Inner Products on the Hilbert Space S₂ of Hilbert–Schmidt Operators

This work presents a rigorous characterization of continuous inner products on the Hilbert space S₂ S 2 of Hilbert–Schmidt operators. We begin by addressing the general framework of continuous sesquilinear forms on a Hilbert space H H and provide a characterization of all continuous inner products, by means of positive operators in B(H) B ( H ) . Next, we establish necessary and sufficient conditions for an operator in B(S₂) B ( S 2 ) to be positive. Identifying a continuous inner product with a positive operator enables us to rigorously describe inner products on S₂ S 2 .