Riesz's lemma

It specifies (often easy to check) conditions that guarantee that a subspace in a normed vector space is dense.be a closed proper vector subspace of a normed spaceis a reflexive Banach space then this conclusion is also true whendenote the canonical metric induced by the norm, call the setfrom the origin the unit sphere, and denote the distance from a pointUsing this new notation, the conclusion of Riesz's lemma may be restated more succinctly as:Using this new terminology, Riesz's lemma may also be restated in plain English as: The proof[3] can be found in functional analysis texts such as Kreyszig.[4] An online proof from Prof. Paul Garrett is available.and so Riesz's lemma holds vacuously for all real numbersis included solely to exclude this trivial case and is sometimes omitted from the lemma's statement.for consideration, in which case the statement of Riesz’s lemma becomes: Whenis reflexive if and only if for every closed proper vector subspaceof all bounded sequences, Riesz’s lemma does not hold for[5] However, every finite dimensional normed space is a reflexive Banach space, so Riesz’s lemma does holds foris continuous, its image on the closed unit ballmust be a compact subset of the real line, proving the claim.or stated in plain English, these vectors are all separated from each other by a distance of more thanwhile simultaneously also all lying on the unit sphere.Such an infinite sequence of vectors cannot be found in the unit sphere of any finite dimensional normed space (just consider for example the unit circle inThis sequence can be constructed by induction for any constantcontains no convergent subsequence, which implies that the closed unit ball is not compact.Riesz's lemma can be applied directly to show that the unit ball of an infinite-dimensional normed spaceThis can be used to characterize finite dimensional normed spaces: ifis finite dimensional if and only if the closed unit ball inNamely, if a topological vector space is finite dimensional, it is locally compact.For a different proof based on Hahn–Banach theorem see Crespín (1994).[7] The spectral properties of compact operators acting on a Banach space are similar to those of matrices.Riesz's lemma is essential in establishing this fact.As detailed in the article on infinite-dimensional Lebesgue measure, this is useful in showing the non-existence of certain measures on infinite-dimensional Banach spaces.Riesz's lemma also shows that the identity operator on a Banach space
Riesz representation theoremRiesz theoremF. Riesz's theoremFrigyes Rieszfunctional analysissubspacenormed vector spaceinner product spacereal numberreflexiveBanach spacemetricunit sphereinfimumReflexive spaceEuclidean normtaxicab normLebesgue space ℓ ∞ ( N ) {\displaystyle \ell _{\infty }(\mathbb {N} )} unit circleinductionlinear spanunit ballcompacttopological vector spacelocally compactboundedHahn–Banach theoremspectral properties of compact operatorsinfinite-dimensional Lebesgue measuremeasuresBanach spacesJames's TheoremreflexivityPlanetMathTao, TerenceRiesz, FredericSz.-Nagy, BélaDover PublicationstopicsglossaryBanachFréchetHilbertHölderNuclearOrliczSchwartzSobolevTopological vectorBarrelledCompleteLocally convexSeparableHahn–BanachRiesz representationClosed graphUniform boundedness principleKrein–MilmanMin–maxGelfand–NaimarkBanach–AlaogluAdjointHilbert–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 theoryAsplundBanach latticeGrothendieck Polarization identityPolynomiallyL-semi-inner productStrictlyUniformlyUniformly smoothInjectiveProjectiveTensor productof Hilbert spacesF-spaceMinkowski functionalsMackeyMetrizableNormedQuasinormedStereotypeBanach–Mazur compactumDual spaceDual normOperatorUltraweakStrongUltrastrongUniform convergenceLinear operatorsBilinearsesquilinearClosedon Hilbert spacesContinuousDensely definedkernelFunctionalspositivePseudo-monotoneSelf-adjointStrictly singularOperator theoryBanach algebrasC*-algebrasOperator spaceSpectrumradiusSpectral theoryof ODEsSpectral theoremPolar decompositionSingular value decompositionAnderson–KadecBanach–MazurBanach–SaksBanach–Schauder (open mapping)Banach–Steinhaus (Uniform boundedness)Bessel's inequalityCauchy–Schwarz inequalityClosed rangeEberlein–ŠmulianFreudenthal spectralGelfand–MazurGoldstinehyperplane separationMackey–ArensMazur's lemmaM. Riesz extensionParseval's identitySchauder fixed-pointAbstract Wiener spaceBanach manifoldbundleBochner spaceConvex seriesDifferentiation in Fréchet spacesDerivativesGateauxfunctionalholomorphicIntegralsBochnerDunfordGelfand–PettisregulatedPaley–WienerLebesgueProjection-valuedVectorWeaklyStronglyAbsolutely convexAbsorbingAffineBalanced/CircledConvexConvex cone (subset)Linear cone (subset)RadialRadially convex/Star-shapedSymmetricZonotopeAffine hullAlgebraic interior (core)Bounding pointsConvex hullExtreme pointInteriorMinkowski 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 numerics