Inner products on the Hilbert space $S_2$ of Hilbert--Schmidt operators

This work presents a rigorous characterization of inner products on the Hilbert space $S_2$ of Hilbert–Schmidt operators. We first deal with a general setting of continuous sesquilinear forms on a Hilbert space $\mathcal{H}$, and provide a characterization of all inner products by means of positive operators in $\mathcal{B}(\mathcal{H})$. Next, we establish necessary and sufficient conditions for an operator in $\mathcal{B}(S_2)$ to be positive. Identifying an inner product with a positive operator enables us to rigorously describe inner products on $S_2$.