Proposition 2

Let $\Phi$ be a hermitian form on $E$, and let $(e_i)$ $(i = 1, \ldots, n)$ be an orthogonal (resp. orthonormal) basis of $E$ for $\Phi$. Then, for every $p \geqslant 0$, the basis of $\bigotimes^p E$ formed by the $e_{i_1} \otimes \cdots \otimes e_{i_p}$ and the basis $(e_H)$ of $\wedge^p E$ (where $H$ runs through the set of parts with $p$ elements of $\{1, n\}$; cf. chap. III, § 5, no. 6) are orthogonal (resp. orthonormal) for the extensions of $\Phi$ to $\bigotimes^p E$ and $\wedge^p E$ respectively ($§ 1$, no. 9). If, moreover, the mappings associated with $\Phi$ are bijective, the basis $(e'_i)$ of $E^*$ dual to $(e_i)$ is orthogonal (resp. orthonormal) for the inverse form $\widehat{\Phi}$ of $\Phi$ ($§ 1$, no. 7).

The assertions concerning $\bigotimes^p E$ and $\wedge^p E$ follow at once from formulas (35) and (37) of the $§ 1$, no. 9. The assertion concerning the inverse form follows from the fact that the matrix of $\widehat{\Phi}$ with respect to $(e'_i)$ is the inverse of the matrix of $\Phi$ with respect to $(e_i)$ ($§ 1$, no. 10).