Solution
= Solution
\b[No.] In $H=\ell^2$, let $x_n=e_n$ be the standard orthonormal sequence. For any $y=(y_j)\in\ell^2$,
$$
\langle e_n,y\rangle=y_n\longrightarrow0,
$$
so $e_n\rightharpoonup0$. However,
$$
\lVert e_n-e_m\rVert=\sqrt2
$$
for $n\ne m$. No subsequence is Cauchy, hence none converges strongly. This is a <weakly null orthonormal sequence>.