Nuclear bilinear form

A bilinear form $ B ( f , g) $ on the Cartesian product $ F \times G $ of two locally convex spaces $ F $ and $ G $ that can be represented as

$$ B ( f, g) = \ \sum _ {i = 1 } ^ \infty \lambda _ {i} \langle f, f _ {i} ^ { \prime } \rangle \langle g, g _ {i} ^ \prime \rangle, $$

where $ \{ \lambda _ {i} \} $ is a summable sequence, $ \{ f _ {i} ^ { \prime } \} $ and $ \{ g _ {i} ^ \prime \} $ are equicontinuous sequences (cf. Equicontinuity) in the dual spaces $ F ^ { \prime } $ and $ G ^ \prime $ of $ F $ and $ G $, respectively, and $ \langle a, a ^ \prime \rangle $ denotes the value of the linear functional $ a ^ \prime $ on the vector $ a $. All nuclear bilinear forms are continuous. If $ F $ is a nuclear space, then for any locally convex space $ G $ all continuous bilinear forms on $ F \times G $ 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 $ F $ 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 $ D $ 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 $ D ^ \prime $ consists of all generalized functions on the line. In the special case when $ F = G = D $, the kernel theorem is equivalent to the following assertion: Every continuous bilinear functional on $ D \times D $ has the form

$$ B ( f, g) = \langle f ( t _ {1} ) g ( t _ {2} ), F \rangle = $$

$$ = \ \int\limits _ {- \infty } ^ \infty F ( t _ {1} , t _ {2} ) f ( t _ {1} ) g ( t _ {2} ) dt _ {1} dt _ {2} , $$

where $ f ( t), g ( t) \in D $ and $ F = F ( t _ {1} , t _ {2} ) $ 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 $ B ( f, g) $ on $ D \times D $ can be identified with a continuous linear operator $ A: D \rightarrow D ^ \prime $ by using the equality

$$ B ( f, g) = \langle g, Af \rangle, $$

and this leads to Schwartz' kernel theorem: For any continuous linear mapping $ A: D \rightarrow D ^ \prime $ there is a unique generalized function $ F ( t _ {1} , t _ {2} ) $ such that

$$ A: f ( t _ {1} ) \mapsto \int\limits _ {- \infty } ^ \infty F ( t _ {1} , t _ {2} ) f ( t _ {2} ) dt _ {2} $$

for all $ f \in D $. In other words, $ A $ is an integral operator with kernel $ F $.

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)