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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. $\endgroup$ – Max Mar 21 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 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 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 at 14:43

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