  We prove an analogue of Alexander's Theorem for holomorphic mappings of the
unit ball in a complex Hilbert space: Every holomorphic mapping which takes a
piece of the boundary of the unit ball into the boundary of the unit ball and
whose differential at some point of this boundary is onto is the restriction of
an automorphism of the ball. We also show that it is enough to assume that the
mapping is only Gateaux-holomorphic.
