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After getting myself confused about the distinction between the various notions of basis in infinite dimensions, I wrote up my attempt to disentangle myself at basis in functional analysis (also redirects from Hamel basis, topological basis, and Schauder basis. Hmm, now I think about it, maybe “topological basis” is too close to “basis of a topology”). I may still be confused about stuff, of course.
@Andrew and others: I note that C*-algebras are not included in the contents of Functional Analysis. Is there a reason for this? I would have thought that was a relevant entry, but classification of a topic under a particular heading is often a question of taste.
As far as I’m concerned there’s no reason for this. I think that it’s just historical. The $C^*$-algebra pages were around before the functional analysis ones, and it’s a bigger topic (in terms of weight of nlab-pages). I wrote the original contents and I’m much less aware of $C^*$-algebras than most functional analysis people are (probably because I’m not a true functional analyst).
Hmm, now I think about it, maybe “topological basis” is too close to “basis of a topology”
Indeed, topological basis already redirects to topological base.
Personally, I would like to have it redirect to your article; I never use ‘basis’ to mean a base. But it’s clear that other contributors to the nLab, particularly Urs, do use ‘basis’ to mean a base for a topology.
2 and 3 (Tim and Andrew): The entry on functional analysis mentions operator algebras which in turn contain $C^\star$-algebras. The top contents page of Mathematics in $n$lab has operator algebras and functional analysis as separate subjects and I think it is OK. While operator algebras have heavy analysis content much of the research is about structural porperties similar and partly motivated by theorems in areas like ring theory. Description of many math departements and of many multidisciplinary conferences have them separately while some have them together. I think it is practical to emphasis each separately in nlab while each should have mention of the other inside. I see no problem with that.
The entry functional analysis - contents does not mention operator algebras nor c-star algebras not links with non-comm geom. There is a category:functional analysis, but only one entry under it, and so on. I was just curious about whether C-star algebras were still considered as part of Funct. Anal. as in the good old days of Simmons book which I used for spectral theory and Banach Algebras as a student. More seriously, and Zoran I think you are much more expert on this than me, should there be some greater amount of structure used to collect up the Funct. Anal. stuff. (Also Stuff on shape in Funct. Anal. could be separated off… but that is another direction. What do you think?
Oh, I have not been aware of that contents page, just functional analysis, where operator algebras are mentioned. I will add operator algebras and spectral theory there. I agree that eventually the special topics like the algebraic theories in this setup could be separated off. At this moment, there is too little functional analysis in nlab anyway (big areas like e.g. “operator spaces” are missing) so some weird extras can feature for a bit, until a more systematic list is collected and realized in nlab. I think that counting operator algebras as part of functional analysis in this or that level of strictness depends on practioner.
Entries on non-commutative geometry will probably need organising, but that is a very diverse area ’out there’ beyond the Lab, so perhaps we should wait and see the growth. I am having discussions with people at Cardiff about exactly that area, and there would seem to be some interesting developments coming soon. I hope they come off!
I hope you explained them that having collectively improving but permanent online notes is useful aid to work, rather than predominantly an altruistic task. People are afraid that they will be loosing time working in nlab. While I had been hurted myself by not restraining from that work in crictial moments; in long run and distributing the “work” more wisely within contributor’s own time, and more functionally as part of one’s own endeavors, one saves time, we believe.
I do not get paid for doing research anyway so it makes no difference from that point of view, but you are right n-Lab activities do provide a very valuable aid to research… except I seem to try to do far too many things at once! :-(
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