Let H be a Hilbert space and $\{x_n\}\subset H$ a sequence such that $\langle x_n,x\rangle\to 0$ when $n\to \infty$.
I am trying to prove the sequence is bounded using Banach-Steinhaus using the following sequence of operators: $T_n: H\to \Bbb R, x\to \langle x_n,x\rangle$.
It is straightforward that $T_n\in \mathcal L(H,\Bbb R)$ and that $\forall x\in H, |T_n(x)|< \infty$.
To conclude I need to relate $||T_n||$ to $||x_n||$.
By Cauchy-Schwarz we have $||T_n||\le ||x_n||$ but is there equality here? Othewise my choice of $||T_n||$ won't be relevant.
Thank you for your help