  Using a "Hodge decomposition" of symplectic isotopies on a compact symplectic
manifold $(M,\omega)$, we construct a norm on the identity component in the
group of all symplectic diffeomorphisms of $(M,\omega)$ whose restriction to
the group $Ham(M,\omega)$ of hamiltonian diffeomorphisms is bounded from above
by the Hofer norm. Moreover, $Ham(M,\omega)$ is closed in $Symp(M,\omega)$
equipped with the topology induced by the extended norm. We give an application
to the $C^0$ symplectic topology. We also discuss extensions of Oh's spectral
distance.
