Let $V$ together with $\langle\cdot,\cdot\rangle$ be a \reftext{def:complex-inner-product-space-2026a}{complex inner product space}, let $0_{V}$ be its \reftext{lem:vector-space-basic-identities-2026a}{zero vector}, let $u,v,w\in V$, and let $\lambda$ be a \reftext{def:complex-numbers-2026a}{complex number} with \reftext{def:complex-conjugate-2026a}{conjugate} $\overline{\lambda}$. Then the following hold.

\textbf{1. (Additivity in the first argument)} $\langle u+v,w\rangle=\langle u,w\rangle+\langle v,w\rangle$.

\textbf{2. (Conjugate homogeneity in the first argument)} $\langle\lambda u,v\rangle=\overline{\lambda}\,\langle u,v\rangle$.

\textbf{3. (Zero vector)} $\langle 0_{V},v\rangle=0$ and $\langle v,0_{V}\rangle=0$.

\textbf{4. (Definiteness)} $\langle v,v\rangle=0$ if and only if $v=0_{V}$.