Definition 3

Given two real functions $f,\ g$ belonging to $\mathscr{H}(\mathfrak{F}, \mathbf{R})$, which are $\geq 0$ on a set in $\mathfrak{F}$, one says that $f$ is negligible relative to $g$, or that $g$ is preponderant over $f$ (along $\mathfrak{F}$), and one writes $f \ll g$ or $g \gg f$ if, for every $\varepsilon > 0$, there exists a set $X \in \mathfrak{F}$ such that $f(t) \leq \varepsilon g(t)$ for every $t \in X$.

Given two normed spaces $V_1,\ V_2$ and two functions $f_1,\ f_2$ belonging to $\mathscr{H}(\mathfrak{F}, V_1)$ and $\mathscr{H}(\mathfrak{F}, V_2)$ respectively, one says that $f_1$ is negligible relative to $f_2$ (along $\mathfrak{F}$), and one writes $f_1 \ll f_2$ or $f_2 \gg f_1$, if $\|f_1\| \ll \|f_2\|$.

For every scalar $\lambda \neq 0$ the relation $f_1 \ll f_2 \lambda$ is equivalent to $f_1 \ll f_2$. The relation $f_1 \ll f_2$ implies $f_1 \preccurlyeq f_2$ but is not equivalent to it.

Note that the relation $f_1 \preccurlyeq f_2$ by no means implies the relation "$f_1 \ll f_2$ or $f_1 \asymp f_2$": one has $\sin x \preccurlyeq 1$ as $x$ tends to $+\infty$, but neither of the relations $\sin x \ll 1$ or $\sin x \asymp 1$ is true.