10

If it must be, why is this so? In the maths courses I have taken normed vector spaces always have been over $\mathbb{R}$ or $\mathbb{C}$, but I don't see that this has to be so. I am asking because I was proving that if a normed vector space X has a Schauder basis then it is separable, and I had a proof along the lines of this question How to prove that if a normed space has Schauder basis, then it is separable? What about the converse? The problem I had is this proof requires that we have a sequence of rationals converging to the scalars of the vector space. But if the vector space is not $\mathbb{R}$ or $\mathbb{C}$ then why should this work?

Someone has asked this in a comment to the second answer of the question above, but I do not quite understand the response given: "The fiel has to be restricted, notably because of the definition of a normed space: an absolute value is needed" I can see that we do use absolute values on the scalars of the vector space but I'm not then sure why this restricts them to being in $\mathbb{R}$ or $\mathbb{C}$, other than I don't know how to define an absolute value on something other than a real or complex number (but wikipedia seems to suggest it can be done).

Community
  • 1
stargazeypie
  • 103

1 Answers1

11

There is a theorem that if $k$ is a normed field, one of the following two things holds:

1) $k$ is a subfield of $\mathbb{C}$.

2) $k$ satisfies the non-archimedean property: $|x+y| \leq \max\{|x|,|y|\}$ (e.g. $k=\mathbb{Q}_p$).

We can certainly define normed spaces over non-archimedean fields, but there are enough major differences in the theory that we should probably treat these as different objects. For example, we are no longer guaranteed to have local compactness for finite dimensional spaces.

Also, we get a little bit of algebra mixed in with our analysis, by way of the residue field $\tilde{k} := \{x\in k \mid |x|\leq 1\}/\{x\in k \mid |x|<1\}$.

I recommend reading about non-archimdean analysis and Berkovich spaces if you're curious about this active and growing area of research.

Andrew Dudzik
  • 31,007
  • For more introductory references see: https://math.stackexchange.com/questions/2567987/do-we-adopt-the-term-normed-space-which-is-over-any-ordered-field – Chilote Dec 15 '17 at 16:52