Let $H$ be a Hilbert space and $T:H\to H$ a compact operator. Let $(e_n)$ be an orthonormal sequence in H.
Prove $Te_n \rightarrow 0$.
Hint: what do you know about $\langle Te_n,f\rangle$ for $f\in H$?
I'm taking a course in Hilbert space and operators and stumbled on this problem that I can't see how to solve.
Can somebody please help me solve this?
The hint tells you to note that $\langle Te_n,f\rangle \to 0$, as $$ \langle Te_n,f\rangle=\langle e_n,T^*f\rangle\to 0 $$ since $e_n$ converges to $0$ weakly. So $Te_n$ converges to $0$ weakly.
Now for the sake of contradiction, suppose it does not converge strongly to $0$, then along a subsequence $$ \lim_{n'\to \infty}\|Te_{n'}\|>\delta>0. $$ But then by compactness, there is a further subsequence such that $Te_{n''}\to g$ for some $g$ strongly. But $\|g\|>\delta$ by the above, while we still have $Te_{n''}$ converges weakly to $0$, contradicting that weak and strong limits should agree.