Proposition 3

Let $V_1, V_2, V$ be three normed spaces over the same valued field, and $(x, y) \mapsto [x.y]$ a bilinear map from $V_1 \times V_2$ into $V$. If $f_1$ and $f_2$ are functions in $\mathscr{H}(\mathfrak{F}, V_1)$ and $\mathscr{H}(\mathfrak{F}, V_2)$ respectively, and $g_1, g_2$ are two functions in $\mathscr{H}(\mathfrak{F}, \mathbf{R})$ such that $f_1 \preccurlyeq g_1$ and $f_2 \preccurlyeq g_2$ then $[f_1.f_2] \preccurlyeq g_1 g_2$.

Indeed (Gen. Top., IX, p. 173, th. 1) there exists a number $a > 0$ such that