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.)
Some time ago I attended a seminar in functional analysis, in which the speaker talked about the problem of finding Lipschitz maps between quasi-Banach spaces, of which the prototypical example is the space $\ell^p$ when $0<p<1$. Those are metric vector spaces that are not locally convex.
The fact that there are seminars on those things means that there are researchers working on them.
EDIT (Re to your answer). Actually, there are some spaces arising in harmonic analysis that are not locally convex. I only know the weak $L^1$ spaces $L^{1, \infty}$ of the Marcinkiewicz interpolation theorem, but surely there are others. I know very little of those things.
EDIT 2. I found the seminar I was talking about. Here you have the abstract.