Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 4 / 22F / v / Solution

Apply the proved projection result to $V=\overline{X}$ and write $f=v+w$ with $w\perp V$. By part (iii), $A_{n}v\to 0$, while part (iv) gives $U^{k}w=w$ and hence $A_{n}w=w$. ThusThe limit is the orthogonal projection onto the fixed vectors of $U$. This is norm convergence, not just weak convergence.