  "Co-Frobenius" coalgebras were introduced as dualizations of Frobenius
algebras. Recently, it was shown in \cite{I} that they admit left-right
symmetric characterizations analogue to those of Frobenius algebras: a
coalgebra $C$ is co-Frobenius if and only if it is isomorphic to its rational
dual. We consider the more general quasi-co-Frobenius (QcF) coalgebras; in the
first main result we show that these also admit symmetric characterizations: a
coalgebra is QcF if it is weakly isomorphic to its (left, or equivalently
right) rational dual $Rat(C^*)$, in the sense that certain coproduct or product
powers of these objects are isomorphic. These show that QcF coalgebras can be
viewed as generalizations of both co-Frobenius coalgebras and Frobenius
algebras. Surprisingly, these turn out to have many applications to fundamental
results of Hopf algebras. The equivalent characterizations of Hopf algebras
with left (or right) nonzero integrals as left (or right) co-Frobenius, or QcF,
or semiperfect or with nonzero rational dual all follow immediately from these
results. Also, the celebrated uniqueness of integrals follows at the same time
as just another equivalent statement. Moreover, as a by-product of our methods,
we observe a short proof for the bijectivity of the antipode of a Hopf algebra
with nonzero integral. This gives a purely representation theoretic approach to
many of the basic fundamental results in the theory of Hopf algebras.
