Definition:Semi-Inner Product

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

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

Let $V$ be a vector space over $\F$

A semi-inner product is a mapping $\left \langle {\cdot, \cdot} \right \rangle : V \times V \to \mathbb F$ that satisfies the following properties:

If $\mathbb F$ is a subfield of the field of real numbers $\R$, it follows from Complex Number Equals Conjugate iff Wholly Real that $\overline{\left \langle {y, x} \right \rangle} = \left \langle {y, x} \right \rangle$ for all $x, y \in V$.

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

Also see

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