tangent and conormal bundles of a Lagrangian

Question

Suppose we have a Lagrangian submanifold $L$ of the symplectic manifold $T^*\mathbb{R}^{n}$ (endowed with symplectic form $\omega$), and a point $p\in L$. I know that there's a map $T_pL\rightarrow N^{*}_{p}L$, $X\mapsto\omega(X,\cdot)$. Why is this map bijective? In what way is it an isomorphism?

Answer 1

Let $X\in T_pL$ such that $\omega (X, Y)=0$ for all $Y \in N_pL$. Then $w(X, Y)$ for all $T_p\mathbb R^{2n}$ as $L$ is Lagrangian (thus $\omega(X, Y) =0$ for all $Y\in T_pL$) and $T_p\mathbb R^{2n} = T_pL \oplus N_pL$. As $\omega$ is nondegenerate, this implies $X=0$ and the map $T_pL \to N_p^*L$ is injective. Then it is bijective as they have the same dimension. This is an isomorphism of linear space, as the map is linear.