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}, which is a \reftext{def:vector-space-2026a}{complex vector space} by \ref{lem:cn-vector-space-2026a}, and let $\langle\cdot,\cdot\rangle$ be the \reftext{def:standard-inner-product-cn-2026a}{standard inner product} on it. Then the following hold.

\textbf{1. (Inner product)} $\langle\cdot,\cdot\rangle$ satisfies the four conditions of \ref{def:complex-inner-product-space-2026a}; consequently $\mathbb{C}^{n}$ together with $\langle\cdot,\cdot\rangle$ is a \reftext{def:complex-inner-product-space-2026a}{complex inner product space}.

\textbf{2. (Induced norm)} The \reftext{def:inner-product-norm-2026a}{norm induced by the inner product} satisfies

with the \reftext{def:complex-modulus-2026a}{modulus} of a complex number and the \reftext{def:finite-sum-field-2026a}{finite sum} taken in the field of \reftext{def:real-numbers-c54-2026c}{real numbers}.