Definition:Inner Product/Complex Field

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Definition

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 $(4)$.


Complex Inner Product Space

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

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


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


Also known as

Some texts refer to $\innerprod \cdot \cdot$ as a scalar product

As this term is disambiguous, it is not used by $\mathsf{Pr} \infty \mathsf{fWiki}$.

Some texts refer to $\innerprod \cdot \cdot$ as an innerproduct.


Also defined as

Some texts define an inner product only for vector spaces over $\R$ or $\C$.

This ensures that for all $v \in V$, the inner product norm:

$\norm v = \sqrt {\innerprod v v}$

is a scalar.

$\mathsf{Pr} \infty \mathsf{fWiki}$ prefers the more general definition, and lists additional requirement on $\Bbb F$ in theorems where it is needed, such as the Gram-Schmidt Orthogonalization theorem.


Also denoted as

$\innerprod x y$ is also denoted as $\left \langle {x; y} \right \rangle$.

If there is more than one vector space under consideration, then the notation $\innerprod x y_V$ for a vector space $V$ is commonplace.


Also see


Sources