Let $p \in \llbracket 1, d \rrbracket$. We denote by $\widetilde{\operatorname{Gr}}(p, E)$ the set of oriented subspaces of dimension $p$ of $E$. For all oriented vector subspace $(V, C)$ of dimension $p$ of $E$, $\Psi(V, C)$ denotes the unique element of $\mathcal{A}_p(E, \mathbb{R})$ such that for all $e \in C$ we have $\Omega_p(e) = \operatorname{vol}_p(e) \Psi(V, C)$. We equip $\mathcal{A}_p(E, \mathbb{R})$ with the inner product introduced in Part III. (a) Verify that $\Psi : (V, C) \mapsto \Psi(V, C)$ is an injection from $\widetilde{\operatorname{Gr}}(p, E)$ into the unit sphere of $\mathcal{A}_p(E, \mathbb{R})$. (b) Show that $\Psi(\widetilde{\operatorname{Gr}}(p, E))$ is a compact subset of $\mathcal{A}_p(E, \mathbb{R})$.
Let $p \in \llbracket 1, d \rrbracket$. We denote by $\widetilde{\operatorname{Gr}}(p, E)$ the set of oriented subspaces of dimension $p$ of $E$. For all oriented vector subspace $(V, C)$ of dimension $p$ of $E$, $\Psi(V, C)$ denotes the unique element of $\mathcal{A}_p(E, \mathbb{R})$ such that for all $e \in C$ we have $\Omega_p(e) = \operatorname{vol}_p(e) \Psi(V, C)$.
We equip $\mathcal{A}_p(E, \mathbb{R})$ with the inner product introduced in Part III.
(a) Verify that $\Psi : (V, C) \mapsto \Psi(V, C)$ is an injection from $\widetilde{\operatorname{Gr}}(p, E)$ into the unit sphere of $\mathcal{A}_p(E, \mathbb{R})$.
(b) Show that $\Psi(\widetilde{\operatorname{Gr}}(p, E))$ is a compact subset of $\mathcal{A}_p(E, \mathbb{R})$.