  We develop a general theory of Hopf image of a Hopf algebra representation,
with the associated concept of inner faithful representation, modelled on the
notion of faithful representation of a discrete group. We study several
examples, including group algebras, enveloping algebras of Lie algebras,
pointed Hopf algebras, function algebras, twistings and cotwistings, and we
present a Tannaka duality formulation of the notion of Hopf image.
