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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

MY AXIOMS

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    $\begingroup$ I guess this is what you are looking for: mathoverflow.net/q/6701/14319 $\endgroup$ – Jack Dec 13 '16 at 3:20
  • $\begingroup$ Sorry with <= defined as per normal, i.e. a <= b --> a = b OR a < b $\endgroup$ – user150203 Dec 13 '16 at 3:20
  • $\begingroup$ @Jack - Thanks for that, I believe if I've read it correctly that it may be possible.... I know (or I think I know) that if I can define an Inner Product over an arbitrary Vector Space V over an arbitrary ordered field F, then this invokes that the norm can be defined (again I believe), i.e. || u || = sqrt( < u, u> ) (assuming sqrt is defined for F) Whether I can extend the norm definitions as I have I'm not sure. I can certainly defined a Symmertic Billinear Form Operator to act like a norm, however I won't have the Triangle inequality satisfied.... $\endgroup$ – user150203 Dec 13 '16 at 3:39
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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

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  • $\begingroup$ My apologies for not getting back sooner. Thanks so much for your help. $\endgroup$ – user150203 Feb 5 '18 at 1:12

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