# Topological vector spaces vs. locally convex spaces

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.)

• Rudins book actually discusses examples. – Michael Greinecker Jun 9 '14 at 10:29

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.

• But working on such spaces doesn't automatically explains why they are indeed useful, applicable mathematical objects. I think I wasn't clear about what kind of example I was looking for: I'm interested in those kind of example that don't just arise out of a desire to generalize things, but that arise out of a need to solve some specific problem - like the usual $L^p$ spaces that can be viewed as arising in the context of solving PDEs. (Please see also the comment I wrote to the other answer). – user36772 Jun 9 '14 at 11:14
• Thanks for your additional explanations - could you please provide a different link ? the above one isn't working (I can't even connect to icmat.es) – user36772 Jun 9 '14 at 13:24
• Neither can I. Must be a temporary problem with their server. – Giuseppe Negro Jun 9 '14 at 13:25

The usual example of nonlocally convex topological vector space is $L^p$ with $0<p<1$. See Why do we consider Lebesgue spaces for $p$ greater than and equal to $1$ only?.

• But are these spaces also useful ? To me this seems to be rather the type of artifical example that proves "that it can be done", but which doesn't arise in a natural setting. (Or differently said: Asking when $L^p$ for $0<p<1$ would make sense seems to me to be a rather academic question). I would of course accept this example, if such a space would pop up in some other context so that there would be some motivation for studying this space. (Please see also the comment I wrote to the other answer). – user36772 Jun 9 '14 at 11:18
• Regardless of the potential usefulness of the $L^{0<p<1}$ (and other) spaces, the idea of topological vector space is natural and serves to generalize the basic properties of the Euclidean spaces to function spaces. The local convexity appears after further study. – Martín-Blas Pérez Pinilla Jun 10 '14 at 6:22