Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 21I / b

Compactness gives a stronger statement than the one requested. Since $T(B_H)$ has compact closure, finitely many open balls of radius $\delta$ cover it. Choose their centres from $T(B_H)$ and let $E$ be their finite-dimensional linear span. ThenFor the orthogonal projection $P_E$, best approximation in a Hilbert space givesIn particular this holds for every vector $e_n$ in the given orthonormal basis.

The operator $P_{E}T$ has image in $E$ and hence has finite rank, whileThus every compact operator is an operator norm limit of finite-rank operators. This is the finite-rank approximation theorem for compact operators on a Hilbert space.