Degenerate bilinear form

Symmetric nondegenerate forms are important generalizations of inner products, in that often all that is required is that the mapThere is the closely related notion of a unimodular form and a perfect pairing; these agree over fields but not over general rings.Symmetric nondegenerate forms are important generalizations of inner products, in that often all that is required is that the mapGeometrically, an isotropic line of the quadratic form corresponds to a point of the associated quadric hypersurface in projective space.Such a line is additionally isotropic for the bilinear form if and only if the corresponding point is a singularity.
Degeneracymathematicslinear algebravector spacebilinear formdual spaceisomorphismfinite-dimensionalif and only ifinner productssymplectic formsSymmetricmanifoldtangent spacesRiemannian manifoldpseudo-Riemannian manifolddeterminantmatrixsingularnon-singularquadratic formisotropic quadratic formdefinite quadratic formunimodular formperfect pairingfieldscomplex numberssplit-complex numbersdual numbersinjectivesurjectivecontinuous functionsintervalDirac delta functionalsubspaceisotropic linequadric hypersurfaceprojective spacesingularityalgebraically closed fieldHilbert's NullstellensatzIndefinite inner product spaceDual systemLinear formDualitylinearDual topologyOperator topologiesPolar setPolar topologyTopologies on spaces of linear mapsTopologiesNorm topologyDual normUltraweak/Weak-*operatorin Hilbert spacesMackey Strong dualUltrastrongBanach–AlaogluMackey–ArensTranspose of a linear mapSaturated familyTotal setBiorthogonal systemTopological vector spacesBanach spaceCompletenessContinuous linear operatorLinear functionalFréchet spaceLinear mapLocally convex spaceMetrizabilityTopological vector spaceAnderson–KadecClosed graph theoremF. Riesz'sHahn–Banachhyperplane separationVector-valued Hahn–BanachOpen mapping (Banach–Schauder)Bounded inverseUniform boundedness (Banach–Steinhaus)Bilinear operatorAlmost openBoundedContinuousClosedCompactDensely definedDiscontinuousTopological homomorphismFunctionalBilinearSesquilinearSeminormSublinear functionTransposeAbsolutely convex/diskAbsorbing/RadialAffineBalanced/CircledBanach disksBounding pointsComplemented subspaceConvexConvex cone (subset)Linear cone (subset)Extreme pointPrevalent/ShyRadialRadially convex/Star-shapedAffine hullAlgebraic interior (core)Convex hullLinear spanMinkowski additionAsplundB-complete/PtakBanachCountablyBarrelledBK-spaceUltra-BornologicalBraunerCompleteConvenient(DF)-spaceDistinguishedF-spaceFK-AK spaceFK-spaceFréchetGrothendieckHilbertInfrabarreledInterpolation spaceK-spaceLB-spaceLF-spaceMackey(Pseudo)MetrizableMontelQuasibarrelledQuasi-completeQuasinormedPolynomiallyReflexiveSchwartzSemi-completeStereotypeStrictlyUniformlyQuasi-UltrabarrelledUniformly smoothWebbedWith the approximation property