Schwartz kernel theorem

Schwartz kernel theorem In mathematics, the Schwartz kernel theorem is a foundational result in the theory of generalized functions, published by Laurent Schwartz in 1952. It states, in broad terms, that the generalized functions introduced by Schwartz (Schwartz distributions) have a two-variable theory that includes all reasonable bilinear forms on the space {displaystyle {mathcal {D}}} of test functions. The space {displaystyle {mathcal {D}}} itself consists of smooth functions of compact support.

Contents 1 Statement of the theorem 1.1 Note 2 Integral kernels 3 Smooth manifolds 4 Generalization to nuclear spaces 5 See also 6 References 7 Bibliography 8 External links Statement of the theorem Let {displaystyle X} and {displaystyle Y} be open sets in {displaystyle mathbb {R} ^{n}} . Every distribution {displaystyle kin {mathcal {D}}'(Xtimes Y)} defines a continuous linear map {displaystyle Kcolon {mathcal {D}}(Y)to {mathcal {D}}'(X)} such that {displaystyle leftlangle k,uotimes vrightrangle =leftlangle Kv,urightrangle }         (1) for every {displaystyle uin {mathcal {D}}(X),vin {mathcal {D}}(Y)} . Conversely, for every such continuous linear map {displaystyle K} there exists one and only one distribution {displaystyle kin {mathcal {D}}'(Xtimes Y)} such that (1) holds. The distribution {displaystyle k} is the kernel of the map {displaystyle K} .

Note Given a distribution {displaystyle kin {mathcal {D}}'(Xtimes Y)} one can always write the linear map K informally as {displaystyle Kv=int _{Y}k(cdot ,y)v(y)dy} so that {displaystyle langle Kv,urangle =int _{X}int _{Y}k(x,y)v(y)u(x)dydx} . Integral kernels The traditional kernel functions {displaystyle K(x,y)} of two variables of the theory of integral operators having been expanded in scope to include their generalized function analogues, which are allowed to be more singular in a serious way, a large class of operators from {displaystyle {mathcal {D}}} to its dual space {displaystyle {mathcal {D}}'} of distributions can be constructed. The point of the theorem is to assert that the extended class of operators can be characterised abstractly, as containing all operators subject to a minimum continuity condition. A bilinear form on {displaystyle {mathcal {D}}} arises by pairing the image distribution with a test function.

A simple example is that the natural embedding of the test function space {displaystyle {mathcal {D}}} into {displaystyle {mathcal {D}}'} - sending every test function {displaystyle f} into the corresponding distribution {displaystyle [f]} - corresponds to the delta distribution {displaystyle delta (x-y)} concentrated at the diagonal of the underlined Euclidean space, in terms of the Dirac delta function {displaystyle delta } . While this is at most an observation, it shows how the distribution theory adds to the scope. Integral operators are not so 'singular'; another way to put it is that for {displaystyle K} a continuous kernel, only compact operators are created on a space such as the continuous functions on {displaystyle [0,1]} . The operator {displaystyle I} is far from compact, and its kernel is intuitively speaking approximated by functions on {displaystyle [0,1]times [0,1]} with a spike along the diagonal {displaystyle x=y} and vanishing elsewhere.

This result implies that the formation of distributions has a major property of 'closure' within the traditional domain of functional analysis. It was interpreted (comment of Jean Dieudonné) as a strong verification of the suitability of the Schwartz theory of distributions to mathematical analysis more widely seen. In his Éléments d'analyse volume 7, p. 3 he notes that the theorem includes differential operators on the same footing as integral operators, and concludes that it is perhaps the most important modern result of functional analysis. He goes on immediately to qualify that statement, saying that the setting is too 'vast' for differential operators, because of the property of monotonicity with respect to the support of a function, which is evident for differentiation. Even monotonicity with respect to singular support is not characteristic of the general case; its consideration leads in the direction of the contemporary theory of pseudo-differential operators.

Smooth manifolds Dieudonné proves a version of the Schwartz result valid for smooth manifolds, and additional supporting results, in sections 23.9 to 23.12 of that book.

Generalization to nuclear spaces Much of the theory of nuclear spaces was developed by Alexander Grothendieck while investigating the Schwartz kernel theorem and published in Grothendieck 1955. We have the following generalization of the theorem.

Schwartz kernel theorem:[1] Suppose that X is nuclear, Y is locally convex, and v is a continuous bilinear form on {displaystyle Xtimes Y} . Then v originates from a space of the form {displaystyle X_{A^{prime }}^{prime }{widehat {otimes }}_{epsilon }Y_{B^{prime }}^{prime }} where {displaystyle A^{prime }} and {displaystyle B^{prime }} are suitable equicontinuous subsets of {displaystyle X^{prime }} and {displaystyle Y^{prime }} . Equivalently, v is of the form, {displaystyle v(x,y)=sum _{i=1}^{infty }lambda _{i}leftlangle x,x_{i}^{prime }rightrangle leftlangle y,y_{i}^{prime }rightrangle } for all {displaystyle (x,y)in Xtimes Y} where {displaystyle left(lambda _{i}right)in l^{1}} and each of {displaystyle {x_{1}^{prime },x_{2}^{prime },ldots }} and {displaystyle {y_{1}^{prime },y_{2}^{prime },ldots }} are equicontinuous. Furthermore, these sequences can be taken to be null sequences (i.e. converging to 0) in {displaystyle X_{A^{prime }}^{prime }} and {displaystyle Y_{B^{prime }}^{prime }} , respectively.

See also Fredholm kernel Injective tensor product Nuclear operator Nuclear space Projective tensor product Rigged Hilbert space Trace class References ^ Schaefer & Wolff 1999, p. 172. Bibliography Grothendieck, Alexander (1955). "Produits Tensoriels Topologiques et Espaces Nucléaires" [Topological Tensor Products and Nuclear Spaces]. Memoirs of the American Mathematical Society Series (in French). Providence: American Mathematical Society. 16. ISBN 978-0-8218-1216-7. MR 0075539. OCLC 1315788. Hörmander, L. (1983). The analysis of linear partial differential operators I. Grundl. Math. Wissenschaft. Vol. 256. Springer. doi:10.1007/978-3-642-96750-4. ISBN 3-540-12104-8. MR 0717035.. Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. Wong (1979). Schwartz spaces, nuclear spaces, and tensor products. Berlin New York: Springer-Verlag. ISBN 3-540-09513-6. OCLC 5126158. External links G. L. Litvinov (2001) [1994], "Nuclear bilinear form", Encyclopedia of Mathematics, EMS Press hide vte Functional analysis (topics – glossary) Spaces BanachBesovFréchetHilbertHölderNuclearOrliczSchwartzSobolevtopological vector Properties barrelledcompletedual (algebraic/topological)locally convexreflexiveseparable Theorems Hahn–BanachRiesz representationclosed graphuniform boundedness principleKakutani fixed-pointKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Operators adjointboundedcompactHilbert–Schmidtnormalnucleartrace classtransposeunboundedunitary Algebras Banach algebraC*-algebraspectrum of a C*-algebraoperator algebragroup algebra of a locally compact groupvon Neumann algebra Open problems invariant subspace problemMahler's conjecture Applications Hardy spacespectral theory of ordinary differential equationsheat kernelindex theoremcalculus of variationsfunctional calculusintegral operatorJones polynomialtopological quantum field theorynoncommutative geometryRiemann hypothesisdistribution (or generalized functions) Advanced topics approximation propertybalanced setChoquet theoryweak topologyBanach–Mazur distanceTomita–Takesaki theory Categories: Generalized functionsTransformsTheorems in functional analysis