Let $V$ together with $\langle\cdot,\cdot\rangle$ be a \reftext{def:complex-inner-product-space-2026a}{complex inner product space}, let $\lVert\cdot\rVert$ be the \reftext{def:inner-product-norm-2026a}{norm induced by the inner product}, and write $u-v=u+(-v)$ with the additive inverse of \ref{lem:vector-space-basic-identities-2026a}. Then the following hold.

\textbf{1. (Cauchy-Schwarz in norm form)} For all $u,v\in V$, the inequality of \ref{thm:cauchy-schwarz-complex-2026a} takes the form

with the \reftext{def:complex-modulus-2026a}{modulus} on the left.

\textbf{2. (Norm)} $\lVert\cdot\rVert$ is a \reftext{def:complex-normed-space-2026a}{norm} on $V$; that is, it satisfies positivity, absolute homogeneity and the triangle inequality.

\textbf{3. (Induced metric)} The map assigning to each pair $u,v$ of elements of $V$ the real number $d(u,v)=\lVert u-v\rVert$ is a \reftext{def:metric-space-2026a}{metric} on $V$.