Let $\mathbb{C}$ be the field of \reftext{def:complex-numbers-2026a}{complex numbers}, let $V$ be a \reftext{def:vector-space-2026a}{complex vector space}, let $0_{V}$ be its \reftext{lem:vector-space-basic-identities-2026a}{zero vector}, and for $\lambda\in\mathbb{C}$ let $\overline{\lambda}$ denote its \reftext{def:complex-conjugate-2026a}{complex conjugate}.

An \textbf{inner product} on $V$ is a map assigning to each pair $u,v$ of elements of $V$ a complex number $\langle u,v\rangle$, subject to the following conditions for all $u,v,w\in V$ and all $\lambda\in\mathbb{C}$.

\textbf{1. (Conjugate symmetry)} $\langle u,v\rangle=\overline{\langle v,u\rangle}$.

\textbf{2. (Additivity in the second argument)} $\langle u,v+w\rangle=\langle u,v\rangle+\langle u,w\rangle$.

\textbf{3. (Homogeneity in the second argument)} $\langle u,\lambda v\rangle=\lambda\langle u,v\rangle$.

\textbf{4. (Positive definiteness)} $\langle v,v\rangle$ is a \reftext{def:real-numbers-c54-2026c}{real number} satisfying $0\le\langle v,v\rangle$, and $\langle v,v\rangle=0$ only if $v=0_{V}$.

A \textbf{complex inner product space} is a complex vector space together with an inner product on it.