Solution
= Solution
Take an <orthonormal basis> $\{|j\rangle\}$ and insert its <completeness relation>:
$$
\operatorname{Tr}(|u\rangle\langle v|)
=\sum_j\langle j|u\rangle\langle v|j\rangle
=\langle v|\left(\sum_j|j\rangle\langle j|\right)|u\rangle
=\boxed{\langle v|u\rangle}.
$$
No normalization or <orthogonality> of $u,v$ is required. This is the <trace> of a <rank-one operator>; in a <Hilbert space> the sum converges by the <Cauchy-Schwarz inequality>.