Given $E$ Banach space and a sequence of continuous operators $(T_n)_n$ with $T_n:E\to E$.
If $T_n\to T$, is it true that $T_n^2\to T^2$?
I think it is true because I don't find any counterexample, but I don't know how to prove it. Thank you!
2026-03-29 15:35:58.1774798558
Convergence of $T_n^2\to T^2$
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If $T_n\to T$ means convergence in the operator norm, then we have
$$||T_n^2-T^2||=||T_n^2-T_nT+T_nT-T^2|| \le (||T_n||+||T||)||T_n-T||.$$
Can you proceed ?