# Bound for matrix inner product based on singular values

Regarding the matrix inner product based on singular values, Lewis (1995) "The convex analysis of unitarily invariant matrix functions" states the result by von Neumann that $$\langle X,Y \rangle \leq \langle \sigma_X ,\sigma_Y \rangle$$. Does anyone know any easy proof or reference for it. I couldn't understand the reference cited in the paper.