An inner product space is often simply described as a vector space with the addition of an inner product, but when it comes to the formal definition, the basefield seems to always be restricted to the fields of real and complex numbers. The Wikipedia article on inner product spaces remarks that this is done for "various technical reasons". I understand that the concept of non-negativity makes difficulties; I'm also not sure how conjugate symmetry could be expressed in more abstract terms. What I would like to know is: how would a more abstract formal definition of an inner product space look like? Is it possible at all?