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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.

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    $\begingroup$ They are not Frechet: Otherwise the identity map $(B(H),SOT) \to (B(H),$operator norm$)$ would be continuous due to the closed graph theorem. By the same reason, they are not ultrabornological. $\endgroup$
    – Jochen
    Mar 29, 2017 at 10:14
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    $\begingroup$ You should be able to show it is not first countable, the same way one does for the weak topology on $H$. $\endgroup$
    – Nate Eldredge
    Mar 29, 2017 at 22:14

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