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}{induced norm}, and let $d$ be the function assigning to each pair $u,v$ of elements of $V$ the real number $d(u,v)=\lVert u-v\rVert$, which is a \reftext{def:metric-space-2026a}{metric} on $V$ by claim 3 of \ref{lem:inner-product-norm-is-norm-2026a}.

The space $V$ together with $\langle\cdot,\cdot\rangle$ is a \textbf{complex Hilbert space} if the metric space $(V,d)$ is \reftext{def:complete-metric-space-2026a}{complete}.