Linear form
If V is a vector space over a field k, the set of all linear functionals from V to k is itself a vector space over k with addition and scalar multiplication defined pointwise.denote the vector space of real-valued polynomial functions of degree(Lax (1996) proves this last fact using Lagrange interpolation).In three dimensions, the level sets of a linear functional are a family of mutually parallel planes; in higher dimensions, they are parallel hyperplanes.This method of visualizing linear functionals is sometimes introduced in general relativity texts, such as Gravitation by Misner, Thorne & Wheeler (1973).This forms the foundation of the theory of numerical quadrature.[6] Linear functionals are particularly important in quantum mechanics.A state of a quantum mechanical system can be identified with a linear functional.Every non-degenerate bilinear form on a finite-dimensional vector space V induces an isomorphism V → V∗ : v ↦ v∗ such thatIn an infinite dimensional Hilbert space, analogous results hold by the Riesz representation theorem.called the dual basis defined by the special property thatHere the superscripts of the basis functionals are not exponents but are instead contravariant indices.can be expressed as a linear combination of basis functionals, with coefficients ("components") ui,In three dimensions (n = 3), the dual basis can be written explicitlyModules over a ring are generalizations of vector spaces, which removes the restriction that coefficients belong to a field.The space of linear forms is always denoted Homk(V, k), whether k is a field or not.The existence of "enough" linear forms on a module is equivalent to projectivity.[8] Dual Basis Lemma — An R-module M is projective if and only if there exists a subsetendowed with a complex structure; that is, there exists a real vector subspaceThis relationship was discovered by Henry Löwig in 1934 (although it is usually credited to F. Murray),[11] and can be generalized to arbitrary finite extensions of a field in the natural way.is the closed unit ball then the supremums above are the operator norms (defined in the usual way) ofThis conclusion extends to the analogous statement for polars of balanced sets in general topological vector spaces.[13] Continuous linear functionals have nice properties for analysis: a linear functional is continuous if and only if its kernel is closed,[14] and a non-trivial continuous linear functional is an open map, even if the (topological) vector space is not complete.is maximal if and only if it is the kernel of some non-trivial linear functional onis a affine hyperplane if and only if there exists some non-trivial linear functionalAny two linear functionals with the same kernel are proportional (i.e. scalar multiples of each other).[15] Any (algebraic) linear functional on a vector subspace can be extended to the whole space; for example, the evaluation functionals described above can be extended to the vector space of polynomials on all ofHowever, this extension cannot always be done while keeping the linear functional continuous.of f to the whole space X that is dominated by p, i.e., there exists a linear functional F such that[19] Moreover, Alaoglu's theorem implies that the weak-* closure of an equicontinuous subset of
mathematicslinear mapvector spacescalarsreal numberscomplex numberspointwisedual spacetopological dual spacecolumn vectorsrow vectorsmatrix productszero functionsurjectiveSamplingkernelNet present valuecash flowdiscount ratemain diagonalfunctional analysisvector spaces of functionsintegrationRiemann integralLagrange interpolationequation of a linelinearaffine-linearhyperplaneslevel setsgeneral relativityGravitationnumerical quadraturequantum mechanicsHilbert spacesisomorphicbra–ket notationgeneralized functionsdistributionstest functionsEuclidean spaceinner productbilinear formisomorphismdot productHilbert spaceRiesz representation theoremcontinuous dual spaceSchauder basisorthogonaldual basisKronecker deltacontravariantlinear combinationLevi-Civita symbolHodge star operatorModulesright moduleprojectivitymoduleprojectiveLinear complex structureComplexificationrealificationcomplex structurevector subspacealgebraic dual space
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{\displaystyle \mathbb {R} }
-linear operatoradditivehomogeneous over
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real partimaginary partbijectivefinite extensions of a fieldtopological vector spaceboundedunit lengthnormed spacesupremumsoperator normspolarsbalanced setstopological vector spacesantilinearContinuous linear operatorvector spacescontinuouscontinuous dualBanach spacelocally convexanalysisopen mapbalancedHahn–Banach theoremsublinear functionlinear functionallinear subspaceequicontinuous(pre)polarweak-*balanced hullconvex hullconvex balanced hullAlaoglu's theoremDiscontinuous linear mapLocally convex topological vector spacePositive linear functionalMultilinear formAxler, SheldonUndergraduate Texts in MathematicsSpringerBishop, RichardConway, JohnGraduate Texts in MathematicsSpringer-VerlagHalmos, Paul RichardKatznelson, YitzhakAmerican Mathematical SocietyLax, PeterMisner, Charles W.Thorne, Kip S.Wheeler, John A.Rudin, WalterMcGraw-Hill Science/Engineering/MathSchaefer, Helmut H.Schutz, Bernard