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)