Nuclear bilinear form

A bilinear form on the Cartesian product of two locally convex spaces and that can be represented as

where is a summable sequence, and are equicontinuous sequences (cf. Equicontinuity) in the dual spaces and of and , respectively, and denotes the value of the linear functional on the vector . All nuclear bilinear forms are continuous. If is a nuclear space, then for any locally convex space all continuous bilinear forms on are nuclear (the kernel theorem). This result is due to A. Grothendieck [1]; the form stated is given in [2]; for other statements see [3]. The converse holds: If a space satisfies the kernel theorem, then it is a nuclear space.

For spaces of smooth functions of compact support, the kernel theorem was first obtained by L. Schwartz [4]. Let be the nuclear space of all infinitely-differentiable functions with compact support on the real line, equipped with the standard locally convex topology of Schwartz, so that the dual space consists of all generalized functions on the line. In the special case when , the kernel theorem is equivalent to the following assertion: Every continuous bilinear functional on has the form

where and is a generalized function in two variables. There are similar statements of the kernel theorem for spaces of smooth functions in several variables with compact support, for spaces of rapidly-decreasing functions, and for other specific nuclear spaces. Similar results are valid for multilinear forms.

A continuous bilinear form on can be identified with a continuous linear operator by using the equality

and this leads to Schwartz' kernel theorem: For any continuous linear mapping there is a unique generalized function such that

for all . In other words, is an integral operator with kernel .

References

[1]	A. Grothendieck, "Produits tensoriels topologiques et espaces nucléaires" , Amer. Math. Soc. (1955)
[2]	A. Pietsch, "Nuclear locally convex spaces" , Springer (1972) (Translated from German)
[3]	I.M. Gel'fand, N.Ya. Vilenkin, "Generalized functions. Applications of harmonic analysis" , 4 , Acad. Press (1964) (Translated from Russian)
[4]	L. Schwartz, "Théorie des noyaux" , Proc. Internat. Congress Mathematicians (Cambridge, 1950) , 1 , Amer. Math. Soc. (1952) pp. 220–230
[5]	L. Schwartz, "Espaces de fonctions différentielles à valeurs vectorielles" J. d'Anal. Math. , 4 (1954–1955) pp. 88–148

References

[a1]	F. Trèves, "Topological vectorspaces, distributions and kernels" , Acad. Press (1967)
[a2]	L. Schwartz, "Théorie des distributions" , 1–2 , Hermann (1966)