Let $H$ be a Hilbert space. We say that a linear operator $T \colon H \to H$ is compact if it maps bounded sets to precompact ones, that is, if for every bounded sequence $(a_n)$ in $H$, $(Ta_n)$ has got a convergent subsequence. We can characterize compactness in terms of weak convergence: $T$ is compact if and only if it maps weakly convergent sequences into norm convergent ones.
Up to now I would have casually said:
$T$ is compact iff $T$ is continuous when its domain is equipped with weak topology and its range with norm topology.
and I would not have been alone: I've heard this from more than one of my professors. Well, this turns out to be false, as I read on Problem VI.34 of Reed & Simon's Methods of modern mathematical physics:
Show that in a Hilbert space $H$, a map $T \colon H \to H$ is continuous when its domain is given the weak topology and its range the norm topology if and only if $T$ has finite rank!
(the exclamation point is part of the original text).
At the moment this problem is beyond my grasp, since I know almost nothing of the necessary topological tools (that is - nets, Moore-Smith convergence, and the like). But I'm curious about this and wanted to share my curiosity with the community.
Has somebody got any idea on how to prove this, or even some hint that could help me to catch this intuitively?