Definition talk:Banach Space

Two of my sources make the claim that the Banach space is "over the real or complex field" or "on the real or complex numbers".

Is this an unnecessary limitation of the definition, or is it a fact that the underlying set is necessarily one of those two sets? (Obviously it can't be the rationals, and I am also prepared to bet that Hamiltonians are out because of their non-commutativity (?) but beyond that I'm clueless.

So, can it indeed be directly inferred from the definition that the underlying set of every Banach space is indeed "isomorphic to" either $\R^n$ or $\C^n$ for some $n$? Or are there instances of complete normed vector spaces which are specifically not so isomorphic? --prime mover (talk) 15:16, 10 January 2023 (UTC)


 * The field needs to be complete for a Banach space. Of course, function spaces over $\R$ and $\C$ can be made into Banach spaces of infinite dimension.


 * I haven't heard of Banach spaces over other fields than $\R$ or $\C$. Maybe it is a niche and many books cut the corners. There seems to be "$p$-adic functional analysis" so I guess it is possible to do it with the $p$-adics as well. &mdash; Lord_Farin (talk) 15:23, 10 January 2023 (UTC)


 * Incidentally, it's worth checking on Definition:Banach Algebra which does specify $\R$ or $\C$, but from the same source (Borwein and Borowski) as the definition for Banach space without that limitation. --prime mover (talk) 15:36, 10 January 2023 (UTC)


 * I think the important thing here is that you have a normed field (or division ring, as is here, but I think this is a rarer formulation), which can be strictly larger than $\R$ or $\C$. For a silly example, consider the rational functions of indeterminates $\set {X_t : t \in \map {\mathcal P} \R}$ with coefficients in $\Q$ - I think this will be a field of cardinality $2^{2^{\aleph_0}}$, and I'm sure you can put some crazy norm on it. $\R^n$ and $\C^n$ for $n > 1$ aren't fields with the standard ring structures so wouldn't be admissible here. I will have a hunt through the literature. The thing is, the only fields that are realistically of any interest are $\R$, $\C$ and the $p$-adics (maybe one or two others I haven't thought about, but even vector spaces over the $p$-adics are a push), so there isn't much reason for authors to formulate anything in a more general context. Caliburn (talk) 16:00, 10 January 2023 (UTC)


 * No, good catch, you're right, $\R^n$ and $\C^n$ are not fields, but they don't need to be, $\R^n$ and $\C^n$ are the vector spaces whose underlying sets are $\R$ and $\C$ was what I think I meant. That's the point I was trying to make. --prime mover (talk) 16:04, 10 January 2023 (UTC)


 * The situation complicates in infinite dimensions. While vector spaces of equal dimension will be isomorphic as vector spaces, we also have the topology to think about. In infinite dimensions there are non-equivalent norms on the same vector space that give distinct topologies. (while in finite dimensions all norms are equivalent) Caliburn (talk) 17:04, 10 January 2023 (UTC)