Definition:Semi-Inner Product

Definition
Given a vector space $V$ 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 ($\forall x,y,z \in V, a \in \Bbb F$):


 * $(1) \quad \left \langle {x, y} \right \rangle = \overline{\left \langle {y, x} \right \rangle}$, 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 $
 * $(4) \quad \left \langle {x, x} \right \rangle \ge 0$

If $\mathbb F$ is a field not contained in $\C$ then $(1)$ above is replaced by:


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

Semi-inner Product Space
A semi-inner product space is a vector space together with an associated semi-inner product.