Let E,F be any two Banach space and let $\mathcal{B}(E,F)$ be the space of all bounded linear operators from E to F. I can show that this space is a complete space with respect to the norm and strong operator topology and any Cauchy sequence in the weak operator topology is uniformly bounded, but:

  1. Can we show that it's complete under the weak operator topology? I know that the answer is YES when $E,F$ are Hilbert spaces.
  2. If not, how can we find a sequence of bounded operators $\{T_n\}$ that converges to an unbounded operator in the Weak Operator Topology?
  3. Moreover, is there an analogue of Banach-Alaoglu theorem on $\mathcal{B}(E,F)$? More specifically, does the closed unit ball in the norm topology compact with respect to the Weak Operator Topology?

Thanks in advance for your help!

  • $\begingroup$ maybe for 3. (and 2.) you can reduce to what you know by considering $g \in F^*$ then for $T \in B(E,F)$ : $gT \in E^*$ $\endgroup$ – reuns Apr 20 '16 at 3:12
  • $\begingroup$ and if $T$ is unbounded then you can always find a sequence $T_n \in B(E,F)$ converging weakly to $T$ on the subspaces where $T$ is bounded, but with $T_n x$ diverging for some $x$, hence $T_n$ doesn't converge to $T$ in weak-* $\endgroup$ – reuns Apr 20 '16 at 3:20

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