Orlicz space

In mathematical analysis, and especially in real, harmonic analysis and functional analysis, an Orlicz space is a type of function space which generalizes the Lp spaces.The spaces are named for Władysław Orlicz, who was the first to define them in 1932.One such space L log+ L, which arises in the study of Hardy–Littlewood maximal functions, consists of measurable functions f such that the Here log+ is the positive part of the logarithm.These spaces are called Orlicz spaces by an overwhelming majority of mathematicians and by all monographies studying them, because Władysław Orlicz was the first who introduced them, in 1932.[1] Some mathematicians, including Wojbor Woyczyński, Edwin Hewitt and Vladimir Mazya, include the name of Zygmunt Birnbaum as well, referring to his earlier joint work with Władysław Orlicz.However in the Birnbaum–Orlicz paper the Orlicz space is not introduced, neither explicitly nor implicitly, hence the name Orlicz space is preferred.By the same reasons this convention has been also openly criticized by another mathematician (and an expert in the history of Orlicz spaces), Lech Maligranda.[2] Orlicz was confirmed as the person who introduced Orlicz spaces already by Stefan Banach in his 1932 monograph.[3] μ is a σ-finite measure on a set X,This might not be a vector space (i.e., it might fail to be closed under scalar multiplication).The vector space of functions spanned byIn other words, it is the smallest linear space containingIn other words, it is the largest linear space contained in, let Ψ be the Young complement of Φ; that is, Note that Young's inequality for products holds: The norm is then given by Furthermore, the spaceis precisely the space of measurable functions for which this norm is finite.is the space of all measurable functions for which this norm is finite., the small and the large Orlicz spaces foris not a vector space, and is strictly smaller thanSuppose that X is the open unit interval (0,1), Φ(x) = exp(x) – 1 – x, and f(x) = log(x).are topological dual Banach spaces.open and bounded with Lipschitz boundary, we have for This is the analytical content of the Trudinger inequality: Foropen and bounded with Lipschitz boundarysuch that Similarly, the Orlicz norm of a random variable characterizes it as follows: This norm is homogeneous and is defined only when this set is non-empty., this coincides with the p-th moment of the random variable.Other special cases in the exponential family are taken with respect to the functionsnorm is said to be "sub-Gaussian" and a random variable with finitenorm characterizes the limiting behavior of the probability distribution function: so that the tail of the probability distribution function is bounded above bynorm may be easily computed from a strictly monotonic moment-generating function.For example, the moment-generating function of a chi-squared random variable X with K degrees of freedom is
mathematical analysisharmonic analysisfunctional analysisLp spacesBanach spacesWładysław OrliczHardy–Littlewood maximal functionspositive partSobolev spacesOrlicz sequence spacesEdwin HewittVladimir MazyaZygmunt BirnbaumStefan Banachσ-finite measureYoung functionconvexlower semicontinuousalmost everywherevector spaceYoung's inequality for productsmonotone convergence theoremBanach spacecompletenormedtopological dual Banach spacesboundedLipschitz boundaryTrudinger inequalityrandom variablemomentsub-Gaussiansub-exponentialmoment-generating functionchi-squaredZygmund, AntoniTrigonometric Series, Volume 1Encyclopedia of MathematicsEMS PresstopicsglossaryBanachFréchetHilbertHölderNuclearSchwartzSobolevTopological vectorBarrelledLocally convexReflexiveSeparableHahn–BanachRiesz representationClosed graphUniform boundedness principleKrein–MilmanMin–maxGelfand–NaimarkBanach–AlaogluAdjointCompactHilbert–SchmidtNormalTrace classTransposeUnboundedUnitaryBanach algebraC*-algebraSpectrum of a C*-algebraOperator algebraGroup algebra of a locally compact groupVon Neumann algebraInvariant subspace problemMahler's conjectureHardy spaceSpectral theory of ordinary differential equationsHeat kernelIndex theoremCalculus of variationsFunctional calculusIntegral linear operatorJones polynomialTopological quantum field theoryNoncommutative geometryRiemann hypothesisDistributionGeneralized functionsApproximation propertyBalanced setChoquet theoryWeak topologyBanach–Mazur distanceTomita–Takesaki theory