Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 22I / c / Solution

Let $P_N$ be the orthogonal projection of ${\ell }^2$ onto the span of its first $N$ standard basis vectors. Then $P_{N}y\to y$ for every $y\in {\ell }^2$, and $\Vert I-P_N\Vert \le 1$.

Let $T$ be compact and letwhere $B$ is the closed unit ball. The set $K$ is compact. Given $\epsilon >0$, choose a finite $\epsilon /3$-net $y_1,\dots ,y_m$ in $K$. Pointwise convergence lets us choose $N$ such thatfor every $j$. If $y\in K$ and $\Vert y-y_j\Vert <\epsilon /3$, thenThus $P_N\to I$ uniformly on $K$, andEach $P_{N}T$ has image in an $N$-dimensional space, so it has finite rank. This coordinate-projection approximation of a compact operator proves the result.