Inner product for vector space over arbitrary field

The definition of an inner product in Linear Algebra Done Right by Sheldon Axler assumes that the vector space is over either the real or complex field. PlanetMath makes the same assumption.

Is there a definition of an inner product over, for example, finite fields? I sometimes find finite fields easier to reason about, so it would be nice to have a definition of an inner product for vector spaces over them.

Solution 1:

You could define an inner product on an ordered field, as you need to satisfy the positive-definiteness axiom. However, without a suitable order on the field, this axiom is meaningless. In order to generalise the conjugation in the definition of a hermitian inner product, you can introduce an involution on a field. As long as you can introduce an order and an involution, you should be able to generalise the definition easily enough.

As an aside, it's worth noting that when you do away with the positive-definiteness axiom, what you have is a symmetric bilinear form, which you can define on most (all? I'm not 100%!) fields.