Let me first give a little rapid background prior to formulating the question.
Let D be a Schwartz space of infinitely differentiable functions and D′ is the space of distributions acting on D.
The Schwartz Kernel Theorem states that for any linear continuous operator A:D→D′ there exists a distribution K∈[D×D]′ such that
where f⊗g is a tensor product of functions f and g.
Grotendieck in “Produits tensoriels topologiques et espaces nucléaires” , Amer. Math. Soc. (1955) obtained a following generalization:
Let F be a nuclear locally convex space. Then for every E locally convex space all continuous bilinear form B(f,φ) are nuclear.
A bilinear form B(f,φ) on the Cartesian product F×E two locally convex spaces is called nuclear if it can be represented in the form
where fk and φk are the sequences in the dual space D′ and λk is a summable sequence.
Now the question is how to see that if an operator A is continuous between D and D′,i.e. A∈L(D,D′), then the bilinear form B(f,φ)=(Af,φ) is nuclear?