Remark

— 1) The corollary to prop. 12 can also be deduced from th. 1 of I, p. 17 and prop. 18 of GT, IX, § 2.10. 2) We keep the notations of prop. 12. Let $p$ be a continuous semi-norm on $E$;

for all $y \in F$, put $q(y) = \inf_{u(x)=y} p(x)$, so that $q$ is a continuous semi-norm on $F$ (II, p. 4). Let $\phi$ be a lower semi-continuous mapping of $F$ in the interval $]0, +\infty[$ of $\overline{\mathbf{R}}$. We show that there exists a continuous section s of u such that $p \circ s < q + \phi$.

Let $s_0$ be a continuous section of $u$ (prop. 12) and $N$ the kernel of $u$. Let $y_0 \in F$, then there exists $z_0 \in N$ such that $p(s_0(y_0) + z_0) < q(y_0) + \phi(y_0)$. There exists an open neighbourhood $W$ of $y_0$ in $F$ such that $p(s_0(y) + z_0) < q(y) + \phi(y)$ for all $y \in W$. Hence there is an open covering $(W_i)_{i \in I}$ of $F$ and constant mappings $t_i : F \to N$ such that $p(s_0(y) + t_i(y)) < q(y) + \phi(y)$ for all $y \in W_i$. As $F$ is metrisable, there exists a locally finite continuous partition of unity subordinated to the covering $(W_i)_{i \in I}$, say $(g_i)_{i \in I}$ (GT, IX, § 4.5, th. 4 and § 4.4, cor. 1). The mapping $s$ of $F$ in $E$ defined by $s(y) = s_0(y) + \sum_{i \in I} g_i(y) \cdot t_i(y)$ fulfils the stated conditions.