Definition:Inner Product

Definition

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

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

An inner product is a mapping $\left \langle {\cdot, \cdot} \right \rangle : V \times V \to \mathbb F$ that satisfies the following properties:

$(1)$	$:$	Conjugate Symmetry	$\displaystyle \forall x, y \in V:$	$\displaystyle \quad \left \langle {x, y} \right \rangle = \overline{\left \langle {y, x} \right \rangle} $
$(2)$	$:$	Bilinearity	$\displaystyle \forall x, y \in V, \forall a \in \mathbb F:$	$\displaystyle \quad \left \langle {a x + y, z} \right \rangle = a \left \langle {x, z} \right \rangle + \left \langle {y, z} \right \rangle $
$(3)$	$:$	Non-Negative Definiteness	$\displaystyle \forall x \in V:$	$\displaystyle \quad \left \langle {x, x} \right \rangle \in \R_{\ge 0} $
$(4)$	$:$	Positiveness	$\displaystyle \forall x \in V:$	$\displaystyle \quad \left \langle {x, x} \right \rangle = 0 \implies x = \mathbf 0_V $

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

If $\mathbb F$ is a subfield of the field of real numbers $\R$, it follows from Complex Number equals Conjugate iff Wholly Real that $\overline{\left \langle {y, x} \right \rangle} = \left \langle {y, x} \right \rangle$ for all $x, y \in V$.

Then $(1)$ above may be replaced by:

$(1^\prime)$

$:$

$\displaystyle \forall x, y \in V:$

$\displaystyle \left \langle {x, y} \right \rangle = \left \langle {y, x} \right \rangle $

An inner product space is a vector space together with an associated inner product.

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

The most well-known example of an inner product is the dot product (see Dot Product is Inner Product).

\((1)\)	$:$	Conjugate Symmetry	\(\displaystyle \forall x, y \in V:\)	\(\displaystyle \quad \left \langle {x, y} \right \rangle = \overline{\left \langle {y, x} \right \rangle} \)
\((2)\)	$:$	Bilinearity	\(\displaystyle \forall x, y \in V, \forall a \in \mathbb F:\)	\(\displaystyle \quad \left \langle {a x + y, z} \right \rangle = a \left \langle {x, z} \right \rangle + \left \langle {y, z} \right \rangle \)
\((3)\)	$:$	Non-Negative Definiteness	\(\displaystyle \forall x \in V:\)	\(\displaystyle \quad \left \langle {x, x} \right \rangle \in \R_{\ge 0} \)
\((4)\)	$:$	Positiveness	\(\displaystyle \forall x \in V:\)	\(\displaystyle \quad \left \langle {x, x} \right \rangle = 0 \implies x = \mathbf 0_V \)