On a superaddivity property of Gram's determinant (Q1364864)

The following result is proven: Let \((.,.)_1\), \((.,.)_2\) be two inner products on the linear space \(H\). Then one has the inequality \[ [\Gamma((.,.)_1+ \Gamma(.,.)_2; a_1,\dots, a_n)]^{1/n}\geq [\Gamma((.,.)_1; a_1,\dots, a_n)]^{1/n}+ [\Gamma((.,.)_2; a_1,\dots, a_n)]^{1/n} \] for all \(a_i\in H\), \(1\leq i\leq n\), \(n\geq 1\), where \(\Gamma\) denotes the Gram determinant, that is \[ \Gamma((.,.); a_1,\dots, a_n)= \text{det}[(a_i, a_j)]_{1\leq i,j\leq n}. \]