# Inner product

In linear algebra, an inner product $\langle \cdot, \cdot \rangle$ in a vector space V is a function from $V \times V$ to $\mathbb{R}$ satisfying the following axioms for all vectors $\vec{u}, \vec{v}, \vec{w} \in V$:[1]

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

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

• $\langle \alpha \vec{u} + \beta \vec{v}, \vec{w}\rangle = \alpha \langle \vec{u}, \vec{w} \rangle + \beta \langle \vec{v}, \vec{w}\rangle$
• $\langle \vec{u}, \alpha \vec{v} + \beta \vec{w}\rangle = \alpha \langle \vec{u}, \vec{v} \rangle + \beta \langle \vec{u}, \vec{w}\rangle$

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

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

## References

1. Anton, Howard and Chris Rorres. Elementary Linear Algebra: Applications Version. 9th ed. N.p.:John Wiley & Sons, Inc., 2005. p. 296