Barreled extended locally convex spaces and Uniform boundedness principle

For an extended locally convex space $(X,\tau )$, in [8], the authors studied the finest locally convex topology (flc topology) ${\tau }_F$ on $X$ coarser than $\tau$. One can often prove facts about $(X,\tau )$ by applying classical locally convex space theory on $(X,{\tau }_F)$. This paper employs the flc topology to analyze barreled extended locally convex spaces and establish the uniform boundedness principle in the extended setting. One of the key results of this paper is the relationship between the barreledness of an extended locally convex space $(X,\tau )$ and the barreledness of the associated finest locally convex space $(X,{\tau }_F)$. This is achieved by examining the lower semi-continuous seminorms on these spaces.