Let $V$ together with $\langle\cdot,\cdot\rangle$ be a \reftext{def:complex-inner-product-space-2026a}{complex inner product space} and let $v\in V$. By condition 4 of \ref{def:complex-inner-product-space-2026a} the number $\langle v,v\rangle$ is a \reftext{def:real-numbers-c54-2026c}{real number} with $0\le\langle v,v\rangle$.

The \textbf{norm induced by the inner product} assigns to $v$ the unique real number $\lVert v\rVert$ with $0\le\lVert v\rVert$ and $\lVert v\rVert^{2}=\langle v,v\rangle$; such a number exists and is unique by \ref{thm:nonnegative-real-has-unique-square-root-2026a}.