Exercise 15, § 2

Let E be a finite-dimensional normed space of dimension n over R or C. We equip the dual E* with the norm defined by $\|x^*\| = \sup_{\|x\| \leq 1} |\langle x, x^* \rangle|$ (TG, X, p. 23). Prove that there exists a basis $(e_i)$ of E such that, if $(e_i^*)$ is the dual basis, one has $\|e_i\| = \|e_i^*\| = 1$ for all i. (Let $(a_i)$ be a basis of E formed by vectors of norm 1; consider, for every system of n vectors $x_i = \sum_j \xi_{ij} a_j$ of norm 1, the determinant $\det(\xi_{ij})$ and consider such a system for which the absolute value of this determinant is maximal.)