For all $e \in E^p$, we consider $\Omega_p(e) : E^p \rightarrow \mathbb{R}$ defined for all $u \in E^p$ by $$\Omega_p(e)(u) = \operatorname{det}(\operatorname{Gram}(e, u)).$$ (a) Verify that the map $(x_1, \ldots, x_p) \mapsto [x_1, \ldots, x_p]$ belongs to $\mathscr{A}_p(\mathbb{R}^p, \mathbb{R})$. (b) Verify that if $F$ is a vector space over $\mathbb{R}$ and if $f : F \rightarrow \mathbb{R}^p$ is linear, then $g : F^p \rightarrow \mathbb{R}$ defined for $u = (u_1, \ldots, u_p) \in F^p$ by $g(u) = [f(u_1), \ldots, f(u_p)]$ is an element of $\mathscr{A}_p(F, \mathbb{R})$.
For all $e \in E^p$, we consider $\Omega_p(e) : E^p \rightarrow \mathbb{R}$ defined for all $u \in E^p$ by
$$\Omega_p(e)(u) = \operatorname{det}(\operatorname{Gram}(e, u)).$$
(a) Verify that the map $(x_1, \ldots, x_p) \mapsto [x_1, \ldots, x_p]$ belongs to $\mathscr{A}_p(\mathbb{R}^p, \mathbb{R})$.
(b) Verify that if $F$ is a vector space over $\mathbb{R}$ and if $f : F \rightarrow \mathbb{R}^p$ is linear, then $g : F^p \rightarrow \mathbb{R}$ defined for $u = (u_1, \ldots, u_p) \in F^p$ by $g(u) = [f(u_1), \ldots, f(u_p)]$ is an element of $\mathscr{A}_p(F, \mathbb{R})$.