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

(Af,φ)=K(f⊗φ),

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 callednuclearif it can be represented in the form

B(f,φ)=∞∑k=1λk(f,fk)(φ,φk),

where fk and φk are the sequences in the dual space D′ and λk is a summable sequence.

Now thequestionis 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?

**Answer**

**Attribution***Source : Link , Question Author : Rauan Akylzhanov , Answer Author : Community*