Yes, if you understand the tensor product topology in the right way. Since the spaces are nuclear, inductive and projective tensor products coincide. The result is theorem 51.6 of Treves: Topological vector spaces, distributions, and kernels.

Yes, if you understand the tensor product topology in the right way. Since the spaces are nuclear, inductive and projective tensor products coincide. The result is theorem 51.6 of Treves: Topological vector spaces, distributions, and kernels.

Added later:

Attention: The description of seminorms on the tensor-product given in the question is not sufficient to specify a locally convex topology on the tensor product. There are many satisfying this description; all between the the projective one and the the inductive one and even more. See the source I have given, or many other books.