Let $(M,\omega)$ be a connected symplectic manifold. Use the convention that the Hamiltonian vector field $X_H\in\mathfrak X(M)$ of a function $H\in C^\infty(M)$ is defined by
is a well-defined injective map whose image is precisely the space of Hamiltonian vector fields on $M$. Moreover, the Poisson bracket descends to $C^\infty(M)/\mathbb R$, and for all $F,H\in C^\infty(M)$,