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(First proper post in this forum, hope it's in roughly the right place; if not, passing mods are welcome to move it.)
Having foolhardily claimed elsewhere that I might have something to add to the nLab page on Banach spaces I have finally got round to getting my hands on a relevant textbook...
... but before starting, a question about terminology. As it stands, the category whose objects are Banach spaces and whose morphisms are the short linear maps between them is being denoted by Ban. Now in a couple of sources I used to skim over several years ago, this category was usually denoted by Ban_1, and isomorphism in this category was signified by attaching a subscript `1' to the usual isomorphism sign; the notation Ban was used instead for the category of Banach spaces and continuous linear maps between them. This convention is also the one used in the book which I want to use as a cited reference. (For a lot of functional analysts, "isomorphism of Banach spaces" habitually means "bicontinuous linear isomorphism of the underlying TVSes" and that habit seems to be ingrained - for me at least.)
So what are people's thoughts? I am more concerned with what will make the page most useful to people, than with debates about the drawbacks of long-established conventions.
(At least one person reading this ought to recognize "You don't wanna do that!" as a cry I wish I to avoid making or hearing too often...)
Hi Yemon,
First, welcome to the forum :)
Second, when it comes to notation, the nLab has been known to “play Bourbaki” and use the opportunity of these pages to clarify things or choose a particular notation.
If there are multiple choices of notation appearing in the literature, then it is a good habit to give a nod to those other notations. For example, a statement like, “What we refer to as $Ban$ here is denoted $Ban_1$ in…”
Last, but definitely not least, PLEASE do not let notation come in the way of adding content to the nLab. Choose any notation you feel most comfortable with and if there is a need to change it later, that is what the Lab Elves are for :)
I’d love to read anything you have to say and if someone decides later that a different notation would be better, I’m happy to help modify things later.
I don’t mind such an alteration, but I’m curious what others will say. Would $Ban_{\leq 1}$ get the intended idea across even more so?
One general comment on notation:
notation conventions on a wiki can’t work like in a book. Even if you tried to establish a convention, there is no guarantee (and in fact just a slim chance) that everyone who will come after you on this or on some other entry will stick to your convention or even be aware of it.
So I think the pragmatic way is this: in the controled context of one single entry, establish your notation conventions well-visibly somewhere, and alert the reader if possible that there might be other conventions. Then go ahead and use whichever convention you deem most appropriate.
If the notation applies to bunch of entries, one could also think of creating a separate page with just the notation conventions, so that one can briefly refer to that.
With my functional analytic hat on, I would be happy to see $Ban$ and $Ban_1$. I know that category theorists like to regard $Ban_1$ as “the one true category of Banach spaces”, but there’s enough around here that like a bit of functional analysis that both categories are likely to be used a fair bit in nLab entries that both will need a notation, so $Ban$ and $Ban_1$ seems a good way to refer to them both cleanly.
Other than that, I second Urs’ main point: be clear in the specific article and don’t worry too much about the rest of the lab.
It seems to me that even functional analysts who interpret ‘isomorphism of Banach spaces’ to mean a bicontinuous linear bijection still regard this as a technical term that doesn’t capture the most general notion of a way that two Banach spaces can be completely equivalent as Banach spaces. Restricting to the case of a single underlying vector space, I know that people will say that two different norms are ‘equivalent’, but sometimes they clarify with an adverb and only metrically equivalent norms are really the same. Generalising again, only a linear surjective isometry really shows that two Banach spaces are the same structure, right? Otherwise nobody would particularly care whether a map is an isometry.
So writing from the nPOV, I think that we can say, hey, for us ‘isomorphism’ isn’t just a term that you can assign any meaning you want to; it means a way to view two objects as the same. So a Banach space is not just a normaable TVS, and a true isomorphism of Banach spaces must be a linear surjective isometry. This only specifies the groupoid of Banach spaces, but the obvious category of Banach spaces is one whose core is this groupoid. At the same time, of course we explain how functional analysts use terminology as well as the other categories that people study from time to time. Incidentally, the category whose morphisms are continuous linear maps arises nicely as the full image of the faithful functor $Ban_1 \to TVS$, so it is a natural object of study from the $n$POV, not something that we should feel at all embarrassed to talk about.
As for notation, I don’t particularly mind ‘$Ban_1$’, although I wonder if there’s something other than just ‘$Ban$’ for its full image in $TVS$? Maybe ‘$Ban_b$’ for ‘bounded’? Is there anything in the literature?
Another thought: Maybe my understanding of functional analysis is all wrong and people really don’t much care whether a map is an isometry. Perhaps caring about this is really evil in some function-analytic sense, and accepting any bicontinuous linear bijection of Banach spaces (whether or not it’s an isometry) as a way in which they are the same structure is like accepting any equivalence of categories (whether or not it’s an isomorphism) as a way in which they are the same. Perhaps category theorists are forcing functional analysis into a mould where it doesn’t belong. This doesn’t seem likely to me, since I do see functional analysts talking about isometries, but then I also see category theorists talking about isomorphisms of categories. In the end, this is not a matter of mathematical fact but of practice and philosophy.
One more idea: Even accepting my argument about what the real groupoid $Ban_\sim$ of Banach spaces is, and seeing that this is the core of $Ban_1$ but not of $Ban_b$, still there is an important †-category whose underlying category is $Ban_b$. As the unitary morphisms in this $\dagger$-category form $Ban_\sim$, just as the isomorphisms in $Ban_1$ do, it is just as good an ’extra structure’ on top of $Ban_\sim$.
So the argument could be: If you really just want a category, then it must be $Ban_1$. But if you want a $\dagger$-category, then you want $Ban_b$. (I have just promoted the notation ‘$Ban_b$’ to mean the $\dagger$-category, rather than merely the category.)
Question: Can $Ban_1$ be recovered from $Ban_b$ (including its $\dagger$)? I don’t see how. Is there a concept that can include all of this structure at once?
Hi Toby,
Glad to have your input. I'll need to reflect on your posts before replying properly, but can I just ask what your dagger operation is supposed to be on $Ban_b$? If it is the usual "take the dual Banach space" functor then this is of course not involutive; there are Banach spaces which are not reflexive.
It seems to me that the category of Banach spaces and continuous linear maps would be better called the category of “Banachable topological vector spaces.” A Banach space is a TVS equipped with a complete norm generating the topology, so a morphism (and in particular an isomorphism) of Banach spaces should preserve the norm, whereas a continuous map is merely preserving the underlying topology.
Mike, you’re right. I tried to be honest about this in my diagram of LCTVS properties. It’s like the difference between a metric space and a metrisable one.
Andrew,
would it make sense to add a tad of a guiding comment to that (otherwise beautiful) entry? Maybe at least a sentence saying: “The following table shows … “.
I need to remind myself of the status of that diagram. The intention was that it be included in some surrounding page (probably LCTVS) once it was finished. I need to read back in the related nForum discussion to remind myself what was going on when I did the diagram - I remember I was having fun talking functional analysis with Tim van Beek, but it stalled, maybe the semester started or I went on holiday or something. Now that Yemon’s interested in expanding some of the entries, I’ll try to get back “into the groove”.
(Have a bit of an essay crisis at the moment - due in on Thursday, so I’ll be a bit more coherent after that.)
On the general point of having an explanation on the diagram, I’m against that because the source of that page can be automatically generated from another page using Graphviz, much like the knot pictures I did a week or so ago can be generated from their TikZ source. So if the source gets modified, at the moment it’s really easy to update the diagram. But if we add stuff to the actual diagram, it gets harder to keep things in step. But we can always include the diagram in a container page that just says “This diagram is of …”. Not sure if this paragraph is coherent …
Ah, okay, I didn’t rememeber that this was to be included elsewhere.
Speaking of Tim van Beek: too bad that he is no longer hanging around. It had been good to have him here.
Thanks for the feedback and kind remarks, everybody. Here are some haphazard thoughts of my own:
A part of me agrees with Mike that the category of "Banach spaces and continuous linear maps" should have been called the category of "completely normable TVSes and continuous linear maps". Perhaps I will try to use that terminology; but the fact remains that the earlier inaccurate terminology has arguably stuck.
(See also the marked paucity of textbooks which talk about "contra-homology", despite Peter Hilton's efforts. Or the fact we use "limit" rather than Freyd's preferred "root"...)
I am interested in Toby's idea that, in some inchoate sense, a lot of functional analytic treatment of/preoccupation with Banach spaces has really been about TVSes where the topology comes from some choice of Banach space structure, picked for convenience like a basis. Yet as he says, one does care when certain Banach spaces are "isometrically isomorphic", sometimes for technical reasons and sometimes for deeper ones (it seems). I'm as much in the dark as you guys, in some respects.
My own contention is that the two categories - which I am tempted to call Bang (short linear maps) and Bant (all continuous linear maps) - are both interesting for different reasons, and one wants to have both at hand rather than insist on one or other being "the" true category. (For a start, Bant has a closed structure, which seems to be absent from Bang - the set Bang(X,Y) is not a vector space, and trying to equip it with the structure of a Banach space just seems to lead back to Bant. The definition of spectrum of an operator seems to live in Bant (or Vect) not in Bang; as does the statement that finite-dimensional Banach spaces are nuclear. On the other hand, it is Bang which is small complete, small co-complete, supports decent comonads, and so on.)
Comment/question at projective Banach space.
Have left a meandering reply to David at projective Banach space and tried to start adding some of the missing detail. Thanks also to Urs for cleaning up the page and putting in all the links I’d rather lazily omitted.
@ Yemon:
The dagger structure must be the identity on objects, so what I wrote doesn’t actually work. I basically copied my comment from a previously understood (never written down by me, but previously thought through in my head) statement about $Hilb$, where it is accurate.
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