Definition:Positiveness
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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