Banach bundle

In mathematics, a Banach bundle is a vector bundle each of whose fibres is a Banach space, i.e. a complete normed vector space, possibly of infinite dimension.Suppose that for each point x ∈ M, the fibre Ex = π−1(x) has been given the structure of a Banach space.The collection of all Banach bundles can be made into a category by defining appropriate morphisms.These two morphisms are required to satisfy two conditions (again, the second one is redundant in the finite-dimensional case): One can take a Banach bundle over one manifold and use the pull-back construction to define a new Banach bundle on a second manifold.Specifically, let π : E → N be a Banach bundle and f : M → N a differentiable map (as usual, everything is Cp).
Banach bundle (non-commutative geometry)mathematicsvector bundleBanach spacecompletenormed vector spaceBanach manifoldtopological spacesurjectivecontinuous mapopen coverhomeomorphismdiagram commutesinvertiblecontinuous linear mapisomorphismcategorytopological vector spacesmorphismequivalence classconnected spaceclosedtangent spacetangent bundleidentity functioncotangent bundleBochner spacesheat equationSobolev spaceCartesian productcross sectionsdifferentiable functionpull-backLang, SergeManifoldsGlossaryTopological manifoldDifferentiable/Smooth manifoldDifferential structureSmooth atlasSubmanifoldRiemannian manifoldSmooth mapSubmersionPushforwardDifferential formVector fieldAtiyah–Singer indexDarboux'sFrobeniusGeneralized StokesHopf–RinowNoether'sSard'sWhitney embeddingDiffeomorphismGeodesicExponential mapin Lie theoryFoliationImmersionIntegral curveLie derivativeSectionCollapsingAlmostComplexContactFiberedFinslerG-structureHadamardHermitianHyperbolicKählerKenmotsuLie groupLie algebraManifold with boundaryNilmanifoldOrientedParallelizablePoissonQuaternionicHypercomplexPseudo−Sub−RiemannianSymplecticTensorsDistributionLie bracketbundleTorsionVector flowClosed/ExactCovariant derivativeCotangent spaceDe Rham cohomologyVector-valuedExterior derivativeInterior productPullbackRicci curvatureRiemann curvature tensorTensor fielddensityVolume formWedge productBundlesAdjointAffineAssociatedCotangentFibrationStableNormalPrincipalSpinorSubbundleTangentTensorVectorConnectionsCartanEhresmannGeneralizedKoszulLevi-CivitaParallel transportClassification of manifoldsGauge theoryHistoryMorse theoryMoving frameSingularity theoryDiffeologyDiffietyFréchet manifoldK-theoryOrbifoldSecondary calculusover commutative algebrasStratifoldSupermanifoldStratified spaceAnalysisAbstract Wiener spaceClassical Wiener spaceBochner spaceConvex seriesCylinder set measureInfinite-dimensional vector functionMatrix calculusVector calculusDerivativesDifferentiable vector–valued functions from Euclidean spaceDifferentiation in Fréchet spacesFréchet derivativeFunctional derivativeGateaux derivativeDirectionalGeneralizations of the derivativeHadamard derivativeHolomorphicQuasi-derivativeBesov measureCanonical GaussianClassical Wiener measureMeasureset functionsProjection-valuedBochnerWeaklyStronglymeasurable functionRadonifying functionIntegralsDirect integralDunfordGelfand–Pettis/WeakRegulatedPaley–WienerCameron–Martin theoremInverse function theoremNash–Moser theoremFeldman–Hájek theoremNo infinite-dimensional Lebesgue measureSazonov's theoremStructure theorem for Gaussian measuresCrinkled arcCovariance operatorFunctional calculusBorel functional calculusContinuous functional calculusHolomorphic functional calculusConvenient vector spaceChoquet theoryHilbert manifoldAsplundBanachBanach latticeGrothendieck HilbertInner product spacePolarization identityPolynomiallyReflexiveL-semi-inner productStrictlyUniformlyUniformly smoothInjectiveProjectiveTensor productof Hilbert spacesBarrelledF-spaceFréchetLocally convexMinkowski functionalsMackeyMetrizableNormedQuasinormedStereotypeBanach–Mazur compactumDual spaceDual normOperatorUltraweakStrongUltrastrongUniform convergenceLinear operatorsBilinearsesquilinearBoundedCompacton Hilbert spacesContinuousDensely definedkernelHilbert–SchmidtFunctionalspositivePseudo-monotoneNuclearSelf-adjointStrictly singularTrace classTransposeUnitaryOperator theoryBanach algebrasC*-algebrasOperator spaceSpectrumC*-algebraradiusSpectral 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 extensionParseval's identityRiesz's lemmaRiesz representationSchauder fixed-pointGateauxfunctionalGelfand–PettisMeasuresLebesgueAbsolutely convexAbsorbingBalanced/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 numericsFunctional analysistopicsHölderOrliczSchwartzSobolevTopological vectorSeparableUniform boundedness principleMin–maxUnboundedBanach 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 variationsIntegral linear operatorJones polynomialTopological quantum field theoryNoncommutative geometryRiemann hypothesisGeneralized functionsApproximation propertyBalanced setWeak topologyBanach–Mazur distanceTomita–Takesaki theory