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Added complete topological vector space including various variants (quasi-complete, sequentially complete, and some others). Hopefully got all the redirects right!
I only have Schaefer’s book at home so couldn’t check “locally complete” - I know that Jarchow deals with this in his book. Kriegl and Michor naturally only consider it in the context of smootheology so I’m not sure what the “best” characterisation is. There’s also a notational conflict with “convenient” versus “locally complete”. As Greg Kuperberg pointed out, in some places “convenient” means “locally complete and bornological” whereas in others it means just “locally complete” (in the contexts where convenient is used the distinction is immaterial as the topology is not considered an integral part of the structure).
I added these whilst working on the expansion of the TVS relationships diagram. That brought up a question on terminology. In the diagram, we have entries “Banach space” and “Hilbert space” (and “normed space” and “inner product space”). These don’t quite work, though, as for a topological vector space the correct notion of a normed space should be normable space as the actual choice of norm is immaterial for the TVS properties. I’m wondering whether or not this is something to worry about. Here’s an example of where it may be an issue: a nuclear Banach space is automatically finite dimensional. That implies that its topology can be given by a Hilbert structure. However, the Hilbertian norm may not be the one that was first thought of. But that’s a subtlety that’s tricky to convey on a simple diagram. So I’d rather have “normable” than “normed”. Does anyone else have an opinion on this?
If “normable” is fine, then the important question is: what’s a better way of saying “Hilbertisable”, or “Banachable”? Length doesn’t matter here, as I’m putting the expanded names in tooltips and only using abbreviations in the diagram.
So I’d rather have “normable” than “normed”.
It would better fit to “metrisable”. And I would suggest B-space and H-space for spaces that admit a norm resp. a scalar product that induce a complete metric, but unfortunatly B-space is usually used as a synonym for Banach-space, so that won’t do.
…so couldn’t check “locally complete”…
Ugh, I’m confused. Köthe, in his first volume, defines an absolutly convex bounded set B in a lc-space to be a Banach-disk iff is a Banach space. Then he defines to be locally complete if every bounded set is contained in a Banach disk. Well, that seems to be just more complicated, but equivalent to, your definition, correct? Take a bounded set, take its absolute convex hull, take the closure, that’s still bounded. Is there a difference to impose that that is a Banach disk or that it is contained in a Banach disk?
Then Köthe states that locally complete is equivalent to sequentially complete, without proof. I’m not sure if it is as trivial as I think or if I’m missing something subtle.
Hmm, that seems to go against what Kriegl and Michor say. They say that “locally complete” is equivalent to Mackey-Complete, which is something about sequences that are fast-converging rather than just all sequences.
Hm, that rings a bell: Everything is taken from Köthe:
Definition: Ultrabornological means inductive limit of Banach spaces. (There seem to be other and definitly inequivalent definitions in the literature!!)
Definition: A sequence is locally convergent aka Mackey convergent if there is a bounded, closed, absolutly convex set K such that it converges in . It is fast convergent if there is a that is not only bounded and closed, but compact.
Theorem: Bornological is equivalent to: every absolutly convex set that absorbs all locally convergent sequences is a neighbourhood of zero.
Ultrabornological is equivalent to: every absolutly convex set that absorbs all fast convergent sequences is a neighbourhood of zero.
Ok, taking these definitions it really is trivial that sequentially complete is equivalent to locally complete, given that all Cauchy sequences are bounded.
Don’t
“2.2. Lemma. Mackey Completeness.”
and
“2.14. Theorem. Convenient vector spaces.”
in Michor/Krigl agree with Köthe?
I think that I agree with all the definitions. It’s the “locally complete implies sequentially complete” bit that I’m stuck on.
The problem with what you say in (5) is that a Cauchy sequence in is bounded in , but not necessarily Cauchy in any . For example, the obvious attempt would be to take to be the convex set generated by the sequence. But then almost certainly, the sequence lies on the unit sphere of the associated Banach space (obviously, this isn’t guaranteed, but I’m pretty sure one could concoct such an example).
Ah, interesting! (Right now I’m not as sure as I pretended, BTW, I would feel safer if I could make it more explicit).
Gee, I need to think this through…
I’ve uploaded the first version of the new diagram to lattice of TVS properties. It’s a long way from being a lattice, though.
Tschake. (I’m not well versed in Englisch’s onomatopoeia, this is the sound of me smacking my forehead).
So what I really thought was:
together imply sequentially complete. But there does not seem to be a way to get 2. without explicitly assuming it. (Köthe even states that characterizing the spaces with this property is an open problem).
The statement that I misunderstood really says “sequentially complete implies locally complete” instead of “both are equal”.
On a slightly different tack, I’m beginning to think that I need another diagram; this time of all the different types of subset of a LCTVS (or its dual). Things like barrel, bornivourous, totally bounded, weakly relatively compact, equicontinuous, and all that jazz.
bornivourous
? Does it eat bounded sets? (absorbs bounded sets?)
H-space for spaces that admit a … scalar product
Don’t use H-space either - this is a pointed topological space X with a homotopy unital map .
Does it eat bounded sets?
Yes, have a look at the definition of ultrabarreled on barreled topological vector space :-)
Added DF space together with a proof that DF+metrisable implies normable (please check!). Lots of links therein to be filled. For the time being, I sent all the different “types” of subset of a lctvs to subsets of lctvs since it might be a bit much to have a page on each. But I’m not convinced that this was right since if we do later want a page on, say, radial sets then we’d have to fix these links. The problem is that some of these names mean different things inside and outside functional analysis (or at least, the special case of being in FA adds enough to warrant separating the two). So perhaps the best is to make the wikilinks on the page of the kind “convex (functional analysis)” and then redirect them for the time being to “subsets of lctvs”.
Also added the definitions of “normed space” and “normable space” to Banach space (and added suitable redirects). For the moment, I don’t think that these deserve their own pages.
PS Had a bit of fun with the CSS at DF space as well.
The layout is very cool! The background color highlighting for proposition and proof is something the nLab should keep IMHO.
It’d look even better if Jacques allowed some CSS3 rules through the sanitiser.
Created subsets of lctvs, though perhaps “subsets of tvs” would be a better name. At the moment, I just made a list (and a huge number of redirects! I did it automatically from the list so there may be some that shouldn’t be there).
PS Had a bit of fun with the CSS at DF space as well.
Looks nice. Or at least it did until I added the sidebar now. ;-) :-/
Maybe I should remove it again, but also maybe have a look at it: is there a way to make the boxes automatically see the sidebar and not collide with it?
I added an explicit background colour to the sidebar (previously it was transparent) so that the colour doesn’t shine through. It’s not perfect, though, as we lose the outline of the definition box. Really, the box ought to see the sidebar. In fact, it does see the sidebar because the text is shoved to one side. However, the box itself doesn’t know about the sidebar. Strange.
Well, feel free to remove the sidebar again.
Are all your cool graphical tables linked prominently? In the sidebar I can only find one. Don’t you have a second one meanwhile?
The sidebar stays. I’d rather have the page functional (ha ha) than look nice. I’ve just looked up the rules on CSS and it is behaving how it is meant to and there are reasons for the behaviour. It doesn’t look bad, so I’ll leave it as it is until I learn enough CSS that I know how to improve it.
There are currently two graphical tables, but only one should be considered “stable” and that’s the one at TVS relationships (included in TVS). I’m working on another version of that which is meant to be a lattice. It’s on the lab (at least, the version as of yesterday is on the lab; I’ve been working on it a bit more today) but not linked anywhere yet as it isn’t finished yet. I’ve also proposed yet another diagram, of the different types of subset of a lctvs, but haven’t done anything about that as yet.
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