Category:Examples of Inner Products

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This category contains examples of Inner Product.

Complex Inner Product

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


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

\((1)\)   $:$   Conjugate Symmetry      \(\ds \forall x, y \in V:\) \(\ds \quad \innerprod x y = \overline {\innerprod y x} \)      
\((2)\)   $:$   Linearity in first argument      \(\ds \forall x, y \in V, \forall a \in \GF:\) \(\ds \quad \innerprod {a x + y} z = a \innerprod x z + \innerprod y z \)      
\((3)\)   $:$   Non-Negative Definiteness      \(\ds \forall x \in V:\) \(\ds \quad \innerprod x x \in \R_{\ge 0} \)      
\((4)\)   $:$   Positiveness      \(\ds \forall x \in V:\) \(\ds \quad \innerprod x x = 0 \implies x = \mathbf 0_V \)      


That is, a (complex) inner product is a complex semi-inner product with the additional condition $(4)$.