Consider the SOT and WOT operator topologies on $B(H)$, the bounded operators on a hilbert space $H$. I'm interested in the properties of the topological vector spaces induces on $B(H)$. Are they Fréchet? Or does there exist a closed graph theorem for mappings between these spaces?
This question came up when looking for a question to this post and hoping to apply the argument I developed in another post to solve the first one. Both questions have only been answered partially and I'm hoping to gain more insight on the matter by opening a new post asking a more specific question.