Weak Banach–Saks theorem in a Hilbert space
= Weak Banach–Saks theorem in a Hilbert space
If $x_n\rightharpoonup x$ in a <Hilbert space>, some subsequence $(x_{n_k})$ has <Cesaro means> converging in norm to $x$. For $x=0$, choose the subsequence so successive vectors have summably small inner products; expansion of the squared norm of its averages then proves convergence.
Thus <Hilbert spaces> have the weak form of the <Banach-Saks property>.