Let $X$ and $Y$ be Banach spaces with $X$ reflexive. I now have a bounded operator $T \colon X \to Y$ that also has the property that if $(x_i)_{i \in I}$ is a bounded net that weakly converges, then the net $(Tx_i)_{i \in I}$ is convergent in norm. I now have to prove that $T$ is a compact but I have no idea on how to start.
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The proofs from this post should work. Also, correct me if I'm wrong: isn't a weakly-convergent net necessarily bounded? – Ben Grossmann Dec 11 '19 at 10:43
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@Omnomnomnom consider $( t^{-1})_{t\in I}$ with $I=(0,+\infty)$ and the usual order so that the limit is zero – daw Dec 11 '19 at 11:02
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@daw ah that's a great example, thanks – Ben Grossmann Dec 11 '19 at 11:30