TITLE: compact-open topology on $B(H)$
QUESTION [18 upvotes]: In topology, it is common to use the compact-open topology on the set of continuous maps between two given topological spaces.
Let now $H$ be a Hilbert space and $B(H)$ the set of continuous linear maps from $H$ to itself. In functional analysis, there are many topologies that people like to use on $B(H)$. Does any one of them agree with the compact-open topology? If yes, which one?

REPLY [2 votes]: As an introductory remark, there has been a considerable amount of work done on topologies on spaces of operators in Hilbert spaces, in particular with regard to their relevance to the theory of von Neumann algebras.  In my opinion there are two essential criteria which one should apply: the topology should be complete and its dual should be a natural space of operators.  The compact open topology fails the second  one---its dual is the space of finite rank operators.
(Incidentally, the most immediate natural  candidate, the norm topology, also passes the first one but fails the second---in this case, the dusl is too large).  Further candidates (which have already been mentioned)---the weak, strong, ultraweak and ultrastrong topologes---also fail this test.  A suitable family of natural topologies which pass both tests with flying colours (whereby the dual spaces are the space of nuclear operators) was introduced 40 years ago in the Comptes Rendues paper "Topologies dans l'espace des operateurs sur les espaces de
Hilbert" (1973).  These are the finest locally convex topologies which agree with the topology of compact  convergence on the unit ball.  (There are four, since we can consider convergence with resepct to the weak or strong  topologies and also their symmetric versions, i.e., ones for which the operation of taking adjoints is continuous).  Some of their basic properties can be found in the above article.
This shows that the answer to your question is pretty well always no and goes on to answer the (implicit?) continuation---what is the "correct" topology on $B(H)$?