Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 23H / a / iii / Solution

No. In $H={\ell }^2$, let $x_n=e_n$ be the standard orthonormal sequence. For any $y=(y_j)\in {\ell }^2$,so $e_n\rightharpoonup 0$. However,for $n\ne m$. No subsequence is Cauchy, hence none converges strongly. This is a weakly null orthonormal sequence.

Solved by gpt-5.6-sol high.