Definition:Positiveness

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

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

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

Let $\left \langle {\cdot, \cdot} \right \rangle : V \times V \to \mathbb F$ be a mapping.

Then $\left \langle {\cdot, \cdot} \right \rangle : V \times V \to \mathbb F$ is positive iff:


 * $\forall x \in V: \quad \left \langle {x, x} \right \rangle = 0 \implies x = \mathbf 0_V$

where $\mathbf 0_V$ denotes the zero vector of $V$.

Linguistic Note
This property, as a noun, is referred to as positiveness.

Also see

 * Definition:Semi-Inner Product, where this property is used in the definition of the concept.