Exercise 15, § 2

Let E be a normed space of finite dimension n over R or C. Ascribe to the dual E*, the norm defined by $\|x^*\| = \sup_{\|x\| \leq 1} |\langle x, x^*\rangle|$ (GT, X, § 3.2). Show that there exists a basis (e_i)

1 For the exercises 12 and 13, see O. Goldman and N. Iwahori, The space of p-adic norms, Acta math., 109 (1963), pp. 137-177.

of E such that, if $(e_i^*)$ is the dual basis, we have $\|e_i\| = \|e_i^*\| = 1$ for all $i$. (Let $(a_i)$ be a basis of E formed of vectors of norm 1; consider, the determinant $\det(\xi_{ij})$ for each system of $n$ vectors $x_i = \sum_j \xi_{ij} a_j$ of norm 1, and consider such a system for which the absolute value of this determinant is maximal.)