Inner product

In linear algebra, an inner product <math>\langle \cdot, \cdot \rangle</math> in a vector space <math>V</math> is a function from <math>V \times V</math> to <math>\mathbb{R}</math> satisfying the following axioms for all vectors <math>\vec{u}, \vec{v}, \vec{w} \in V</math>:

• <math>\langle \vec{v}, \vec{v}\rangle \geq 0</math>, with <math>\langle \vec{v}, \vec{v}\rangle = 0</math> if and only if <math>\vec{v} = \vec{0}</math>,
• <math>\langle \vec{v}, \vec{w}\rangle = \langle \vec{w}, \vec{v} \rangle</math> (the inner product is commutative),
• <math>\langle \vec{u} + \vec{v}, \vec{w} \rangle = \langle \vec{u}, \vec{w} \rangle + \langle \vec{v}, \vec{w} \rangle</math>, and
• for all <math>k \in \mathbb{R}</math>, <math>\langle k\vec{v}, \vec{w}\rangle = k\langle \vec{v}, \vec{w}\rangle</math>.

One consequence of the inner product axioms is that the inner product is multilinear in both variables; that is:

• <math>\langle \alpha \vec{u} + \beta \vec{v}, \vec{w}\rangle = \alpha \langle \vec{u}, \vec{w} \rangle + \beta \langle \vec{v}, \vec{w}\rangle</math>
• <math>\langle \vec{u}, \alpha \vec{v} + \beta \vec{w}\rangle = \alpha \langle \vec{u}, \vec{v} \rangle + \beta \langle \vec{u}, \vec{w}\rangle</math>

The dot product in the Euclidean vector space <math>\mathbb{R}^n</math> is the best-known example of an inner product.

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