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} with the \reftext{def:standard-inner-product-cn-2026a}{standard inner product} $\langle\cdot,\cdot\rangle$, which is a \reftext{def:complex-inner-product-space-2026a}{complex inner product space} by \ref{lem:standard-inner-product-cn-2026a}, let $\lVert\cdot\rVert$ be the \reftext{def:inner-product-norm-2026a}{induced norm}, and let $d$ be the \reftext{def:metric-space-2026a}{metric} $d(u,v)=\lVert u-v\rVert$ of claim 3 of \ref{lem:inner-product-norm-is-norm-2026a}. Then the following hold.

\textbf{1. (Completeness)} The metric space $(\mathbb{C}^{n},d)$ is \reftext{def:complete-metric-space-2026a}{complete}.

\textbf{2. (Hilbert space)} Consequently $\mathbb{C}^{n}$ together with the standard inner product is a \reftext{def:complex-hilbert-space-2026a}{complex Hilbert space}. In particular, taking $n=2$, the \reftext{def:qubit-state-space-2026a}{qubit state space} is a complex Hilbert space.