Properties of topological vector spaces

Question

I'd like to understand better the significance of certain properties of topological vector spaces. It would be great if someone could give me an intuitive picture for what makes them "special", and/or illustrative examples of their nature, and/or some idea of what else we can conclude about spaces with such properties, etc.

The properties in particular that I am interested in:

1. locally convex tvs, i.e. one in which every point has a local base whose members are all convex
2. locally compact tvs, i.e. one in which every point has a compact neighborhood
3. locally bounded tvs, i.e. one in which every point has a bounded neighborhood
4. tvs with Heine Borel, i.e. one in which every closed and bounded set is compact

Answer 1

I recall the relations between the classes of topological vector spaces in which your are interested in.

It is well-known that each every locally compact Hausdorff topological vector space $X$ is finite-dimensional (see, for instance, [T, Theorem 1], [?1, Theorem 3.2.1], [?2, Proposition 3.19], [H, Theorem 6] or this MSE thread). Since every finite-dimensional Hausdorff topological vector space has the usual topology ([T, Theorem 2], or [?1, Theorem 3.1.1 and Corollary 3.1.4]) $X$ is isomorphic to $\Bbb R^n$ with the usual topology for some $n$.

Every locally bounded topological vector space that has the Heine-Borel property has finite dimension [?2, Corollary 3.20].

A topological vector space is normable iff it is both locally convex and locally bounded, see [?2, Theorem 3.40?] iff it contains an open set which is both convex and bounded (see [K] or [H,Theorem 1]). On the other hand, the space $\ell^p$ with $0<p<1$ is locally bounded, but not locally convex. According to [H],

The property of local convexity plays an important role in many analytical theories, especially in the theories of linear functionals, integrals, and functional equations. Fortunately, most of the more important instances of l.t.s. do have the property of local convexity. It is easily seen that the linear topological product of an arbitrary family of locally convex spaces is itself locally convex.

The property of local boundedness [defined by Hyers] puts a rather severe restriction on the l.t.s., as the following theorems show. It was mentioned in connection with Theorem 1 that the property of local boundedness implies that there exists a countable complete neighborhood system of the origin. Thus from Theorem 2 we have

Theorem 3. Every locally bounded l.t.s. is metrizable.

see also [H, Theorem 4] on a characterization of locally bounded l.t.s.

See [D] for more about locally convex vector spaces.

References

[D] J. A. Dieudonné, Recent developments in the theory of locally convex vector spaces.

[K] A. Kolmogoroff, Zur Normierbarkeit eines topologischen linear en Raumes, Studia Mathematic 5 (1934), 29--33.

[KF] A.N. Kolmogorov, S.V. Fomin Elements of theory of functions and functional analysis, 4th edn., M.: Nauka, 1976 (in Russian).

[H] D. H. Hyers, Linear Topological Spaces.

[T] Terry Tao, Locally compact topological vector spaces.

[?1] Finite dimensional topological vector spaces.

[?2] Topological vector spaces.