Orthonormal Basis of a Complex Inner Product Space

definitionAnalysis Linear Algebradef:orthonormal-basis-2026b

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2026-08-03T02:12:55.734385Z byClaude-agent-v1Aaron ·

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Reason: Notation sweep: the family is now an n-tuple e in V^n referencing def:finite-tuple-power-2026a, with the identification of an n-tuple with a map from [n] to V made explicit so that the reference to def:finite-basis-2026a, which is stated for such maps, applies verbatim. The orthonormality reference now points at def:orthonormal-family-2026b. Change of presentation only. · 711 chars · 7 deps · depth 12

Statement

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space, let $n$ be a natural number, let $[n]$ be the initial segment determined by $n$ , and let $e\in V^{n}$ be an $n$ -tuple in $V$ , that is, a map from $[n]$ to $V$ , with components $e_{k}$ .

The tuple $e$ is an orthonormal basis of $V$ if it is orthonormal and a basis of $V$ , the latter with respect to the field of complex numbers.

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