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.