Definition:Semi-Inner Product

Definition
Let $V$ be a vector space over a subfield $\mathbb F$ of $\C$.

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


 * $\forall x, y, z \in V, a \in \mathbb F$:
 * $(1): \quad \left \langle {x, y} \right \rangle = \overline{\left \langle {y, x} \right \rangle}$, commonly referred to as conjugate symmetry
 * $(2): \quad \left \langle {a x, y} \right \rangle = a \left \langle {x, y} \right \rangle$
 * $(3): \quad \left \langle {x + y, z} \right \rangle = \left \langle {x, z} \right \rangle + \left \langle {y, z} \right \rangle$
 * $(4): \quad \left \langle {x, x} \right \rangle \in \R_{\ge 0}$

If $\mathbb F$ is a subfield of $\R$, 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:


 * $(1^\prime): \quad \left \langle {x, y} \right \rangle = \left \langle {y, x} \right \rangle$, that is: a semi-inner product is symmetric.

Also see

 * Inner Product