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 $\left \langle {\cdot, \cdot} \right \rangle : V \times V \to \mathbb F$ that satisfies the following properties:

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:

Also known as

 * Innerproduct.

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

Also see

 * Definition:Semi-Inner Product, a slightly more general concept.


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