Square-integrable function

In mathematics, a square-integrable function, also called a quadratically integrable function orfunction 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 linesquare integrable{\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 assquare integrable on{\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.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 thespaces, 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 thespaces are complete under their respectiveOften the term is used not to refer to a specific function, but to equivalence classes of functions that are equal almost everywhere.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 byIt 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 bydenotes 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 productThe space of square integrable functions is the[3] Bounded functions, defined on
mathematicscomplexmeasurable functionintegralabsolute valueLebesgue integrablevector spaceLebesgue measure L p {\displaystyle L^{p}} spaceinner productHilbert spacecompletealmost everywhereequivalence classinner product spacecomplex conjugatecomplete metric spaceCauchy spaceCauchyBanach spaceGiovanni SansoneLp spacesBanachHilbert spacesMeasureLebesgueMeasure spaceMeasurable spacefunctionMinkowski distanceSequence spacesL1 spacesIntegrable functionLebesgue integrationTaxicab geometryL2 spacesBessel'sCauchy–SchwarzEuclidean distanceParseval's identityPolarization identityPythagorean theorem L ∞ {\displaystyle L^{\infty }} spacesBounded functionChebyshev distanceInfimum and supremumEssentialUniform normConvergence almost everywhereConvergence in measureFunction spaceIntegral transformLocally integrable functionSymmetric decreasing rearrangementBabenko–BecknerChebyshev'sClarkson'sHanner'sHausdorff–YoungHölder'sMarkov'sMinkowskiYoung's convolutionMarcinkiewicz interpolation theoremPlancherel theoremRiemann–LebesgueRiesz–Fischer theoremRiesz–Thorin theoremIsoperimetric inequalityBrunn–Minkowski theoremMilman's reverseMinkowski–Steiner formulaPrékopa–Leindler inequalityVitale's random Brunn–Minkowski inequalityBochner spaceFourier analysisLorentz spaceProbability theoryQuasinormReal analysisSobolev space*-algebraC*-algebraVon NeumannAsplundBanach latticeGrothendieck HilbertPolynomiallyReflexiveL-semi-inner productStrictlyUniformlyUniformly smoothInjectiveProjectiveTensor productof Hilbert spacesBarrelledF-spaceFréchetLocally convexMinkowski functionalsMackeyMetrizableNormedQuasinormedStereotypeBanach–Mazur compactumDual spaceDual normOperatorUltraweakStrongUltrastrongUniform convergenceLinear operatorsAdjointBilinearsesquilinearBoundedClosedCompacton Hilbert spacesContinuousDensely definedkernelHilbert–SchmidtFunctionalspositivePseudo-monotoneNormalNuclearSelf-adjointStrictly singularTrace classTransposeUnitaryOperator theoryBanach algebrasC*-algebrasOperator spaceSpectrumradiusSpectral theoryof ODEsSpectral theoremPolar decompositionSingular value decompositionAnderson–KadecBanach–AlaogluBanach–MazurBanach–SaksBanach–Schauder (open mapping)Banach–Steinhaus (Uniform boundedness)Bessel's inequalityCauchy–Schwarz inequalityClosed graphClosed rangeEberlein–ŠmulianFreudenthal spectralGelfand–MazurGelfand–NaimarkGoldstineHahn–Banachhyperplane separationKrein–MilmanMackey–ArensMazur's lemmaM. Riesz extensionRiesz's lemmaRiesz representationSchauder fixed-pointAbstract Wiener spaceBanach manifoldbundleConvex seriesDifferentiation in Fréchet spacesDerivativesGateauxfunctionalholomorphicIntegralsBochnerDunfordGelfand–PettisregulatedPaley–WienerFunctional calculusMeasuresProjection-valuedVectorWeaklyStronglyAbsolutely convexAbsorbingAffineBalanced/CircledConvexConvex cone (subset)Linear cone (subset)RadialRadially convex/Star-shapedSymmetricZonotopeAffine hullAlgebraic interior (core)Bounding pointsConvex hullExtreme pointInteriorLinear spanMinkowski additionAbsolute continuity AC b a ( Σ ) {\displaystyle ba(\Sigma )} c spaceBanach coordinate BKBesov B p , q s ( R ) {\displaystyle B_{p,q}^{s}(\mathbb {R} )} Birnbaum–OrliczBounded variation BVBs spaceContinuous C(K) with K compact HausdorffHardy HpMorrey–Campanato L λ , p ( Ω ) {\displaystyle L^{\lambda ,p}(\Omega )} Schwartz S ( R n ) {\displaystyle S\left(\mathbb {R} ^{n}\right)} Segal–Bargmann FSequence spaceSobolev Wk,pSobolev inequalityTriebel–LizorkinWiener amalgam W ( X , L p ) {\displaystyle W(X,L^{p})} Differential operatorFinite element methodMathematical formulation of quantum mechanicsOrdinary Differential Equations (ODEs)Validated numericsPrehilbert spaceOrthogonal complementOrthonormal basisHilbert projection theoremCompact operator on Hilbert spaceSesquilinear formCn(K) with K compact & n<∞