Square-integrable function

In mathematics, a square-integrable function, also called a quadratically integrable function or $L^{2}$ function or square-summable function,[1] is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line $(-\infty ,+\infty )$ is defined as follows.

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

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

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

An equivalent definition is to say that the square of the function itself (rather than of its absolute value) is Lebesgue integrable. 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 of (equivalence classes of) square integrable functions (with respect to Lebesgue measure) forms the $L^{p}$ space with $p=2.$ Among the $L^{p}$ spaces, the class of square integrable functions is unique in being compatible with an inner product, which allows notions like angle and orthogonality to be defined. Along with this inner product, the square integrable functions form a Hilbert space, since all of the $L^{p}$ spaces are complete under their respective $p$ -norms.

Often the term is used not to refer to a specific function, but to equivalence classes of functions that are equal almost everywhere.

Properties

The square integrable functions (in the sense mentioned in which a "function" actually means an equivalence class of functions that are equal almost everywhere) form an inner product space with inner product given by

\langle f,g\rangle =\int _{A}{\overline {f(x)}}g(x)\,\mathrm {d} x

where

$f$ and $g$ are square integrable functions,
${\overline {f(x)}}$ is the complex conjugate of $f(x),$
$A$ is the set over which one integrates—in the first definition (given in the introduction above), $A$ is $(-\infty ,+\infty );$ in the second, $A$ is $[a,b].$

Since $|a|^{2}=a\cdot {\overline {a}},$ square integrability is the same as saying

\langle f,f\rangle <\infty .\,

It can be shown that square integrable functions form a complete metric space under the metric induced by the inner product defined above. A complete metric space is also called a Cauchy space, because sequences in such metric spaces converge if and only if they are Cauchy. A space that is complete under the metric induced by a norm is a 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, because the space is complete under the metric induced by the inner product.

This inner product space is conventionally denoted by $\left(L_{2},\langle \cdot ,\cdot \rangle _{2}\right)$ and many times abbreviated as $L_{2}.$ Note that $L_{2}$ 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 $\langle \cdot ,\cdot \rangle _{2}$ specify the inner product space.

The space of square integrable functions is the $L^{p}$ space in which $p=2.$

Examples

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

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

Non-examples

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

References

Todd, Rowland. "L^2-Function". MathWorld--A Wolfram Web Resource.
Giovanni Sansone (1991). Orthogonal Functions. Dover Publications. pp. 1–2. ISBN 978-0-486-66730-0.
"Lp Functions" (PDF). Archived from the original (PDF) on 2020-10-24. Retrieved 2020-01-16.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[:1-1] Todd, Rowland. "L^2-Function". MathWorld--A Wolfram Web Resource.

[2] Giovanni Sansone (1991). Orthogonal Functions. Dover Publications. pp. 1–2. ISBN 978-0-486-66730-0.

[:0-3] "Lp Functions" (PDF). Archived from the original (PDF) on 2020-10-24. Retrieved 2020-01-16.

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
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 $ba(\Sigma )$ c space Banach coordinate BK Besov $B_{p,q}^{s}(\mathbb {R} )$ Birnbaum–Orlicz Bounded variation BV Bs space Continuous C(K) with K compact Hausdorff Hardy H^p Hilbert H Morrey–Campanato $L^{\lambda ,p}(\Omega )$ ℓ^p $\ell ^{\infty }$ L^p $L^{\infty }$ weighted Schwartz $S\left(\mathbb {R} ^{n}\right)$ Segal–Bargmann F Sequence space Sobolev W^k,p Sobolev inequality Triebel–Lizorkin Wiener amalgam $W(X,L^{p})$
Applications	Differential operator Finite element method Mathematical formulation of quantum mechanics Ordinary Differential Equations (ODEs) Validated numerics

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

Square-integrable function

Properties

Examples

Non-examples

See also

References