Vector spaces without distances
13.1 A little philosophy
There are at least two ways that the notion of a finite dimensional vector space over ℝ or ℂ can be generalised. The first is that of the analyst who considers infinite dimensional spaces. The second is that of the algebraist who considers finite dimensional vector spaces over more general objects than ℝ or ℂ.
It appears that infinite dimensional vector spaces are not very interesting unless we add additional structure. This additional structure is provided by the notion of distance or metric. It is natural for analysts to invoke metric considerations when talking about finite dimensional spaces, since they expect to invoke metric considerations when talking about infinite dimensional ...