Definition 5

*A family $(f_i)_{i \in I}$ of real-valued functions defined on a set $X$ is said to be uniformly bounded above (resp. uniformly bounded below) in $X$, if there exists a finite number $a$ such that $f_i(x) \leq a$ [resp. $f_i(x) \geq a$] for all $x \in X$ and all $i \in I$. The family $(f_i)$ is said to be uniformly bounded in $X$ if it is uniformly bounded above and below in $X$.

Thus $(f_i)$ is uniformly bounded above in $X$ if and only if the upper envelope of this family is bounded above in $X$. $(f_i)$ is uniformly bounded in $X$ if and only if the upper envelope of the family $(|f_i|)$ is bounded above in $X$ [i.e. if and only if there is a finite real number $a \geq 0$ such that $|f_i(x)| \leq a$ for all $x \in X$ and all $i \in I$].