# Square-integrable function

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Square-integrable_function
> Markdown URL: https://mediated.wiki/source/Square-integrable_function.md
> Source: https://en.wikipedia.org/wiki/Square-integrable_function
> Source revision: 1328256312
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

Function whose squared absolute value has finite integral

This article's lead section may be too long. Please read the length guidelines and help move details into the article's body. (January 2024) (Learn how and when to remove this message)

In [mathematics](/source/Mathematics), a **square-integrable function**, also called a **quadratically integrable function** or **L 2 {\displaystyle L^{2}} function** or **square-summable function**,[1] is a [real](/source/Real_number)- or [complex](/source/Complex_number)-valued [measurable function](/source/Measurable_function) for which the [integral](/source/Integral) of the square of the [absolute value](/source/Absolute_value) is finite. Thus, square-integrability on the real line ( − ∞ , + ∞ ) {\displaystyle (-\infty ,+\infty )} is defined as follows.

f : R → C square integrable ⟺ ∫ − ∞ ∞ | f ( x ) | 2 d x < ∞ {\displaystyle f:\mathbb {R} \to \mathbb {C} {\text{ square integrable}}\quad \iff \quad \int _{-\infty }^{\infty }|f(x)|^{2}\,\mathrm {d} x<\infty }

One may also speak of quadratic integrability over bounded intervals such as [ a , b ] {\displaystyle [a,b]} for a ≤ b {\displaystyle a\leq b} .[2]

f : [ a , b ] → C square integrable on [ a , b ] ⟺ ∫ a b | f ( x ) | 2 d x < ∞ {\displaystyle f:[a,b]\to \mathbb {C} {\text{ square integrable on }}[a,b]\quad \iff \quad \int _{a}^{b}|f(x)|^{2}\,\mathrm {d} x<\infty }

An equivalent definition is to say that the square of the function itself (rather than of its absolute value) is [Lebesgue integrable](/source/Lebesgue_integrable). For this to be true, the integrals of the positive and negative portions of the real part must both be finite, as well as those for the imaginary part.

The [vector space](/source/Vector_space) of (equivalence classes of) square integrable functions (with respect to [Lebesgue measure](/source/Lebesgue_measure)) forms the [L p {\displaystyle L^{p}} space](/source/Lp_space) with p = 2. {\displaystyle p=2.} Among the L p {\displaystyle L^{p}} spaces, the class of square integrable functions is unique in being compatible with an [inner product](/source/Inner_product_space), which allows notions like angle and orthogonality to be defined. Along with this inner product, the square integrable functions form a [Hilbert space](/source/Hilbert_space), since all of the L p {\displaystyle L^{p}} spaces are [complete](/source/Complete_metric_space) under their respective [p {\displaystyle p} -norms](/source/Lp_space#Lp_spaces_and_Lebesgue_integrals).

Often the term is used not to refer to a specific function, but to equivalence classes of functions that are equal [almost everywhere](/source/Almost_everywhere).

## Properties

The square integrable functions (in the sense mentioned in which a "function" actually means an [equivalence class](/source/Equivalence_class) of functions that are equal almost everywhere) form an [inner product space](/source/Inner_product_space) with [inner product](/source/Inner_product) given by ⟨ f , g ⟩ = ∫ A f ( x ) ¯ g ( x ) d x , {\displaystyle \langle f,g\rangle =\int _{A}{\overline {f(x)}}\ g(x)\mathrm {d} x,} where

- f {\displaystyle f} and g {\displaystyle g} are square integrable functions,

- f ( x ) ¯ {\displaystyle {\overline {f(x)}}} is the [complex conjugate](/source/Complex_conjugate) of f ( x ) , {\displaystyle f(x),}

- A {\displaystyle A} is the set over which one integrates—in the first definition (given in the introduction above), A {\displaystyle A} is ( − ∞ , + ∞ ) {\displaystyle (-\infty ,+\infty )} , in the second, A {\displaystyle A} is [ a , b ] {\displaystyle [a,b]} .

Since | a | 2 = a ⋅ a ¯ {\displaystyle |a|^{2}=a\cdot {\overline {a}}} , square integrability is the same as saying ⟨ f , f ⟩ < ∞ . {\displaystyle \langle f,f\rangle <\infty .\,}

It can be shown that square integrable functions form a [complete metric space](/source/Complete_metric_space) under the metric induced by the inner product defined above. A complete metric space is also called a [Cauchy space](/source/Cauchy_space), because sequences in such metric spaces converge if and only if they are [Cauchy](/source/Cauchy_sequence). A space that is complete under the metric induced by a norm is a [Banach space](/source/Banach_space). Therefore, the space of square integrable functions is a Banach space, under the metric induced by the norm, which in turn is induced by the inner product. As we have the additional property of the inner product, this is specifically a [Hilbert space](/source/Hilbert_space), because the space is complete under the metric induced by the inner product.

This inner product space is conventionally denoted by ( L 2 , ⟨ ⋅ , ⋅ ⟩ 2 ) {\displaystyle \left(L_{2},\langle \cdot ,\cdot \rangle _{2}\right)} and many times abbreviated as L 2 . {\displaystyle L_{2}.} Note that L 2 {\displaystyle L_{2}} denotes the set of square integrable functions, but no selection of metric, norm or inner product are specified by this notation. The set, together with the specific inner product ⟨ ⋅ , ⋅ ⟩ 2 {\displaystyle \langle \cdot ,\cdot \rangle _{2}} specify the inner product space.

The space of square integrable functions is the [L p {\displaystyle L^{p}} space](/source/Lp_space) in which p = 2. {\displaystyle p=2.}

## Examples

The function 1 x n , {\displaystyle {\tfrac {1}{x^{n}}},} defined on ( 0 , 1 ) , {\displaystyle (0,1),} is in L 2 {\displaystyle L^{2}} for n < 1 2 {\displaystyle n<{\tfrac {1}{2}}} but not for n = 1 2 . {\displaystyle n={\tfrac {1}{2}}.} [1] The function 1 x , {\displaystyle {\tfrac {1}{x}},} defined on [ 1 , ∞ ) , {\displaystyle [1,\infty ),} is square-integrable.[3]

Bounded functions, defined on [ 0 , 1 ] , {\displaystyle [0,1],} are square-integrable. These functions are also in L p , {\displaystyle L^{p},} for any value of p . {\displaystyle p.} [3]

### Non-examples

The function 1 x , {\displaystyle {\tfrac {1}{x}},} defined on [ 0 , 1 ] , {\displaystyle [0,1],} where the value at 0 {\displaystyle 0} is arbitrary. Furthermore, this function is not in L p {\displaystyle L^{p}} for any value of p {\displaystyle p} in [ 1 , ∞ ) . {\displaystyle [1,\infty ).} [3]

## See also

- [Inner product space](/source/Inner_product_space)

- [L p {\displaystyle L^{p}} space](/source/Lp_space) – Function spaces generalizing finite-dimensional p norm spaces

## References

1. ^ [***a***](#cite_ref-:1_1-0) [***b***](#cite_ref-:1_1-1) Todd, Rowland. ["L^2-Function"](http://mathworld.wolfram.com/L2-Function.html). *MathWorld--A Wolfram Web Resource*.

1. **[^](#cite_ref-2)** [Giovanni Sansone](/source/Giovanni_Sansone) (1991). *Orthogonal Functions*. Dover Publications. pp. 1–2. [ISBN](/source/ISBN_(identifier)) [978-0-486-66730-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-66730-0).

1. ^ [***a***](#cite_ref-:0_3-0) [***b***](#cite_ref-:0_3-1) [***c***](#cite_ref-:0_3-2) ["Lp Functions"](https://web.archive.org/web/20201024063542/http://faculty.bard.edu/belk/math461/LpFunctions.pdf) (PDF). Archived from [the original](http://faculty.bard.edu/belk/math461/LpFunctions.pdf) (PDF) on 2020-10-24. Retrieved 2020-01-16.

v t e Lp spaces Basic concepts Banach & Hilbert spaces Lp spaces Measure Lebesgue Measure space Measurable space/function Minkowski distance Sequence spaces L1 spaces Integrable function Lebesgue integral Taxicab geometry L2 spaces Bessel's Cauchy–Schwarz Euclidean distance Hilbert space Parseval's identity Polarization identity Pythagorean theorem Square-integrable function L ∞ {\displaystyle L^{\infty }} spaces Bounded function Chebyshev distance Infimum and supremum Essential Uniform norm Maps Almost everywhere Convergence almost everywhere Convergence in measure Function space Integral transform Locally integrable function Measurable function Symmetric decreasing rearrangement Inequalities Babenko–Beckner Chebyshev's Clarkson's Hanner's Hausdorff–Young Hölder's Markov's Minkowski Young's convolution Results Marcinkiewicz interpolation theorem Plancherel theorem Riemann–Lebesgue Riesz–Fischer theorem Riesz–Thorin theorem For Lebesgue measure Isoperimetric inequality Brunn–Minkowski theorem Milman's reverse Minkowski–Steiner formula Prékopa–Leindler inequality Vitale's random Brunn–Minkowski inequality Applications & related Bochner space Fourier analysis Lorentz space Probability theory Quasinorm Real analysis Sobolev space *-algebra C*-algebra Von Neumann

v t e Banach space topics Types of Banach spaces Asplund Banach list Banach lattice Grothendieck Hilbert Inner product space Polarization identity (Polynomially) Reflexive Riesz L-semi-inner product (B Strictly Uniformly) convex Uniformly smooth (Injective Projective) Tensor product (of Hilbert spaces) Banach spaces are: Barrelled Complete F-space Fréchet tame Locally convex Seminorms/Minkowski functionals Mackey Metrizable Normed norm Quasinormed Stereotype Function space Topologies Banach–Mazur compactum Dual Dual space Dual norm Operator Ultraweak Weak polar operator Strong polar operator Ultrastrong Uniform convergence Linear operators Adjoint Bilinear form operator sesquilinear (Un)Bounded Closed Compact on Hilbert spaces (Dis)Continuous Densely defined Fredholm kernel operator Hilbert–Schmidt Functionals positive Pseudo-monotone Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary Operator theory Banach algebras C*-algebras Operator space Spectrum C*-algebra radius Spectral theory of ODEs Spectral theorem Polar decomposition Singular value decomposition Theorems Anderson–Kadec Banach–Alaoglu Banach–Mazur Banach–Saks Banach–Schauder (open mapping) Banach–Steinhaus (Uniform boundedness) Bessel's inequality Cauchy–Schwarz inequality Closed graph Closed range Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Gelfand–Naimark Goldstine Hahn–Banach hyperplane separation Kakutani fixed-point Krein–Milman Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Parseval's identity Riesz's lemma Riesz representation Robinson-Ursescu Schauder fixed-point Sobczyk's theorem Analysis Abstract Wiener space Banach manifold bundle Bochner space Convex series Differentiation in Fréchet spaces Derivatives Fréchet Gateaux functional holomorphic quasi Integrals Bochner Dunford Gelfand–Pettis regulated Paley–Wiener weak Functional calculus Borel continuous holomorphic Measures Lebesgue Projection-valued Vector Weakly / Strongly measurable function Types of sets Absolutely convex Absorbing Affine Balanced/Circled Bounded Convex Convex cone (subset) Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx)) Linear cone (subset) Radial Radially convex/Star-shaped Symmetric Zonotope Subsets / set operations Affine hull (Relative) Algebraic interior (core) Bounding points Convex hull Extreme point Interior Linear span Minkowski addition Polar (Quasi) Relative interior Examples Absolute continuity AC b a ( Σ ) {\displaystyle ba(\Sigma )} c space Banach coordinate BK Besov B p , q s ( R ) {\displaystyle B_{p,q}^{s}(\mathbb {R} )} Birnbaum–Orlicz Bounded variation BV Bs space Continuous C(K) with K compact Hausdorff Hardy Hp Hilbert H Morrey–Campanato L λ , p ( Ω ) {\displaystyle L^{\lambda ,p}(\Omega )} ℓp ℓ ∞ {\displaystyle \ell ^{\infty }} Lp L ∞ {\displaystyle L^{\infty }} weighted Schwartz S ( R n ) {\displaystyle S\left(\mathbb {R} ^{n}\right)} Segal–Bargmann F Sequence space Sobolev Wk,p Sobolev inequality Triebel–Lizorkin Wiener amalgam W ( X , L p ) {\displaystyle W(X,L^{p})} Applications Differential operator Finite element method Mathematical formulation of quantum mechanics Ordinary Differential Equations (ODEs) Validated numerics

v t e Hilbert spaces Basic concepts Adjoint Inner product and L-semi-inner product Hilbert space and Prehilbert space Orthogonal complement Orthonormal basis Main results Bessel's inequality Cauchy–Schwarz inequality Riesz representation Other results Hilbert projection theorem Parseval's identity Polarization identity (Parallelogram law) Maps Compact operator on Hilbert space Densely defined Hermitian form Hilbert–Schmidt Normal Self-adjoint Sesquilinear form Trace class Unitary Examples Cn(K) with K compact & n<∞ Segal–Bargmann F

---
Adapted from the Wikipedia article [Square-integrable function](https://en.wikipedia.org/wiki/Square-integrable_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Square-integrable_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.