  We consider a general class of infinite dimensional reversible differential
systems. Assuming a non resonance condition on the linear frequencies, we
construct for such systems almost invariant pseudo norms that are closed to
Sobolev-like norms. This allows us to prove that if the Sobolev norm of index
$s$ of the initial data $z_0$ is sufficiently small (of order $\epsilon$) then
the Sobolev norm of the solution is bounded by $2\epsilon$ during very long
time (of order $\epsilon^{-r}$ with $r$ arbitrary). It turns out that this
theorem applies to a large class of reversible semi linear PDEs including the
non linear Schr\"odinger equation on the d-dimensional torus. We also apply our
method to a system of coupled NLS equations which is reversible but not
Hamiltonian.
  We also notice that for the same class of reversible systems we can prove a
Birkhoff normal form theorem that in turn implies the same bounds on the
Sobolev norms. Nevertheless the technics that we use to prove the existence of
quasi invariant pseudo norms is much more simple and direct.
