Let $n$ be a \reftext{def:natural-numbers-2026a}{natural number}, let $\mathbb{C}^{n}$ be the \reftext{def:complex-coordinate-space-cn-2026a}{complex coordinate space}, and for a \reftext{def:complex-numbers-2026a}{complex number} $z$ let $\overline{z}$ denote its \reftext{def:complex-conjugate-2026a}{complex conjugate}.

The \textbf{standard inner product} on $\mathbb{C}^{n}$ assigns to each pair $u,v\in\mathbb{C}^{n}$ the complex number

where $u_{k}$ and $v_{k}$ are the components of $u$ and $v$ and the \reftext{def:finite-sum-field-2026a}{finite sum} is taken in the field of complex numbers.