Definition:Inner Product/Also Defined as

Also defined as
Let $V$ be a vector space over a field $\GF$ that is a subfield of $\R$ or $\C$.

Let $\innerprod \cdot \cdot: V \times V \to \GF$ be an inner product on $\GF$.

Some texts define an inner product only for vector spaces over $\R$ or $\C$.

This ensures that for all $v \in V$, the inner product norm:
 * $\norm v = \sqrt {\innerprod v v}$

is a scalar.

prefers the more general definition, and lists additional requirement on $\Bbb F$ in theorems where it is needed, such as the Gram-Schmidt Orthogonalization theorem.