Let $(M,\omega)$ be a symplectic manifold.
We say that a submanifold $N$ of $M$ is coisotropic if $(T_pN)^{\omega}\subseteq T_pN$, for any $p\in N$, where $(T_pN)^{\omega}$ is the orthogonal of $T_pN$ with respect to $\omega$.
What I am not yet able to prove is that $N$ is coisotropic if and only if for all $f,g\in C^{\infty}(M)$ such that $f$ and $g$ vanish on $N$, $\{f,g\}|_N=0$, where $\{\cdot,\cdot\}$ is the Poisson bracket.
I denote by $I_N$ the ideal of functions vanishing along $N$.
The symplectic form $\omega$ gives a bundle isomorphism $$\omega^{\flat}:TM\rightarrow T^{*}M:v\mapsto\iota_{v}\omega.$$ In terms of this isomorphism, we have $$ TN^{\omega}=(\omega^{\flat})^{-1}(TN^{0}), $$ where $TN^{0}$ is the subbundle consisting of all covectors annihilating $TN$. Note that $TN^{0}$ is locally generated by elements of the form $df|_{N}$, where $f\in I_N$. Hence, $TN^{\omega}$ is locally generated by vector fields of the form $$ (\omega^{\flat})^{-1}(df)|_{N}=X_{f}|_{N} $$ where $f\in I_N$. So we obtain \begin{aligned} TN^{\omega}\subset TN &\Leftrightarrow X_{f}\ \text{is tangent to}\ N\ \ \ \ \ \ \forall\ f\in I_N\\ &\Leftrightarrow dg(X_f)\in I_N\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \forall\ f,g\in I_N\\ &\Leftrightarrow \omega(X_f,X_g)\in I_N\ \ \ \ \ \ \ \ \ \ \ \forall\ f,g\in I_N\\ &\Leftrightarrow \{f,g\}\in I_N\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \forall\ f,g\in I_N. \end{aligned}