For all $e \in E^p$, we call the $p$-volume of $e$ the quantity $$\operatorname{vol}_p(e) = \sqrt{\Omega_p(e)(e)} = \left(\operatorname{det}(\operatorname{Gram}(e, e))\right)^{1/2}.$$ (a) Calculate $\operatorname{vol}_p(b)$ when $b = (b_1, \ldots, b_p)$ is an orthonormal family of vectors of $E$. (b) Suppose here that $p \geqslant 2$. Let $e = (e_1, \ldots, e_p) \in E^p$. We denote by $\operatorname{pr}$ the orthogonal projection onto the orthogonal of the space spanned by the family $e_2^p = (e_2, \ldots, e_p)$. Show that $\operatorname{vol}_p(e) = \|\operatorname{pr}(e_1)\| \operatorname{vol}_{p-1}(e_2^p)$. (c) For all free families $e = (e_1, \ldots, e_p) \in E^p$, show that $\operatorname{vol}_p(e) \leqslant \prod_{i=1}^p \|e_i\|$ with equality if and only if $e$ is a family of pairwise orthogonal vectors.
For all $e \in E^p$, we call the $p$-volume of $e$ the quantity
$$\operatorname{vol}_p(e) = \sqrt{\Omega_p(e)(e)} = \left(\operatorname{det}(\operatorname{Gram}(e, e))\right)^{1/2}.$$
(a) Calculate $\operatorname{vol}_p(b)$ when $b = (b_1, \ldots, b_p)$ is an orthonormal family of vectors of $E$.
(b) Suppose here that $p \geqslant 2$. Let $e = (e_1, \ldots, e_p) \in E^p$. We denote by $\operatorname{pr}$ the orthogonal projection onto the orthogonal of the space spanned by the family $e_2^p = (e_2, \ldots, e_p)$. Show that $\operatorname{vol}_p(e) = \|\operatorname{pr}(e_1)\| \operatorname{vol}_{p-1}(e_2^p)$.
(c) For all free families $e = (e_1, \ldots, e_p) \in E^p$, show that $\operatorname{vol}_p(e) \leqslant \prod_{i=1}^p \|e_i\|$ with equality if and only if $e$ is a family of pairwise orthogonal vectors.