Axiom:Real Inner Product Axioms

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

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

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

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

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

If $\GF$ 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:

Also known as

 * Innerproduct

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

 * 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).


 * Definition:Hilbert Space