Lemma 10

Let $(f_1, f_2, \ldots, f_n)$ be a basis of E. a) *There exists one and only one orthonormal basis $(e_1, e_2, \ldots, e_n)$ of E such that $f_i = e_1 \alpha_{i1} + e_2 \alpha_{i2} + \cdots + e_i \alpha_{ii}$ ($i = 1, 2, \ldots, n$) with $\alpha_{ii} > 0$ for all $i$. b) For fixed $\Phi$, the $e_i$ and $\alpha_{ij}$ depend continuously on $(f_1, \ldots, f_n) \in E^n$.

Let $E_i = f_1 K + f_2 K + \cdots + f_i K$, which has dimension $i$. Let $g_i$ be a nonzero element of $E_i$ orthogonal to $E_{i-1}$ and such that $\Phi(g_i, g_i) = 1$. By induction on $i$, one sees that $(g_1, \ldots, g_i)$ is an orthonormal basis of $E_i$. In particular, $(g_1, \ldots, g_n)$ is an orthonormal basis of $E$. Let $\lambda_i = \Phi(f_i, g_i)$. Since $f_i \notin E_{i-1}$, one has $\lambda_i \neq 0$. Set $e_i = g_i |\lambda_i| \lambda_i^{-1}$. Then

thus $(e_1, \ldots, e_i)$ is also an orthonormal basis of $E_i$; moreover, $\Phi(e_i, f_i) = |\lambda_i| \overline{\lambda_i}^{-1} \Phi(g_i, f_i) = |\lambda_i| > 0$, thus the $e_i$ have the properties of a). Let $(e'_1, \ldots, e'_n)$ be another orthonormal basis of $E$ with the same properties. One sees by induction on $i$ that $(e'_1, \ldots, e'_i)$ must be a basis of $E_i$, therefore $e'_i = e_i \mu_i$ for some $\mu_i \in \mathbf{K}$. Then

and $0 < \Phi(e'_i, f_i) = \overline{\mu_i} \Phi(e_i, f_i)$, therefore $\mu_i > 0$ and $\mu_i^2 = 1$, thus $\mu_i = 1$, whence a). Suppose already proven that the $e_i$ and $\alpha_{ij}$ depend continuously on $(f_1, \ldots, f_n)$ for $i < i_0$, and let us prove that $e_{i_0}$ and the $\alpha_{i_0 j}$ depend continuously on $(f_1, \ldots, f_n)$. For $j < i_0$, $\overline{\alpha}_{i_0 j} = \Phi(f_{i_0}, e_j)$ depends continuously on $(f_1, \ldots, f_n)$ by the induction hypothesis. On the other hand,

thus $\alpha_{i_0 i_0}$ depends continuously on $(f_1, \ldots, f_n)$. Therefore

depends continuously on $(f_1, \ldots, f_n)$.

Henceforth let $E = K^n$ and let us take for $\Phi$ the form

Recall that $\mathbf{U}(n, \mathbf{K})$ denotes the corresponding unitary group. Even when $\mathbf{K}$ is noncommutative, we shall again denote by $\mathbf{T}_1(n, \mathbf{K})$ the group of upper triangular matrices of $\mathbf{M}_n(\mathbf{K})$ whose diagonal entries are all 1.