Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 4 / 22H / b / Solution

A compact operator is a bounded linear operator carrying every bounded set into a relatively compact set. Equivalently, for every bounded sequence $(x_n)$, the sequence $(T{x}_n)$ has a norm-convergent subsequence. It is enough to require this for the closed unit ball, because any bounded set lies in a scalar multiple of that ball.