We fix a symplectic form $\omega$ on $E$. Let $F$ be a vector subspace of $E$. (a) Show that, for every linear form $u : F \rightarrow \mathbb { R }$, there exists a linear form $\widetilde { u } : E \rightarrow \mathbb { R }$ whose restriction to $F$ coincides with $u$. We denote by $F ^ { \omega }$ the vector subspace of $E$ defined by $$F ^ { \omega } = \{ x \in E : \forall y \in F , \omega ( x , y ) = 0 \}$$ and $\psi _ { F }$ the linear map defined by $$\left\lvert \, \begin{aligned} \psi _ { F } : \quad E & \rightarrow F ^ { * } \\ x & \left. \mapsto \varphi _ { \omega } ( x ) \right| _ { F } \end{aligned} \right.$$ where $\left. \varphi _ { \omega } ( x ) \right| _ { F }$ is the restriction of $\varphi _ { \omega } ( x )$ to $F$. (b) Show that the restriction of $\omega$ to $F \times F$ is a symplectic form on $F$ if and only if $F \cap F ^ { \omega } = \{ 0 \}$. (c) What are the kernel and image of $\psi _ { F }$ ? (d) Show that $\operatorname { dim } ( F ) + \operatorname { dim } \left( F ^ { \omega } \right) = \operatorname { dim } ( E )$. (e) Show that, if the restriction of $\omega$ to $F \times F$ is a symplectic form on $F$, then $E = F \oplus F ^ { \omega }$ and the restriction of $\omega$ to $F ^ { \omega } \times F ^ { \omega }$ is a symplectic form on $F ^ { \omega }$.
We fix a symplectic form $\omega$ on $E$. Let $F$ be a vector subspace of $E$.
(a) Show that, for every linear form $u : F \rightarrow \mathbb { R }$, there exists a linear form $\widetilde { u } : E \rightarrow \mathbb { R }$ whose restriction to $F$ coincides with $u$.
We denote by $F ^ { \omega }$ the vector subspace of $E$ defined by
$$F ^ { \omega } = \{ x \in E : \forall y \in F , \omega ( x , y ) = 0 \}$$
and $\psi _ { F }$ the linear map defined by
$$\left\lvert \, \begin{aligned} \psi _ { F } : \quad E & \rightarrow F ^ { * } \\ x & \left. \mapsto \varphi _ { \omega } ( x ) \right| _ { F } \end{aligned} \right.$$
where $\left. \varphi _ { \omega } ( x ) \right| _ { F }$ is the restriction of $\varphi _ { \omega } ( x )$ to $F$.
(b) Show that the restriction of $\omega$ to $F \times F$ is a symplectic form on $F$ if and only if $F \cap F ^ { \omega } = \{ 0 \}$.
(c) What are the kernel and image of $\psi _ { F }$ ?
(d) Show that $\operatorname { dim } ( F ) + \operatorname { dim } \left( F ^ { \omega } \right) = \operatorname { dim } ( E )$.
(e) Show that, if the restriction of $\omega$ to $F \times F$ is a symplectic form on $F$, then $E = F \oplus F ^ { \omega }$ and the restriction of $\omega$ to $F ^ { \omega } \times F ^ { \omega }$ is a symplectic form on $F ^ { \omega }$.