Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 22H / a / i / Solution

Assume $x_n\rightharpoonup x_{\infty }$. For each $n$, define the bounded functionalWeak convergence makes $(T_n(h){)}_n$ convergent, hence bounded, for every $h\in H$. The Uniform boundedness principle givesBy the Riesz representation theorem, $\Vert T_n\Vert =\Vert x_n\Vert$, so $(x_n)$ is bounded.

Taking $h=e_i$ in the definition of weak convergence in a Hilbert space givesfor every $i$. Thus (i) implies (ii).