I have the following question in my functional analysis book I dont understand: $X$ is an infinite dimensional Hilbert space with an orthonormal basis $(e_n)$. Show that if $T \in K(X)$, then $Te_n \rightarrow 0$.
My attempt:
I define $Te_n=\sum\limits_{i=1}^\infty (Te_n,e_i)e_i$, $T$ is compact so there exists for every bounded sequence $x_n \in X$ a convergent subsequence $x_{n_k}$ such that $Tx_{n_k}$ converges in $X$.
I think that $e_n$ is a sequence of some sorts, dont see why actually, and here I strand.
Thanks for the help!
For any $x\in X$ we have $(x,e_n )\longrightarrow 0$ hence $e_n\longrightarrow 0$ weakly and since, every compact oprator $T$ maps weak convergent sequences into strong convergent sequences we obtain that $Te_n \longrightarrow 0.$