  Salas and Garcia introduced the concept of an extended locally convex space
in [D. Salas and S. Tapia-Garcia. Extended seminorms and extended topological
vector spaces. Topology and its Applications, 2016] which extends the idea of
an extended normed space (introduced by Beer in G. Beer. Norms with infinite
values. Journal of Convex Analysis, 2015). This article gives an attractive
formulation of the finest locally convex topology of an extended locally convex
space and provides a systematic study of the resulting locally convex space. As
an application, we characterize the coincidence of the finest locally convex
topologies corresponding to the topologies of uniform and strong uniform
convergences on a bornology for the function space C(X).
