The abstract kernel theorem implies Schwartz kernel theorem. How exactly?

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:DD there exists a distribution K[D×D] such that
(Af,φ)=K(fφ),
where fg 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
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 the question is how to see that if an operator A is continuous between D and D,i.e. AL(D,D), then the bilinear form B(f,φ)=(Af,φ) is nuclear?

Answer

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

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