Let $e = (e_1, \ldots, e_d)$ be an orthonormal basis of $E$, $p$ an integer such that $1 \leqslant p \leqslant d$ and $\mathcal{I}_p = \{\alpha = (i_1, \ldots, i_p) \in \mathbb{N}^p \mid 1 \leqslant i_1 < \cdots < i_p \leqslant d\}$. For all $\omega$ and $\omega^{\prime}$ elements of $\mathcal{A}_p(E, \mathbb{R})$: $$\langle\omega, \omega^{\prime}\rangle = \sum_{\alpha \in \mathcal{I}_p} \omega(e_{\alpha}) \omega^{\prime}(e_{\alpha})$$ We consider $u, v \in E^p$. Show that $$\Omega_p(u)(v) = \langle\Omega_p(u), \Omega_p(v)\rangle$$
Let $e = (e_1, \ldots, e_d)$ be an orthonormal basis of $E$, $p$ an integer such that $1 \leqslant p \leqslant d$ and $\mathcal{I}_p = \{\alpha = (i_1, \ldots, i_p) \in \mathbb{N}^p \mid 1 \leqslant i_1 < \cdots < i_p \leqslant d\}$. For all $\omega$ and $\omega^{\prime}$ elements of $\mathcal{A}_p(E, \mathbb{R})$:
$$\langle\omega, \omega^{\prime}\rangle = \sum_{\alpha \in \mathcal{I}_p} \omega(e_{\alpha}) \omega^{\prime}(e_{\alpha})$$
We consider $u, v \in E^p$. Show that
$$\Omega_p(u)(v) = \langle\Omega_p(u), \Omega_p(v)\rangle$$