I'm taking a course on locally convex spaces and our lecturer mentioned that these form the most general collection of spaces on which one can still prove interesting theorems (like Hahn-Banach - which fails for the more general topological vector spaces) and which still have applications (exactly because interesting theorems hold for locally convex spaces).
But it still seems to me, that a rather large number of books do treat topological vector spaces (like Rudins book on functional analysis), which seems to imply that there is a need for this more general setting - otherwise why bother discussing the most general setting if there isn't any use for that ?
Can one provide me please with some example outside the theory of locally convex spaces which justify the need for a theory of topological convex spaces in a plausible way ? (By "plausible" I mean examples that arise in some natural way - in contrast to artificial examples constructed in such a way that fall out of the locally-convex-spaces-theory.)
(Notice that I didn't mentioned if these spaces had to be hausdorff - I would warmly welcome it, if both cases - hausdorff and not hausdorff - would be covered by examples.)