Question regarding Normed Vector Spaces

Question

I've recently been reading an Analysis textbook (Zakon Analysis - free textbook) and I'm covering an introductory section to Vector Spaces. I've noticed that when it comes to Vector Spaces we can leave it general (i.e. a Vector Space V over a field F). I'm curious about how this extends to Inner Product Spaces and Normed Vector Spaces. I have taken the axioms for Normed Vector Spaces when they apply to the Ordered Field of Real Numbers and have tried to generalise for any Vector Space V over an ordered field F. I was wondering two parts, firstly, can a Normed Vector Space have it's axioms defined for any ordered field F?? and if so, can someone provide some feedback on what I have done, or alternatively could you please point me in the direction of some texts to seek out.

Thanks heaps!

David

Answer 1

It is possible to define normed spaces over arbitrary fields. For details and some references to read see my answer to a similar question here: https://math.stackexchange.com/a/2568042/113061

Regarding inner product spaces, it is possible to construct Hilbert-like spaces with prescribed valued fields. For this topic, in particular, I recommend the papers:

1. On a class of orthomodular quadratic spaces, H. Gross, U.M. Künzi - Enseign. Math, 1985.

2. Banach spaces over fields with a infinite rank valuation - [H.Ochsenius A., W.H.Schikhof] - 1999

3. After that see: Norm Hilbert spaces over Krull valued fields - [H. Ochsenius, W.H. Schikhof] - Indagationes Mathematicae, Elsevier - 2006