The following is a Theorem of Conway's operator theory:
I can not understand how he proves it. I think $\phi(\text{ ball B(H)})$ is compact if $\phi(\text{ ball B(H)})$ is closed subset of compact set $X$, but why is $X$ compact?
2026-03-29 20:51:57.1774817517
Closed unit ball of $B(H)$ with wot topology is compact
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Tychonoff's theorem in general point-set topology states that an arbitrary product of compact topological spaces, endowed with the product topology, is again compact. Since the unit ball in a Hilbert space is weakly compact, $X$ becomes compact. Your conclusion is correct I believe.