Inner Product on a Vector Space over a field besides $\mathbb R$ or $\mathbb C$?

Question

Are there any fields with vector spaces you can define an inner product over besides subfields of $\mathbb C$? I know that you'd want the field to contain an ordered subfield, so it must have characteristic $0$. Is there any inner product on, say, the ordered field of rational functions?

Answer 1

Bilinear forms over any sort of field are well studied. If you are mainly interested in those which look like the ordinary inner product on the reals or on the complex numbers, then some restrictions have to be made.

Firstly, the positive definite axiom does not make any sense unless the field has an order. For ordered fields, the naive coordinatewise inner product produces an inner product for a finite dimensional vector space.

To generalize what goes on in the complex case, you do not need an ordered field ($\Bbb C$ is obviously not ordered), but you are using a field with an involution $x\to \bar{x}$ such that $x\bar{x}$ lies in the positive part of an ordered field for every x in the field. If your field has such an involution, then you can do exactly the same Hermitian product you use for finite dimensional spaces over the complex numbers.