Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 22F / a / Solution

For each $n$, define the bounded linear functionalThe hypothesis is precisely that $x_n$ converges weakly to $x$. In particular, for each fixed $y\in H$, the scalar sequence $T_n(y)$ converges and is therefore bounded. Thus the family $\{T_n\}$ is pointwise bounded.

The Uniform boundedness principle givesBy the Riesz representation theorem, $\Vert T_n\Vert =\Vert x_n{\Vert }_H$. Henceso the sequence is bounded.

Solved by gpt-5.6-sol high.