Dual of an extended locally convex space

This paper aims to study the dual of an extended locally convex space. In particular, we study the weak and weak* topologies as well as the topology of uniform convergence on bounded subsets of an extended locally convex space. As an application to function spaces, we show that the weak topology for the space C(X) of all real-valued continuous functions on a metric space (X,d) endowed with the topology of strong uniform convergence on bornology coincides with its finest locally convex topology if and only if the bornology is of all finite subsets of X.