that vanish outside a set of finite measure is dense in \( L^p(X, \mu) \).<sup>[2](https://faculty.etsu.edu/gardnerr/5210/notes/19-1.pdf)</sup> Convergence of a sequence in \( L^p \) norm can be characterized by uniform integrability and tightness through the Vitali \( L^p \) convergence theorem: if \( f_n \to f \) pointwise a.e. with \( f \in L^p \), then \( f_n \to f \) in \( L^p \) if and only if \( \{\|f\|^p\} \) is uniformly integrable and tight.<sup>[2](https://faculty.etsu.edu/gardnerr/5210/notes/19-1.pdf)</sup>

**Inclusions between the spaces depend on the measure.** On a finite measure space, if \( 1 \le p_1 < p_2 \le \infty \) then \( L^{p_2} \subset L^{p_1} \), with \( \|f\|_{p_1} \le [\mu(X)]^{(p_2 - p_1)/(p_1 p_2)} \|f\|_{p_2} \).<sup>[2](https://faculty.etsu.edu/gardnerr/5210/notes/19-1.pdf)</sup> For the sequence spaces \( \ell^p = L^p(\mathbb{N}) \) with counting measure the inclusions run the other way, \( \ell^1 \subset \ell^2 \subset \ell^\infty \).<sup>[12](https://math.hodapp.com/lectures/higher/functional-analysis/lp-spaces.html)</sup>

## Duality and the special role of p = 2

The dual of \( L^p \) is another \( L^q \) space. One form of the [Riesz representation theorem](https://www.edgepedia.org/riesz-representation-theorem) states that if \( 1/p + 1/q = 1 \), then \( L^q \cong (L^p)^* \); for the sequence spaces, \( (\ell^p)^* \cong \ell^q \).<sup>[5](https://sites.duke.edu/dums/files/2024/03/Notes_for_PDE__Weak_Derivatives_and_Sobolev_Spaces-1-1.pdf)</sup> Historically, the representation problem for \( L^2 \) was solved by Fréchet and by Riesz in two papers published in the same issue of the *Comptes rendus* of the Académie des Sciences in Paris on 24 June 1907; in an important 1910 article in volume 69 of *Mathematische Annalen*, Riesz presented the representation theorem for continuous linear functionals on \( L^p \) for \( p > 1 \), with the representing function \( \Phi \in L^q \) uniquely determined and with the norm of the functional equal to the norm of \( \Phi \).<sup>[4](https://www.treccani.it/enciclopedia/la-seconda-rivoluzione-scientifica-matematica-e-logica-le-origini-dell-analisi-funzionale_(Storia-della-Scienza)/)</sup> Riesz and Fischer had worked on \( L^2 \) convergence starting around 1905, and Riesz subsequently generalized to \( L^p \).<sup>[9](https://www2.cms.math.ca/Events/winter99/abstracts/node89.html)</sup>

\( L^2 \) occupies a privileged position because its norm comes from an inner product, \( \langle f, g \rangle = \int_\Omega f\overline{g} \) for complex-valued functions (with the conjugation omitted for real-valued functions), making it a [Hilbert space](https://www.edgepedia.org/hilbert-space); this is well defined since \( \int_\Omega \|fg\| \le \int_\Omega (\|f\|^2 + \|g\|^2)/2 < \infty \) by Cauchy–Schwarz.<sup>[5](https://sites.duke.edu/dums/files/2024/03/Notes_for_PDE__Weak_Derivatives_and_Sobolev_Spaces-1-1.pdf)</sup><sup> • </sup><sup>[14](https://www.math.toronto.edu/ivrii/PDE-textbook/Chapter5/S5.2.html)</sup> For \( p \in [1, \infty) \) with \( p \ne 2 \), the \( L^p \) norm cannot be induced by any inner product: applying the parallelogram-type identity to indicators of two disjoint sets of equal measure \( m \) forces \( (2^p + 1)^{2/p} = 5 \), which holds in \( [1, \infty) \) only at \( p = 2 \).<sup>[5](https://sites.duke.edu/dums/files/2024/03/Notes_for_PDE__Weak_Derivatives_and_Sobolev_Spaces-1-1.pdf)</sup>

## Weak convergence and weak Lp

In \( L^p \) for \( 1 < p < \infty \), a sequence can converge weakly without converging strongly. For \( \Omega = \mathbb{R}^n \) with \( 1 < p < \infty \), three basic mechanisms produce this behavior, illustrated for \( n = 1 \): oscillation, as in \( f^k(x) = \sin kx \) on \( [0,1] \); concentration, as in \( f^k(x) = k^{1/p} g(kx) \) for fixed \( g \in L^p(\mathbb{R}^1) \), which becomes large near \( x = 0 \); and translation, as in \( f^k(x) = g(x + k) \), which wanders off to infinity. In each case \( f^k \to 0 \) weakly but \( f^k \) does not converge strongly to zero or to anything else.<sup>[8](https://www.math.uh.edu/~dlabate/PagesfromAnalysis.pdf)</sup>

The term weak \( L^p \) names a different object from the weak topology. The space \( L^{p,\infty} \), called weak \( L^p \), is the Lorentz space defined through the decreasing rearrangement of \( f \); it contains \( L^p \), and the inclusion is strict in general, for example \( f(x) = \|x\|^{-n/p} \) on \( \mathbb{R}^n \) lies in \( L^{p,\infty}(\mathbb{R}^n) \) but not in \( L^p(\mathbb{R}^n) \).<sup>[6](https://metaphor.ethz.ch/x/2026/fs/401-3462-00L/functional-analysis-2-2025.pdf)</sup> More general Lorentz spaces \( L^{p,q} \) are defined by a norm \[ \|f\|_{L^{p,q}} = \left( \int_0^{+\infty} \left( u^{1/p} f^{**}(u) \right)^q \, du \right)^{1/q}, \] and \( L^{p,p} = L^p \).<sup>[7](https://www.math.u-bordeaux.fr/~pjaming/enseignement/CoursHA2022.pdf)</sup> Convergence in \( L^p \) implies convergence in weak \( L^p \), and convergence in weak \( L^p \) implies convergence in measure.<sup>[7](https://www.math.u-bordeaux.fr/~pjaming/enseignement/CoursHA2022.pdf)</sup> The name is unrelated to the weak topology on the \( L^p \) space.<sup>[6](https://metaphor.ethz.ch/x/2026/fs/401-3462-00L/functional-analysis-2-2025.pdf)</sup>

## Applications

\( L^p \) spaces supply the integrability framework for several fields. In [Fourier analysis](https://www.edgepedia.org/fourier-analysis), the Riesz–Thorin interpolation theorem and the Hausdorff–Young inequality extend the [Fourier transform](https://www.edgepedia.org/fourier-transform) to \( L^p \) functions for \( 1 \le p \le 2 \).<sup>[13](https://metaphor.ethz.ch/x/2026/fs/401-3462-00L/)</sup>

## References

1. [Lp spaces, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Lp_spaces)
2. [Chapter 19. General Lp Spaces: Completeness, Duality, and Weak Convergence, ETSU lecture notes](https://faculty.etsu.edu/gardnerr/5210/notes/19-1.pdf)
3. [Measure Theory, Integration Theory, and Lp Spaces: Introductory Lp Space Theory, UC Davis](https://www.math.ucdavis.edu/~krc/analysis_notes.pdf)
4. [La seconda rivoluzione scientifica: matematica e logica. Le origini dell'analisi funzionale, Treccani](https://www.treccani.it/enciclopedia/la-seconda-rivoluzione-scientifica-matematica-e-logica-le-origini-dell-analisi-funzionale_(Storia-della-Scienza)/)
5. [Notes for PDE, Weak Derivatives and Sobolev Spaces, Duke University](https://sites.duke.edu/dums/files/2024/03/Notes_for_PDE__Weak_Derivatives_and_Sobolev_Spaces-1-1.pdf)
6. [Functional Analysis II, ETH Zürich lecture notes](https://metaphor.ethz.ch/x/2026/fs/401-3462-00L/functional-analysis-2-2025.pdf)
7. [Cours d'analyse harmonique, Université de Bordeaux lecture notes](https://www.math.u-bordeaux.fr/~pjaming/enseignement/CoursHA2022.pdf)
8. [LP-Spaces, University of Houston analysis notes (D. Labate)](https://www.math.uh.edu/~dlabate/PagesfromAnalysis.pdf)
9. [Jal Choksi, A history of the convergence theorems of (Lebesgue) integration, CMS](https://www2.cms.math.ca/Events/winter99/abstracts/node89.html)
10. [Lp Spaces, Inequalities, and Weak Convergence, QFT.org](https://qft.org/mathematical-methods/analysis-measure-fourier/lp-spaces-inequalities-weak-convergence/)
11. [Notes for Sobolev Spaces Seminar, University of Alabama at Birmingham](https://people.cas.uab.edu/~mswyatt/Sobolev_Notes_June_6.pdf)
12. [Lp spaces and the inequality scale, Math foundations lecture notes](https://math.hodapp.com/lectures/higher/functional-analysis/lp-spaces.html)
13. [Functional Analysis II Spring 2026, ETH Zürich course page](https://metaphor.ethz.ch/x/2026/fs/401-3462-00L/)
14. [S5.2](https://www.math.toronto.edu/ivrii/PDE-textbook/Chapter5/S5.2.html)

---
*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Analysis and mathematical models › Functional analysis*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license