Wikipedia:Reference desk/Archives/Mathematics/2023 June 6

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June 6

Bloch or Banach space?

Is the symbol ${\mathcal {B}}$ or ℬ for Bloch space or Banach space? We say several places it's Bloch, but the refs I've seen say Banach. I have no idea what either one is, so thought I'd check before imposing any more of my OR. — kwami (talk) 07:54, 6 June 2023 (UTC)

Bloch. And they are a type of Banach space - which is something important you'll probably come across if you do a course in maths. NadVolum (talk) 10:44, 6 June 2023 (UTC)

Thanks. I was misreading the definition after the symbol as defining the symbol rather than what it stood for. — kwami (talk) 10:53, 6 June 2023 (UTC)

There are also uses of calligraphic ${\mathcal {B}}$ or script capital ℬ for a Banach space.^[1][2][3][4] While some uses of specific font styles are more conventional than others, there is no norm. Plain italic $B$ is fairly common,^[5][6][7] and you may also find other letters used ("let ${\mathcal {A}}$ and ${\mathcal {B}}$ be associated Bloch spaces"^[8]).  --Lambiam 12:20, 6 June 2023 (UTC)

Thanks. I'll update Wiktionary. A lot of these symbols have no definition in their articles and I'm nominating them for deletion, but this is one I thought I could save. — kwami (talk) 12:29, 6 June 2023 (UTC)

Books sometimes have an index of the various symbols they use with their long names or meaning, but there isn't really all that much standardization, if there was there would have to be a lot more symbols! Does Wiktionary show symbols? Could you give an example of what you mean thanks. NadVolum (talk) 12:36, 6 June 2023 (UTC)

My gosh! it has a lot of the utf-8 symbols as entries. And it has Banach space! I don't know what to think, I thought Wikipedia covered all that. NadVolum (talk) 12:43, 6 June 2023 (UTC)

There's some overlap. People sometimes add lists of basic vocab to language articles here, and then someone will move them over to Wikt. For someone looking up Banach space, the WP entry might be too much detail, or for someone who just needs to refresh their memory, so there should be a simple definition at Wikt. — kwami (talk) 21:54, 6 June 2023 (UTC)

Re. the exapmels you requested, what I was working on yesterday were the entries in the list at wikt:Appendix:Unicode/Letterlike_Symbols. A lot of them were simple descriptions. For example, wikt:ℬ was simply "The upper-case letter B in cursive script", with the comment "(chiefly mathematics)". I've tagged most of those for deletion, so that people who might expand the article will see a red link and recognize that it still needs to be done. But for ℬ I added a short description. Even if the list of uses isn't complete, it could help someone puzzled by the symbols; people can always add more. — kwami (talk) 22:00, 6 June 2023 (UTC)

In this sort of context, this ${\mathcal {B}}$ is just a bound variable, and doesn't have to be any particular sort of object. You could in principle say "let n be a Banach space" or "let ${\mathcal {B}}$ be a natural number", and while it might subvert readers' expectations, it wouldn't actually be wrong. That said, it is indeed useful to know (and adhere to) common usages about what variables to use for what sort of object. --Trovatore (talk) 23:29, 8 June 2023 (UTC)

Oh, I should have looked up Bloch space before I typed the above. Apparently Bloch space is one particular object, so it seems that in this context ${\mathcal {B}}$ is a constant symbol rather than a bound variable. As far as I'm aware there is no particular object called "Banach space". --Trovatore (talk) 23:44, 8 June 2023 (UTC)

For any natural $n,$ the Euclidean space $\mathbb {R} ^{n}$ is a Banach space.  --Lambiam 00:16, 9 June 2023 (UTC)

How is that responsive to anything I said? --Trovatore (talk) 00:29, 9 June 2023 (UTC)

$\mathbb {R} ^{2}$ and $\mathbb {R} ^{3}$ are different spaces. If "Banach space" referred to a single object, they would be the same. You may know what a Banach space is, but some potential readers of your comment may be less familiar with the concept. See also this edit.  --Lambiam 07:37, 9 June 2023 (UTC)

My point is that "Bloch space" and "Banach space" are not parallel locutions, because Bloch space is a unique object whereas there are distinct Banach spaces. So if you use one of these symbols, say ${\mathcal {B}}$, to denote Bloch space, then you're using it as a name for that particular thing, whereas if you use ${\mathcal {B}}$ as a variable to indicate a Banach space, then that's just a dummy variable which could be replaced by any other symbol, though not necessarily without cost in readability. The rough convention in choosing particular dummy variables for certain types of object should not be conflated with the more rigid conventions regarding constant symbols that name particular objects. --Trovatore (talk) 15:44, 9 June 2023 (UTC)