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 ${\vec {u}},{\vec {v}},{\vec {w}}\in V$:

• $\langle {\vec {v}},{\vec {v}}\rangle \geq 0$, 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.