inner product

Any operation on two vectors obeying symmetry, additivity, homogeneity, and positive definiteness.

An inner product on a real vector space $V$ is any operation $\mathbf u\cdot\mathbf v$ returning a real number, subject to four axioms: symmetry, additivity, homogeneity, and positive definiteness ( $\mathbf u\cdot\mathbf u\ge 0$ , zero only for $\mathbf u=\mathbf 0$ ). The ordinary dot product satisfies all four, but so do other operations, like a weighted dot product with fixed positive weights — any operation clearing the four bars earns every tool the dot product gives: length, angle, distance, orthogonality.