Corollary 1

Let $u$ be a compact hermitian endomorphism of E. There exists an orthonormal basis B of E such that $u$ is diagonal in the basis B and its eigenvalues are real.

If $K =\mathbf{C}$, this follows at once from the theorem. Suppose $K =\mathbf{R}$. The space E is an $\mathbf{R}$-structure on $E_{(\mathbf{C})}($cf. A, II, p. 119). The endomorphism $u_{(\mathbf{C})}$ of $E_{(\mathbf{C})}$ is compact (Remark 4 of III, p. 2) and hermitian. Let $B = (e_j)_{j\in J}$ be an orthonormal basis of $E_{(\mathbf{C})}$ such that $u_{(\mathbf{C})}\in \mathscr{D}_B(E_{(\mathbf{C})})$ and $\lambda$ the family of eigenvalues of $u_{(\mathbf{C})}$ (Theorem 1). One has $\lambda \in \mathbf{R}^J($I, p. 106, Proposition 4) ; since the linear mapping $u_{(\mathbf{C})}$ is $\mathbf{R}$-rational, the eigenspace of $u_{(\mathbf{C})}$ corresponding to $\lambda_j$ is $\mathbf{R}$-rational for every $j\in J ($cf. A, V, p. 60, Proposition 6). Hence there exists a basis of it belonging to E, and a fortiori, there exists an orthonormal basis of it in E. The union of these bases is an orthonormal basis $B_{\mathbf{R}}$ of E such that $u\in \mathscr{D}_{B_{\mathbf{R}}}(E)$.