  Using the completed inductive, projective and injective tensor products of
Grothendieck for locally convex topological vector spaces, we develop a
systematic theory of locally convex Hopf algebras with an emphasis on
Pontryagin-type dualities. We describe how classical Hopf algebras, real and
complex Lie groups, as well as compact and discrete quantum groups, can all
give rise to natural examples of this theory in a variety of different ways. We
also show that the space of all continuous functions on a topological group $ G
$ whose topological structures are compactly generated has an $ \varepsilon
$-Hopf algebra structure, and we can recover $ G $ fully as a topological group
from this locally convex Hopf algebra. The latter is done via a generalization
of Gelfand duality, which is of its own interest. Certain projective and
inductive limits are also considered in this framework, and it is shown that
how this can lead to examples seemingly outside of the framework of locally
compact quantum groups in the sense of Kustermans-Vaes. As an illustration, we
propose a version of the infinite quantum permutation group $ S^{+}_{\infty} $,
the free orthogonal group $ O^{+}_{\infty} $, and the free unitary group $
U^{+}_{\infty} $ as certain strict inductive limits, all of which still retain
a nice duality. Combined with our duality theory, this may be seen as an
alternative tentative approach to the Kac program of developing a
Pontryagin-type duality to a wider class, while at the same time, we include
many more interesting examples of classical and quantum groups.
