TITLE: Inner Product Spaces over Finite Fields
QUESTION [42 upvotes]: Inner product spaces are defined over a field $\mathbb{F}$ which is either $\mathbb{R}$ or $\mathbb{C}$.
I want to know what happens if we try to define them over some finite field. Here's an example:
Let $\mathbb{F} = \{0,1,a,b\}$ be a finite field with + and * defined by the following Cayley tables:

Now, define a very simple vector space $\mathcal{V} = \{O, V\}$ over $\mathbb{F}$ as follows:

$\mathcal{V}$ is an Abelian group over addition, with identity $O$. Therefore, $O+O = V+V = O$, and $O+V = V+O =V$.
The scalar multiplication is governed with these rules: For any $e \in \mathbb{F}$, we have $eO = O$. Define $0V = O$, and $1V=aV=bV=V$.

One can easily check that $\mathcal{V}$ is a vector space over $\mathbb{F}$.
Now, we define an inner product for $\mathcal{V}$:

$\langle O,O \rangle = 0$ and $\langle V,V \rangle = 1$;
$\langle V,O \rangle = \langle O,V \rangle = 0$.

It seems that the above example demonstrates an inner product space over a finite field.

Is the above notion ever studied? Does it have any applications?

We avoided "conjugate symmetry" in the definition above, by assuming the conjugate of each member of $\mathbb{F}$ is itself. Can we define conjugation for fields other than $\mathbb{C}$? (Well, I heard the name C*-algebra, but I don't know whether it relates to my question.)
For instance, let $\mathbb{Q}[\sqrt 3] = \{a+b\sqrt 3 \mid a,b \in \mathbb{Q} \}$ be $\mathbb{Q}$ adjoined with $\sqrt 3$. For any $e = a+b\sqrt 3$ in $\mathbb{Q}[\sqrt 3]$, can we define the conjugate of $e$ as $e^* = a-b\sqrt 3$? This satisfies the condition below: Both addition and multiplication of $e$ and $e^*$ are members of the underlying subfield.

REPLY [7 votes]: There is a certain generalization of inner product spaces but it's not quite what you want: Hilbert C* modules are (essentially) Hilbert spaces (i.e. complete inner product spaces) over arbitrary C*-algebras. Finite fields are definitely not C*-algebras, but you can have (among other things) $\mathbb{C}^n$ (and in particular $\mathbb{C}$, the classical case), $\ell^2$, $L^2 [0,1]$, etc. The theory is quite well-developed. You have a Cauchy-Schwarz inequality, an induced norm and metric, and so on. You can read more about it in Wiki: 
http://en.wikipedia.org/wiki/Hilbert_C*-module