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

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).

Conversely, supposeand defineFor every $N$, Bessel inequality and passage to the limit giveHence ${\sum }_i|a_i{|}^2\le C^2$. The partial sums ${\sum }_{i=1}^N{a}_i{e}_i$ are therefore Cauchy; completeness of the Hilbert space gives an element

Let $P_{N}h={\sum }_{i=1}^N⟨h,e_i⟩e_i$. For fixed $N$, coordinate convergence givesThe Hilbertian basis property gives $P_{N}h\to h$, andFirst choose $N$ large and then $n$ large. This provesfor every $h\in H$, so $x_n\rightharpoonup x_{\infty }$. This establishes the coordinate criterion for weak convergence in a separable Hilbert space.