# Definition:Inner Product

## Definition

Let $\C$ be the field of complex numbers.

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

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

An inner product is a mapping $\innerprod \cdot \cdot: V \times V \to \mathbb F$ that satisfies the following properties:

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

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 $\overline {\innerprod y x} = \innerprod y x$ for all $x, y \in V$.

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

 $(1')$ $:$ Symmetry $\displaystyle \forall x, y \in V:$ $\displaystyle \innerprod x y = \innerprod y x$

### Inner Product Space

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

• Innerproduct

## Notation

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