Tensor Products of Hilbert Spaces

But each en belongs either to the nullspace of V or its orthogonal complement, so the nonzero V en’s are an orthonormal set. Since S

∗ and T are Hilbert-Schmidt, it follows that

∑ ‖SV en‖2 < ∞ and ∑ ‖Ten‖2 < ∞, so |ST | is trace-class.