Inductive limit commutes with topological tensor product

Question

Consider $H \left(U \right); U \subset \mathbb{C}$ - space of holomorphic functions with compact-open topology. In this topology, this space is Montel, nuclear and Frechet. I want to take the inductive limit of tensor product of 2 copy of this space and I am wondering:

$$ \varinjlim\limits_{U \supset K} \left( H \left(U \right) \hat{\bigotimes} H \left(U \right) \right) \stackrel{?}{=} H \left(K \right) \hat{\bigotimes} H \left(K \right) = H \left(K \times K\right), $$ where $H \left( K \right)-$ space of germs of holomorphic functions on compact set $K \subset \mathbb{C}$. It doesn't matter which type of tensor product we take because space is nuclear.

Answer 1

Only some ideas, but too long for a comment.

I think there are at least to approaches which might help here:

1. Since the spaces here are Frechet spaces and therefore every partially continuous bilinear map is continuous, you can work with the "inductive tensor product" (which is different from the injective tensor product) introduced by A. Grothendieck which works well with inductive limits - here the question reduces to the question of completeness of the inductive limit on the left hand side, see Proposition 14 on p. 76 of Grothendieck's thesis. The completeness of inductive limits of course is also quite nontrivial.
2. You could try to use the commutativity results of R. Hollstein (Inductive limits and ε-tensor products.) to show the desired results. Since every nuclear space is an $\varepsilon$-space in his sense the problem here is again to show that your spectrum satisfies one of his conditions.