Generalized functions, space of

distribution space

The space dual to the space of test (sufficiently good) functions. The Fréchet–Schwartz spaces (cf. Fréchet space) (type FS) and the spaces strongly dual to them (type DFS) play an important role here. A space of type FS is the projective limit of a directed set of Banach spaces and its dual is a space of type DFS. A space of type DFS is the inductive limit of a directed set of Banach spaces and its dual is a space of type FS. Spaces of types FS and DFS are complete, separable, reflexive, and Montel. In spaces of types FS and DFS weak and strong convergence coincide.

Examples of spaces of test and generalized functions.

1) The spaces and . The space of (rapidly-decreasing) test functions consists of the -functions that together with all their derivatives decrease at infinity faster than any power of . This space is the projective limit of the sequence of Banach spaces , consisting of the -functions with norm

and the inclusion is compact; is of type FS. The dual space (the space of generalized functions of slow growth) is the inductive limit of the sequence of Banach spaces , where the imbedding is compact, so that is of type DFS. If a sequence of generalized functions is (weakly) convergent in , then it converges with respect to the norm of functionals in some . The Fourier transformation is an isomorphism on the spaces and .

2) The spaces and ( an open set in ). The space of test functions consists of the -functions that have compact support in (see Support of a generalized function). It is endowed with the topology of the strong inductive limit of the (increasing) sequence of spaces of type FS, where is a strictly-increasing sequence of open sets that exhausts , , compact, . The space is the projective limit of the (decreasing) sequence of Banach spaces , consisting of the functions with support in and with norm

where the imbedding is compact. Let be the space (strongly) dual to ; and . A sequence of test functions in converges in if it converges in some space . A sequence of generalized functions in converges in if it converges on every element of (weak convergence). For a linear functional on to be a generalized function in it is necessary and sufficient that for any open set there exist numbers and such that

The space is (weakly) complete: If a sequence of generalized functions , is such that for any in the sequence of numbers converges, then the functional

belongs to . A generalized function in has unrestricted "growth" in a neighbourhood of the boundary ; in particular, any function determines a generalized function in by the formula

3) The spaces and . Let be the Banach space of all functions , , that are holomorphic in the tubular neighbourhood , , with norm

the imbedding , , is compact. Let be the inductive limit of the (increasing) sequence of spaces , . The space is of type DFS, and its dual is of type FS. The elements of are Fourier hyperfunctions; is also isomorphic to the space .

References

[1]	L. Schwartz, "Théorie des distributions" , 1–2 , Hermann (1950–1951)
[2]	N. Bourbaki, "Elements of mathematics. Topological vector spaces" , Addison-Wesley (1977) (Translated from French)
[3]	J. Dieudonné, L. Schwartz, "La dualité dans les espaces () et ()" Ann. Inst. Fourier , 1 (1949) pp. 61–101
[4]	A. Grothendieck, "Sur les espaces et " Summa Brasil. Math. , 3 : 6 (1954) pp. 57–123
[5]	I.M. Gel'fand, G.E. Shilov, "Generalized functions" , 2 , Acad. Press (1968) (Translated from Russian)
[6]	K. Yoshinaga, "On a locally convex space introduced by J.S.E. Silva" J. Sci. Hiroshima Univ. Ser. A , 21 (1957) pp. 89–98
[7]	T. Kawai, "On the theory of Fourier hyperfunctions and its applications to partial differential equations with constant coefficients" J. Fac. Sci. Univ. Tokyo Sect. 1A Math. (1970) pp. 467–517
[8]	V.S. Vladimirov, "Generalized functions in mathematical physics" , MIR (1979) (Translated from Russian)

Comments

For functional-analytic properties of distribution spaces see [a1].

For generalized function spaces which are invariant under certain given integral transformations see [a2], [a3], [a4].

Test spaces and spaces of generalized functions which satisfy such invariance requirements can be constructed starting from a separable Hilbert space and an unbounded self-adjoint operator on . The test space ( analyticity space) is defined by . The distribution space ( trajectory space) consists of mappings with the property: for all : . The duality pairing is , where is sufficiently small and depends on .

Both spaces are inductive and projective limits of Hilbert spaces. Many Gel'fand–Shilov spaces are of this type.

For classical examples, topological properties and operator algebras on those spaces see [a3], [a4].

References

[a1]	F. Treves, "Topological vectorspaces, distributions and kernels" , Acad. Press (1967)
[a2]	A.H. Zemanian, "Generalized integral transformations" , Interscience (1968)
[a3]	N.G. de Bruijn, "A theory of generalized functions with applications to Wigner distribution and Weyl correspondence" Niew Archief for Wiskunde (3) , 21 (1973) pp. 205–280
[a4]	S.J.L. van Eijndhoven, J. de Graaf, "Trajectory spaces, generalized functions and unbounded operators" , Lect. notes in math. , 1162 , Springer (1985)
[a5]	P. Antosik, J. Mikusiński, R. Sikorski, "Theory of distributions. The sequential approach" , Elsevier (1973)
[a6]	G. Köthe, "Topological vector spaces" , 1 , Springer (1969)
[a7]	J. Horvath, "Topological vector spaces and distributions" , Addison-Wesley (1966)
[a8]	W. Rudin, "Functional analysis" , McGraw-Hill (1974)