For $\ell \in \mathcal { L } ( E , \mathbb { R } )$, we denote by $\left. \ell \right| _ { F }$ the restriction of $\ell$ to $F$. The restriction map is $r _ { F } : \mathcal { L } ( E , \mathbb { R } ) \rightarrow \mathcal { L } ( F , \mathbb { R } )$, $\ell \mapsto \left. \ell \right| _ { F }$, and $d_{\omega} : E \rightarrow \mathcal{L}(E,\mathbb{R})$, $x \mapsto \omega(x,\cdot)$. The $\omega$-orthogonal is $F ^ { \omega } = \{ x \in E \mid \forall y \in F , \omega ( x , y ) = 0 \}$. Specify the kernel of $r _ { F } \circ d _ { \omega }$. Deduce that $\operatorname { dim } F ^ { \omega } = \operatorname { dim } E - \operatorname { dim } F$.
For $\ell \in \mathcal { L } ( E , \mathbb { R } )$, we denote by $\left. \ell \right| _ { F }$ the restriction of $\ell$ to $F$. The restriction map is $r _ { F } : \mathcal { L } ( E , \mathbb { R } ) \rightarrow \mathcal { L } ( F , \mathbb { R } )$, $\ell \mapsto \left. \ell \right| _ { F }$, and $d_{\omega} : E \rightarrow \mathcal{L}(E,\mathbb{R})$, $x \mapsto \omega(x,\cdot)$. The $\omega$-orthogonal is $F ^ { \omega } = \{ x \in E \mid \forall y \in F , \omega ( x , y ) = 0 \}$.
Specify the kernel of $r _ { F } \circ d _ { \omega }$. Deduce that $\operatorname { dim } F ^ { \omega } = \operatorname { dim } E - \operatorname { dim } F$.