I am currently reading up on nuclear spaces in Yarchow, Locally Convex Spaces, but I got confused an don't seem to find my mistake :( So, in said book, theorem 21.5.9 states:

Let F be a nuclear Frechet space. Then, F'\beta \otimes_pi F = L\beta (F,F)

i.e. the projective tensor product of F with its strong dual is homeomorphic to the operators on F with the strong topology.

Now, that means every operator on F has a trace, right?

So take the the space X=\Pi_\mathbb{N} \mathbb{R}. This is a nuclear Frechet space as it is a countable infinite product of those. Let id_k : \mathbb{R} \rightarrow \mathbb{R} denote the identity on the k'th component of X. We have id_X = \Sigma_k id_k. Now, by above theorem id_X and all id_k must be trace class, but clearly, the sum can not converge to id_X as otherwise the trace must be infinite.

If I read correctly, the strong topology has a 0-basis of L(B,U), maps from some bounded set into an open set. The partial sums f_n=\Sigma_1≤k≤n id_k however satisfy that f_n-id_X is zero on the first n components and thus mapa any set into a chosen open set for n big enough. Thus, the partial sum converges and there is a contradiction.

I simply can't spot my mistake, perhaps you can help me?