Definition 1

Given two real functions $f, g$ belonging to $\mathscr{H}(\mathfrak{F}, \mathbf{R})$, which are $\geq 0$ on a set in $\mathfrak{F}$, we say that $f$ is dominated by $g$, or that $g$ dominates $f$ (along $\mathfrak{F}$*), and write* $f \preceq g$ or $g \succeq f$, if there are a set $X \in \mathfrak{F}$ and a number $k > 0$ such that $f(t) \leq k\ g(t)$ for every $t \in X$ (in other words, if there exists a $k > 0$ such that $\tilde{f} \leq k\ \tilde{g}$)

$^1$ In particular, in all that follows, given a function $f$ in $\mathscr{H}(\mathfrak{F}, V)$ we shall denote by $\|f\|$ the function $t \mapsto \|f(t)\|$ which belongs to $\mathscr{H}(\mathfrak{F}, \mathbf{R})$ and is defined on the same set as $f$: we expressly draw attention to the fact that in this chapter $\|f\|$ is a function and not a number.

Given two normed vector 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 dominated by $f_2$ (along $\mathfrak{F}$), and writes $f_1 \preccurlyeq f_2$ or $f_2 \succeq f_1$, if $\|f_1\| \preccurlyeq \|f_2\|$.

The relation $f_1 \preccurlyeq f_2$ is clearly of local character in $f_1$ and $f_2$; it is thus equivalent to the relation $\tilde{f}_1 \preccurlyeq \tilde{f}_2$ derived by passing to the quotient. When $f$ and $g$ are real functions one must be careful not to confuse the relations $\tilde{f} \preccurlyeq \tilde{g}$ and $\tilde{f} \leq \tilde{g}$.

Note that for every scalar $\lambda \neq 0$ the relation $f_1 \preccurlyeq f_2 \lambda$ is equivalent to $f_1 \preccurlyeq f_2$. If $f_1 \preccurlyeq f_2$ there exists a set $X \in \mathfrak{F}$ such that $f_1(x) = 0$ for every point $x \in X$ where $f_2(x) = 0$.