I researched a lot what the point of nuclear spaces is. From what I understand they were invented by Grothendieck to make a more general statement for the Kernel Theorem by Schwartz. He figured out that the Theorem holds more generally if the respective spaces are nuclear.
Now a nuclear space can be defined by saying that the projective tensor product coincides with the injective tensor product.
I wanted to figure out where exactly in the proof or which part of it really depends on the nuclearity. More generally I wanted to understand how Grothendieck came up with the idea that the spaces have to be nuclear. I also can't really find the proof of the theorem, I looked in Grothendiecks doctoral thesis but it is french so I am not sure which part it is.
I hope there is someone here that could shed some light on this.
Thanks you!
Well, for one thing, A. Grothendieck had a great affection (which turned out to be substantially justifiable) for "the general case". So, in some cases, that explains why he wrote one thing rather than another.
Also, unsurprisingly, sometimes wrangling with "the general case" sheds light on the more-specific cases where things work well.
And, yes, the technical details of the discrepancy between "injective" and "projective" tensor products of Banach (and other) topological vector spaces can be an interesting research line in itself, of course.
From my own viewpoint, I'm more interested in the situations where the two "opposite" tensor products do coincide... which is to say, there is a genuine (in a categorical sense) tensor product of two topological vector spaces. If we have genuine tensor products for two TVS's $X,Y$, then $$ \mathrm{Hom}(X,Y^*) \;=\; \mathrm{Hom}(X,\mathrm{Hom}(Y,\mathbb C)) \;=\; \mathrm{Hom}(X\otimes Y,\mathbb C) $$ as a special case of the Hom-tensor duality (also known as Cartan-Eilenberg adjunction). For $X,Y$ both equal to test functions on $\mathbb R^n$, for example, or Schwartz functions, this exactly asserts the Schwartz Kernel theorem.