Uniform boundedness

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In mathematics, bounded functions are functions for which there exists a lower bound and an upper bound, in other words, a constant which is larger than the absolute value of any value of this function. If we consider a family of bounded functions, this constant can vary between functions. If it is possible to find one constant which bounds all functions, this family of functions is uniformly bounded.

More precisely, if $\mathcal F=\{f_i: X \to Y, i\in I\}$ is a family of functions with $Y$ being a set of real (or complex) numbers, then this family is uniformly bounded if there exists $M\in\mathbb R$ such that for each $i\in I$ and each $x\in X$ the inequality