Prompted by Andrew's question, I compiled some data on relationships between types of locally convex topological spaces. In particular, I made a Hasse diagram of property that began as a simplification of the one in EDM, and then added more conditions. The rule for the Hasse diagram is that I only allow "single-name" properties, not things like Frechet and nuclear, although adverbs are allowed. The graph makes the topic of locally convex spaces look laughably complicated, but that's a little unfair. You could argue that there are more than enough defined properties in the field, but of course mathematicians are always entitled to make new questions. Moreover, if you look carefully, relatively few properties toward the top of the diagram imply most of the others, which is part of the point of my answer. If a topological vector space is Banach or even Frechet, then it automatically has a dozen other listed properties.
I have the feeling that the Hasse diagram has missing edges even for the nodes listed. If someone wants to add comments about that, that would be great. (Or it could make a future MO question.) The harder question is combinations of properties. It might make sense to have a computer-assisted survey to compare all possible combinations of important properties, together with counterexamples and open status. I started a similar computer-assisted survey of complexity classes a few years ago.
