# 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.

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You can indeed equip any finite-dimensional vector space over an arbitrary field with a symmetric bilinear form. You could also weaken the positive-definite axiom slightly and only insist that the form is non-degenerate, i.e. that for any given nonzero vector $x$ there exists some $y$ such that $\langle x,y\rangle \neq 0$. For many applications of an inner product, you really only need this nondegeneracy (but you won't have, for example, a good notion of the "angle between two vectors" without an inner product). –  Brad Oct 19 '12 at 2:29