Let $A_{(n)}:\mathcal B\rightarrow \mathcal B$ be a linear bounded operators such that for any $f\in \mathcal B$, $A_nf\rightarrow Af$. Show that then for any compact operator $B$
$$\|A_nB-AB\|\rightarrow0$$
Any tips how to show this?
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Also: https://math.stackexchange.com/questions/182641/compact-operators-and-uniform-convergence – Jan Feb 25 '20 at 19:55
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Does this answer your question? Convergence of $A_nT$ to $AT$ in operator norm for compact $T$ – Jan Feb 25 '20 at 19:55