Definition:Uniformly Bounded

Definition
Let $X = \left({A, d}\right)$ and $Y = \left({B, \rho}\right)$ be metric spaces.

Let $\mathcal F = \left \langle{f_i}\right \rangle_{i \mathop \in I}$ be a family of mappings $f_i: X \to Y$.

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

That is, if there exists some $M \in \R$ such that:
 * $\forall a, b \in A, i \in I : \rho \left({f_i \left({x}\right), f_i \left({y}\right)}\right) \le M$