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At Urs’ urging, I have created functional analysis - contents. It needs considerable extending; and I’ve yet to include it anywhere.
As hinted by the contents, I plan to move the diagram from TVS to its own page (but still include it on TVS).
Here is a really dumb question for the natural speakers: Is it “barrelled” with 2 ll as in Greg’s original chart or is it “barreled” with one l as I wrote in barreled space?
(That difference is the reason why the link in the content tabel does not work).
Schafer also has ’barreled’, but I never liked that and always felt queasy when writing it. My spellchecker here doesn’t like ’barreled’ either (but then it doesn’t like ’spellchecker’ so I’m not sure I trust it completely) and does like ’barrelled’. Greg’s based in the US so I discount his spelling. My copy of Chambers has ’barrel … -ll- to put in barrels’ which I read as meaning double-l. Finally, the Official Scrabble Words lists ’barrelled’ as a valid Scrabble word but does not list ’barreled’.
(Hmm, the above comes out a lot more “I’m British and I’m right” than I mean it to. It’s just that this precise question is something that I’ve often wondered about but never to the point of actually looking it up. Now that someone has asked about it, that’s given me the motivation to find out and so I’m recording the results of my research!)
Actually, you already had put in a redirect for “barrelled”. It was the word “topological” that was missing (which I’ve now added).
Greg’s based in the US so I discount his spelling.
Yes, the Americans and English have much in common, except the language of course.
Actually, you already had put in a redirect for “barrelled”.
That’s an error, I started with barrelled because that’s on the chart, browsed the book by Francois Treves while writing, noticed that he has barreled and switched to barreled. I’m not advocating the concurrent use of both spellings :-)
Here is another question: One topic that Schäfer mentions for every type of TVS are the permanence properties, which means which constructs inherit the property. Example: bornologic and barreled both “survive” inductive limits, that is the inductive limits of bornological and of barreled spaces are again bornological resp. barreled. Is there a better word or concept for this, that I could use as a subheader under “properties”?
Given that we’re (pretending to be) category theorists, I suppose something along the lines of “Colimit Properties”, meaning properties that are inherited by colimits. There’s probably a fancy name for a subcategory of a complete/cocomplete category that is complete/cocomplete with the same colimits. It’s a bit like a closed subspace of a complete metric space being again complete. So maybe “Coclosed Properties”!
But more seriously, I guess that the inheritances are going to vary by type quite a bit, so maybe to keep it uniform across the pages we should have a subheading “Inheritance Properties” and then say “X is inherited by constructions Y,Z”. That would make it easiest to look up different properties.
One thing I always get tripped up by is the difference between inductive limits and those tricksy inductive sequences.
Another example: Treves defines LF-spaces to be a countable strict inductive limit of Fréchet spaces, and remarks that for a closed liner subspace of a LF-space that is the inductive limit of spaces , it is not true, that the topology on induced by is equal to the topology of the limit .
(And he writes that he himself has made this mistake a couple of times :-)
Now we have a non-commutating diagram, what would be a good way to explain this situation?
Yes, that’s something that’s very subtle! It’s the lynchpin of my argument in my paper on piecewise-smooth loops (well, the part about piecewise-smooth-with-unbounded-endpoints).
I don’t follow what you mean by a “non-commutating diagram”. Which diagram are you referring to?
(Incidentally, I should also record that having now checked in Schafer’s book, the change in Greg’s diagram that I made of ’DF-space’ to ’Dual-Frechet space’ was wrong. I’ll correct it next time I update the diagram if no-one does it before.)
“Inheritence Properties” seems fine to me.
I don’t follow what you mean by a “non-commutating diagram”. Which diagram are you referring to?
Sorry, I was thinking about the arrows “taking the inductive limit” and “restrict to subspace” and as objects .
Ah, I see.
Being good category theorists, we could say that this is due to the fact that these arrows are taking place in different categories. The “taking the inductive limit” is going on in LCTVS, whilst “restrict to subspace” is in Top.
There’s a possible project for us: translate all these functional analysis statements into their correct categorical ones. I suspect that this might be useful in both directions: for FA people trying to understand categorical constructions and for non-FA people trying to figure out which way the various limits and things go!
Now, the subcategory of bornological LCTVS has a reflector (or coreflector, not sure of direction), the “bornologicification” functor. I wonder if barrel(l?)ed LCTVS has something similar. Is there a “barrellification functor”?
Yes, I think so, a lc space with topology T is barreled iff any topology that has local bases consisting of T-closed sets is weaker than T. So you get a barreled space from a lc space by equipping it with the stronges lc topology aka the Mackey topology.
Andrew #3: which Scrabble dictionary are you using? My OSPD has both the British and American spellings. (Hey, just be glad it’s in there at all!)
For what it’s worth: the rule of thumb for the American spelling is: if the verb ends in -vc but the accent is not on the final syllable, then just add -ed or -ing, but if it is, then double the consonant. (I used to write “modelled” and “focussed” but I’ve long since changed to American, just because my life seems easier that way. (-: )
I just like being contrary!
It is, I confess, the UK official scrabble word list.
I must admit, I don’t know many verbs ending in ’-vc’. Can you give me an example?
At Urs’ urging, I have created functional analysis - contents.
Cool. Now you only need to remember to also include it! ;.)
But I have done it now.
Hm, well, unless of course I am misunderstanding your intentions. I think it should be the way it is now for instance at functional analysis.
Looks good!
The whole TOC is about (topological) vector spaces. My background to functional analysis coming from a school of Gelfand and Kirillov says that the functional analysis studies compatible topological and algebraic structures; especially those related to the function spaces of various kind. So topological rings and the study of say Pontrjagin duality for locally compact abelian groups belong to the subject, not only TVS, and they take first few chapters of textbooks like Kirillov. Do others disagree with that point of view ?
Andrew #15: -vc means ending in vowel-consonant. So, “model” and the like. Sorry for the confusion!
@Zoran: I certainly do not disagree! My knowledge of functional analysis is all on the TVS side (indeed, LCTVS), so that’s what I put in the TOC. But others should, of course, feel free to add other topics and pages that they think should be covered.
@Todd: I guessed that there was some underlying logic like that! But the idea of a word ending ’-vc’ and there being some special rule for it amused me.
@Zoran: I agree, functional analysis is about combining algebraic and topological structures, not only vector spaces. Therefore operator algebras as well as topological groups should eventually be added. Well, that’s the central European school that I know :-)
The math departments that I know offer “linear operators on Hilbert spaces”, “banach algebras”, “topological vector spaces” and “topological groups” as classes and file it all under “functional analysis”.
OK, I was already afraid I belong to some obsolete educational class of mathematical readers…we-ll update that in a while. It was a hard week for me and my strengths are about to vanish till I recharge well (not only my own but also doctor’s opinion…).
I’ve put the diagram on its own page and included it in TVS via the include directive. I also fixed the DF bit. I tried a couple of ways to make the links proper wikilinks but the system didn’t like my attempts so I left it as it is. I’m not sure what the best way is to make the titles obviously links. There’s probably a simple CSS selector which we don’t currently have set. What is more complicated is to make the distinction between wikilinks and arbitrary links.
very nice! My candidate for the next nLab digest.
next ? there was some already ?
well, there was 3000 things to think about. The “next” (or first, yeah) nLab digest i am thinking of as continuing this kind of post.
The word digest may not be the best in internet setup. The digest/s of newsgroups were always having all the texts in a big file form, rather than the highlights/choice of the main streams of work&topics. The digests are also more of a report, and we should here look in future and say what some new threads in fact want to become and lure people into thgem as programs. Thus nlab highlights is kind of better catch name in my view; which may not be shared by others of course.
(Zoran’s got a good point about the word “digest”, but perhaps it should be made over in the relevant discussion.)
Back to the diagram: I’ve added the colour “by hand”. It seems that SVG “draws” the text rather than writing it so it isn’t recognised as text. So I’d need to hack the CSS to make it the right colour, and it’s quicker to add it in by a shell script so that’s what I’ve done for now.
Of course, if this kind of diagram is seen as a Good Thing then it’ll be worth figuring out and adding the CSS. Note, however, that because of how SVG treats the text, fancy stuff like underlining won’t be easy. Colour and font is probably all that’s easily customisable.
next ? there was some already ?
The first “digest” is of course the next one to the empty set. (empty -> first digest -> second digest…)
Note, however, that because of how SVG treats the text, fancy stuff like underlining won’t be easy. Colour and font is probably all that’s easily customisable.
Ah, the .dot source is a Graphviz specific data format? Because in SVG itself it should be no problem to add all sorts of fancy layout to text content…
What I meant was that the text in the SVG is rendered as a <text> element and the contents of that seems to be treated as a picture of text rather than the text itself. Thus it doesn’t seem to inherit CSS rules that ought to apply to text. In particular, using Firebug I could see that the <text> element was inheriting the CSS rule that coloured text a certain colour but the contents of the element were not getting coloured, ergo it is not text. That’s why I think that underlining is going to be tricky. Underlining text is easy, but underling a picture of text means saying “draw a line that is as long as the picture and is a little bit below”. I’m not saying it can’t be done, just that it may not be as easy as changing the colour and font.
In particular, using Firebug I could see that the -text-element was inheriting the CSS rule that coloured text a certain colour but the contents of the element were not getting coloured, ergo it is not text.
That’s tricky. I thought that the embedded SVG would not even inherit the CSS rule, but that the whole SVG subtree would be processed by a SVG renderer directly. The question I asked myself was if the easiest way could be to add e.g. style=”text-decoration:underline;” to the text-Element in the SVG manually, but that would mean of course that one could not generate the SVG from the .dot source anymore (without editing it manually afterwards everytime).
It seems that the SVG-Renderer of Firefox does not render the style=”text-decoration:underline;” I added it manually and it is displayed in IE 8, but not in Firefox 3.6.3.
Given that I send the generated svg through a perl script afterwards anyway (to strip out comments), I see no problem in adding extra stuff so long as it is done generically. What one would want, of course, is to do different styles on text-in-links to ordinary text. This is where the “cascading” bit of CSS shows its strength. For this particular diagram, it doesn’t matter as everything is a link.
Good point. Do the comments cause trouble of some sorts? And do you see an underline in firefox when you add the tyle=”text-decoration:underline;” to one of the text-Elements?
Instiki doesn’t allow comments when XHTML is written directly into the entries, so the comments get “escaped” and thus display. I’ll need to experiment a little to learn all the intricacies of SVGs to see what is and isn’t possible with styling - I admit I didn’t try too hard once the obvious thing didn’t work. What one would really like is to be able to put actual wikilink syntax into the SVG and have the Markdown parse it, but that needs an expansion of what’s allowed on instiki in the foreignObject tags in SVG.
Back to the contents of the diagram, I guess what I would like is for it to be a lattice. So the fact that Frechet means “complete, metrisable lctvs” should be visible from the fact that the node for Frechet is the sup of those for complete, metrisable, and lctvs. But where the supremum doesn’t have a name, then we have an extra node. Ideally, this node would link to an example showing that the next best named thing isn’t the supremum.
For example, looking at the diagram then the supremum of “ultrabornological” and “Frechet” is currently “Banach”. I doubt that that is true, so we’d have another node above “ultrabornological” and “Frechet” which linked to an example of a space that was an ultrabornological Frechet space that wasn’t a Banach space.
Of course, that would entail a lot more nodes so then I think we’d have to abbreviate the names. But we can use visual information to our advantage here: boxed nodes are named concepts, round nodes are the extras.
This could go horribly wrong, so I propose forking the current diagram and working on this new version separately. Also, it might be too big to include on the TVS page so having two versions may be no bad thing.
I think that this could be a Really Useful Engine, I mean, Diagram (sorry, been reading too much Thomas Toget to my kids).
…we’d have another node above “ultrabornological” and “Frechet” which linked to an example of a space that was an ultrabornological Frechet space that wasn’t a Banach space.
plus necessary assumptions for the conclusion to be valid :-) Sounds very good, and sounds like a lifetime project, at least for someone like me :-)
This could go horribly wrong…
Oh dear! What could happen in the worst case scenario?
…so I propose forking the current diagram and working on this new version separately.
That’s probably a good idea anyway.
…it might be too big to include on the TVS page…
Isn’t one of the key ideas of SVG (besides being human readable and editable XML) that it is scalable and you can zoom in where you like?
This could go horribly wrong…
Meaning that after trying it, we may decide that it’s really yukky and not at all useful, in which case we’ll want to have the old diagram to return to. Also, while the new one is being designed, it’ll be mathematically wrong and so shouldn’t be so prominent.
Isn’t one of the key ideas of SVG (besides being human readable and editable XML) that it is scalable and you can zoom in where you like?
Yes, but I don’t know enough about how this all works to know if this is a good idea within a larger page, or is best on a standalone page. If you look at the diagrams in the random category on my website, you’ll see a few experiments with different display programs.
(One could argue that all of my website is pretty random, but I mean this bit.)
Moving the chart to a half public playground is definitly a good idea. Zooming in and zooming out will become easiear/a feature of more viewers in the future, I’m sure. (I still think that SVG will succeed as a graphics standard for the Internet). If we keep the simple chart around and point to the bigger one - with the warning that one needs a capable viewer for this - just as you suggested, people should be fine.
A part of this project would be a transcription of counterexamples from Khaleelulla: “Counterexamples in Topological Vector Spaces”. That’s where I looked for “ultrabornological Frechet that is not Banach”: That does not seem to be there (maybe it’s too trivial).
BTW, a short but good explanation of SVG features is this: J. David Eisenberg: “SVG essentials” (O’Reilly)
Thomas Toget
:) I’m just imagining Ringo narrating Thomas in Norwegian….
I have slightly expanded and slightly re-organized functional analysis - contents. Please have a look.
Is capitalization intentional ?
Is capitalization intentional ?
That originates with Andrew, I think.
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