An inner product space that makes a set of vectors orthonormal (Q2772820)

Let \(n \geq 2\) and V be a real vector space of dimension \(d \geq n\). An \(n\)-inner product space is a real-valued function \((\cdot,\cdot |\cdot,\ldots,\cdot)\) on \(V^{n+1}\) such that it is commutative in the two first variables, is additive in the first variable and satisfies the following two conditions: NEWLINENEWLINENEWLINE(i) \((x_1,x_1 |x_2,\dots ,x_n)\geq 0\), and the equality to 0 takes place if and only if \(x_1,x_2,\dots ,x_n\) are linearly dependent; NEWLINENEWLINENEWLINE(ii) \((x_1,x_1|x_2,\dots ,x_n)=(x_{i_1},x_{i_1}|x_{i_2},\dots ,x_{i_n})\) for every permutation \((i_1,\dots ,i_n)\) of \((1,\dots ,n)\). NEWLINENEWLINENEWLINEThe concept of \(2\)-inner product was introduced by \textit{C. Diminnie, S. Gähler} and \textit{A. White} [Demonstratio Math. 6, 525-536 (1973; Zbl 0296.46022)] and developed by \textit{A. Misiak} [Math. Nachr. 140, 299-319 (1989; Zbl 0673.46012)] for general \(n \geq 2\). A comprehensive presentation of the latest results of the theory of \(n\)-inner product spaces is \textit{Yeol Je Cho, Paul C.S. Lin, Seong Sik Kim} and \textit{A. Misiak} [``Theory of 2-Inner Product Spaces'', Huntington: Nova Science Publishers (2001)]. NEWLINENEWLINENEWLINELet now \(\{a_1,a_2,\ldots ,a_n\}\) be a linearly independent set in a real inner product space \((X,<.,.>)\) of dimension \(d \geq n\). It is an interesting problem whether one can explicitly get an inner product \(<.,.>^*\) on \(X\) with respect to which \(\{a_1,a_2,\ldots ,a_n\}\) becomes an orthonormal set. Applying the notion of \(n\)-inner product, the author affirmatively answers this problem.