We prove the global-in-time well-posedness of the one dimensional
Gross-Pitaevskii equation in the energy space, which is a complete metric space
equipped with a newly introduced metric and with the energy norm describing the
Hs regularities of the solutions. We establish a family of conserved
energies for the one dimensional Gross-Pitaevskii equation, such that the
energy norms of the solutions are conserved globally in time. This family of
energies is also conserved by the complex modified Korteweg-de Vries flow
