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.

For a "pedagogical" proof, see A Note on von Neumann's Trace Inequality by Rolf Dieter Grigorieff.

It has been remarked in the literature that

"unexpectedly, finding a decent proof of this seemingly simple result turns out to be anything but trivial". The aim of the present note is to present still a further proof which seems to be elementary enough to correspond to the simplicity of the statement in von Neumann's Theorem.

The original proof by von Neumann is in "*Some matrix-inequalities and metrization of metric-space,*" Tomsk Univ. Rev. **1** (1937) 286-300 -- which I have not found online.