Definition:Semi-Inner Product/Real Field

Definition
Let $\C$ be the field of complex numbers.

Let $\GF$ be a subfield of $\C$.

Let $V$ be a vector space over $\GF$.

A semi-inner product is a mapping $\innerprod \cdot \cdot: V \times V \to \GF$ that satisfies the following properties:

If $\GF$ is a subfield of the field of real numbers $\R$, it follows from Complex Number equals Conjugate iff Wholly Real that $\overline {\innerprod y x} = \innerprod y x$ for all $x, y \in V$.

Then $(1)$ above may be replaced by:

Also see

 * Definition:Inner Product, a semi-inner product with the additional property of positiveness.