Definition:Positiveness

Definition

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

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

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

Let $\innerprod \cdot \cdot : V \times V \to \GF$ be a mapping.

Then $\innerprod \cdot \cdot : V \times V \to \GF$ is positive if and only if:

$\forall x \in V: \quad \innerprod x x = 0 \implies x = \bszero_V$

where $\bszero_V$ denotes the zero vector of $V$.

Also see

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

Linguistic Note

The property of being positive, when considered as a noun, is referred to as positiveness.

Sources

• 1989: R.M. Dudley: Real Analysis and Probability: $\S 5.3$: Hilbert Spaces