Definition 1

Tag 0DLE, permanent. Cite it as [Bourbaki, Tag 0DLE].

Algebra, chapter IX, § 6. Propriétés spéciales aux formes hermitiennes

Let $\Phi$ be a Hermitian form on $E$ . A basis $(e_i)$ of $E$ is said to be orthogonal for $\Phi$ if any two elements of this basis are orthogonal for $\Phi$ .

If in addition $\Phi(e_i, e_i) = 1$ for every $i$ , the basis $(e_i)$ is said to be orthonormal.

Let $(e_i)$ be an orthogonal basis; if we put $\Phi(e_i, e_i) = \alpha_i$ , we have

\Phi(\sum_i \xi_i e_i, \sum_i \eta_i e_i) = \sum_i \xi_i \alpha_i \overline{\eta_i}.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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