Definition:Uniformly Bounded

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Let $X$ be a set and let $Y = \left({A, d}\right)$ be a metric space.

Let $\FF = \family {f_i}_{i \mathop \in I}$ be a family of mappings $f_i: X \to Y$.

Then $\FF$ is said to be uniformly bounded if and only if every mapping $f \in \FF$ can be bounded by the same constant.

That is, if and only if there exists some $M \in \R$ such that:

$\forall x, y \in X, i \in I : \map d {\map {f_i} x, \map {f_i} y} \le M$