  Any finite-dimensional Hopf algebra has a left and a right integral.
Conversely, Larsen and Sweedler showed that, if a finite-dimensional algebra
with identity and a comultiplication with counit has a faithful left integral,
it has to be a Hopf algebra.
  In this paper, we generalize this result to possibly infinite-dimensional
algebras, with or without identity. We have to leave the setting of Hopf
algebras and work with multiplier Hopf algebras. Moreover, whereas in the
finite-dimensional case, there is a complete symmetry between the bialgebra and
its dual, this is no longer the case in infinite dimensions. Therefore we
consider a direct version (with integrals) and a dual version (with
cointegrals) of the Larson-Sweedler theorem.
  We also add some results about the antipode. Furthermore, in the process of
this paper, we obtain a new approach to multiplier Hopf algebras with
integrals.
