Lemma 4

The function $Q : B \to \mathbf{R}$ is continuous.

Let $(f_j)_{j \in J}$ be an orthonormal basis of F. Put $\lambda_j = \|u^*(f_j)\|^2$ for all $j \in J$. Since $u$ belongs to $\mathscr{L}^2(E; F)$ we have $u^* \in \mathscr{L}^2(F; E)$, hence $\sum_j \lambda_j < +\infty$. Further, we have

by Parseval’s formula (V, p. 22) and the definition of the adjoint (V, p. 38). For every $x \in B$, $|\langle u^*(f_j)|x \rangle|^2 \leq \lambda_j$ by Cauchy-Schwarz inequality; consequently, the convergence of the sum in formula (43) is uniform on B, hence lemma 4 (GT, X, § 1, No. 6).