The Bijectivity of the Antipode Revisited

We provide a very short approach to several fundamental results for Hopf algebras with nonzero integrals. Besides being short, our approach is the first to prove the bijectivity of the antipode without using the uniqueness of the integrals of Hopf algebras and to obtain the uniqueness of integrals as a corollary in a way similar to the classical theory of the Haar measure on compact groups.