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)