Exercise 12, § 1

a) Show that there exists a number $c_0 > 0$ such that, for every real normed vector space E of dimension $n$ and every integer $k \leq c_0 n$, there exists a Hilbertian norm $x \mapsto \|x\|_2$ on E such that $\|x\|_2 \leq \|x\|$ for every $x \in E$, as well as an orthonormal system $(x_j)_{1 \leq j \leq k}$ of $k$ elements of E (for the Hilbertian structure) of norms $\|x_j\| \leq 2$. (Use Ex. 3 of V, p. 59.) b) Let $n, m$ be two integers $> 0$ such that $n \leq c_0 m$. Let E be a real normed vector space, of dimension $m$. Show that there exists a vector subspace F of E, of dimension $n$, a positive and separating symmetric bilinear form $(x, y) \mapsto \langle x|y \rangle$ on F and an orthonormal basis $\{a_1, a_2, \ldots, a_n\}$ of F such that

(where $\|x\|_2^2 = \langle x|x \rangle$) for every $x \in F$. (Apply a) to the dual E’ of E.)