Exercise 37, § 1

Let $E$ be a finite-dimensional real Hilbert space of dimension $n$ and let $\gamma$ be the positive measure on $E$ such that $\int_E e^{i\langle x|y \rangle} d\gamma(y) = \exp(-\pi \|x\|^2/2)$ for all $x \in E$ (INT, IX, § 6, No. 5). Let $m$ be the unique positive measure of norm 1 on the unit sphere $S$ of $E$ which is invariant under the automorphism group of $E$.

a) Let $p$ be a continuous function on $E$, satisfying $p(t.x) = t.p(x)$ for $x \in S$ and $t$ positive real. Now prove that one has $\int_E pd\gamma = c_n \int_S pdm$ with $c_n = \pi^{1/2} \Gamma(n/2)/\Gamma((n+1)/2)$.

b) Now prove that there exists a constant $C > 0$, independent of $n$, such that one has

for every orthonormal basis $(e_1, ..., e_n)$ of $E$.

c) Let $\varepsilon > 0$ and let $k$ be a positive integer. Deduce from b) and Exercises 12 (V, p. 62) and 36, $d$) that if $n$ is sufficiently large, there exists, for every real normed space $V$ of dimension $n$, a vector subspace $W$ of $V$, of dimension $k$, satisfying the following property: there exists a real Hilbert space $W_1$, of dimension $k$, and a bijective linear mapping $u$ of $W$ onto $W_1$ such that $\sup(\|u\|, \|u^{-1}\|) \leq 1 + \varepsilon$.