Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 225 / 3 / b

For positive trace-class covariance operators $C_i=L_i{L}_i^{\ast }$, their Procrustes distance between covariance operators iswhere $\mathcal{O}(H)$ is the group of unitary operators on the real Hilbert space $H=L^2[0,1]$. Expanding the square gives the equivalent formula

Convergence in the Hilbert-Schmidt norm impliesProducts of two Hilbert-Schmidt operators are trace-class operators, and the Schatten norm Hölder inequality givesBy trace duality, every unitary $R$ satisfiesTaking the supremum over $R$ shows that the supremum terms in the two Procrustes formulas converge. Combining this with convergence of the squared norms proves