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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
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    $\begingroup$ I'm not an expert in the area, but from what I've seen in functional analysis courses, these properties are mainly technical ones : the closer you are to a finite dimensional normed vector space, the better-behaved you are, and so the more things we can prove about you. All of these are very basic and very important properties of f.d. normed vector spaces. $\endgroup$ – Maxime Ramzi Mar 21 '19 at 19:23
  • $\begingroup$ I suggest looking at spaces where they are false, to find out why they are helpful. Our natural concepts of vector spaces are finite dimensional, with the normal topology. All of these hold for them, so we are used to them. But consider the plane where neighborhoods of $0$ include the traditional neighborhoods, but with all points in the first quadrant removed. How would this tvs differ in behavior from a locally convex one. Look for theorems that use locally convex, and consider if the results still hold for this topology, or where they fail. Approach the others similarly. $\endgroup$ – Paul Sinclair Mar 22 '19 at 3:02
  • $\begingroup$ Not an answer but the point of abstraction is to generalize familiar theorems that hold in finite dimensions. If a proof uses the fact a finite-dimensional ball is convex then it most likely generalizes to locally convex spaces, etc. Mathematicians keep asking: What are the minimal assumptions we can make about a space that make a theorem true? $\endgroup$ – Chrystomath Mar 22 '19 at 5:20
  • $\begingroup$ I think the main idea is to find the most general settings where some theorem holds, for example there are barreled space where Banach Steinhaus theorem holds and so on. Nonetheless they are interesting spaces themselves: many properties hold easily because of linearity. Moreover some nice correspondances show up: barreled basis and pseudometrics are a good example. $\endgroup$ – Francesco Bilotta Apr 25 '19 at 14:43
  • $\begingroup$ @PaulSinclair “The plane where neighborhoods of $0$ include the traditional neighborhoods, but with all points in the first quadrant removed” is not a topological vector space (over $\Bbb R$), becasue both maps $\Bbb R\times \Bbb R^2\to\Bbb R^2$, $(\lambda, x)\mapsto \lambda x$ and $\Bbb R^2\times \Bbb R^2\to\Bbb R^2$, $(x, y)\mapsto x+y$ are discontinuous at zero. $\endgroup$ – Alex Ravsky Jun 17 '19 at 4:13
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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.

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