  We prove that the non-squeezing theorem of Gromov holds for
symplectomorphisms on an infinite-dimensional symplectic Hilbert space, under
the assumption that the image of the ball is convex. The proof is based on the
construction by duality methods of a symplectic capacity for bounded convex
neighbourhoods of the origin. We also discuss some examples of
symplectomorphisms on infinite-dimensional spaces exhibiting behaviours which
would be impossible in finite dimensions.
