Definition:Semi-Inner Product/Real Field

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Definition

Let $V$ be a vector space over a real subfield $\GF$.


A (real) semi-inner product is a mapping $\innerprod \cdot \cdot: V \times V \to \GF$ that satisfies the real semi-inner product axioms:

\((1^\prime)\)   $:$   Symmetry      \(\ds \forall x, y \in V:\) \(\ds \innerprod x y = \innerprod y x \)      
\((2)\)   $:$   Sesquilinearity      \(\ds \forall x, y, z \in V, \forall a \in \GF:\) \(\ds \quad \innerprod {a x + y} z = a \innerprod x z + \innerprod y z \)      
\((3)\)   $:$   Non-Negative Definiteness      \(\ds \forall x \in V:\) \(\ds \quad \innerprod x x \in \R_{\ge 0} \)      


Real Semi-Inner Product Space

Let $V$ be a vector space over a real subfield $\GF$.

Let $\innerprod \cdot \cdot : V \times V \to \GF$ be an real semi-inner product on $V$.


We say that $\struct {V, \innerprod \cdot \cdot}$ is a (real) semi-inner product space.


Also see


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