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I got the book “Counterexamples in Topological Vector Spaces” out of our library, and just the sheer number of them made me realise that my goal of getting the poset of properties to be a lattice would produce a horrendous diagram. So I’ve gone for a more modest aim, that of trying to convey a little more information than the original diagram.
Unfortunately, the nLab isn’t displaying the current diagram, though the original one displays just fine and on my own instiki installation then it also displays just fine so I’m not sure what’s going on there. Until I figure that out, you can see it here. The source code is in the nLab: second lctvs diagram dot source.
A little explanation of the design:
(It seems that Instiki’s SVG support has … temporarily … broken. I’ll email Jacques.)
Taken out some properties: I took out those that seemed “merely” topological in flavour: paracompactness, separable, normal. I’m pondering taking out completeness and sequential completeness as well.
But there are relationships between TVS-properties and purly topological properties, like “sequentially complete implies locally complete” etc. These are still interesting (but could be covered by another diagram of course).
Tried to classify the different properties. I picked three main categories: Size, Completeness, Duality. By “Size”, I mean “How close to a Banach space?”.
Does this correspond to the colors of the boxes? I have to admit that I have no clear understanding of what these categories mean and why the spaces get the colors they do, but that’s probably my fault. This is the level that I am stuck at:
Duality = definition uses the dual space?
Size = percentage of my conjectures that will turn out to be false if I think of $\mathbb{R}^n$ instead of the space at hand?
I got the book “Counterexamples in Topological Vector Spaces” out of our library, and just the sheer number of them made me realise that my goal of getting the poset of properties to be a lattice would produce a horrendous diagram.
It is a subject that is both vast and deep, despite the small and simple set of axioms it starts with.
Helmut Schäfer, when lecturing a TVS class, was asked to provide educational objectives for each lecture, like “after this lecture you should understand this and that and be able to do this and that”. He did that only once and said (translated by me): “The educational objectives of this lecture is that you understand in full clarity the material presented here. But it was developed over the course of several decades with the help of several of the most prominent mathematicians of their time, and experience shows that a full understanding when learning the material for the first time is a tall order”.
I agree that there are (significant) relationships between topological and TVS properties, but the diagram risks being very crowded and it seemed a reasonable place to draw the line. As an example, consider completeness. I’d be very interested to know if a result in FA actually uses honest completeness, as opposed to one of the more TVS notions of quasi-completeness or local completeness (the question is a little confused by the fact that for so many spaces, the notions coincide). Thus although completeness is useful, its main use is in establishing a weaker form of completeness.
Let me elaborate a little more on the classification of properties (you’re right, it is by colour):
Is that a bit clearer?
I like the Schaefer quote (and apologise for the lack of accents).
Is that a bit clearer?
Definitly, so “size” is something that I would like to call Banach-(co)detectable, or (co-) probable.
With regard to duality: Not sure about the “these properties are primarily used for making statements about the dual (or about mapping spaces)” - part. That seems fuzzy.
I like the Schaefer quote…
The part of the story that I did not like was that he did not provide educational objectives, but in hindsight it was a very interesting experience: For an expert it is hard to figure out what is difficult for a newbie and what isn’t and what should be emphazised and what should not.
Jacques figured out what was wrong with my diagrams - there was a linefeed that some part of Instiki (Jacques isn’t sure what yet) didn’t like. Removing that made the original diagram reappear and the new one appear. So it is now in the nLab in all its glory at the link in the first comment.
Back on track. I agree with all your remarks. If you can come up with a better way to phrase the “duality” part then please do! My reason for that grouping was a section in Schaefer’s book, just before the part on (semi-)reflexivity where he talks about various families of subsets in the dual. Off the top of my head, I think that they are equicontinuous, precompact, bounded, and weakly bounded. Barrelled means that all these are the same, infrabarrelled means that equicontinuous equals bounded, and so on. So they are all about certain families in the dual space.
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