Proposition 6

With the notation of prop. 3, the relations $f_1 \ll g_1$ and $f_2 \preccurlyeq g_2$ (resp. $f_1 \preccurlyeq g_1$ and $f_2 \ll g_2$) imply $[f_1.f_2] \ll g_1g_2$.

Prop. 4 shows that the relation $f \ll g$ between functions in $\mathscr{H}(\mathfrak{F}, V)$ is transitive; but it is not reflexive: to be precise, the relation $f \ll f$ implies that $f$ vanishes on a set in $\mathfrak{F}$ (in other words, $f$ is equivalent to 0 modulo $R_\infty$); indeed, for an $\varepsilon$ such that $0 < \varepsilon < 1$ there exists an $X \in \mathfrak{F}$ such that $\|f(x)\| \leq \varepsilon \|f(x)\|$ for every $x \in X$, which is possible only if $f(x) = 0$ for every $x \in X$. It follows that the relation "f \ll g and g \ll f" is transitive and symmetric, but not reflexive: so it is not an equivalence relation (it implies that there is a set $X \in \mathfrak{F}$ such that $f(x) = g(x) = 0$ for every $x \in X$).