Uniform boundedness

Let

${\displaystyle {\mathcal {F}}=\{f_{i}:X\to K,i\in I\}}$

be a family of functions indexed by ${\displaystyle I}$, where ${\displaystyle X}$ is an arbitrary set and ${\displaystyle K}$ is the set of real or complex numbers. We call ${\displaystyle {\mathcal {F}}}$ uniformly bounded if there exists a real number ${\displaystyle M}$ such that

${\displaystyle |f_{i}(x)|\leq M\qquad \forall i\in I\quad \forall x\in X.}$

In general let ${\displaystyle Y}$ be a metric space with metric ${\displaystyle d}$, then the set

${\displaystyle {\mathcal {F}}=\{f_{i}:X\to Y,i\in I\}}$

is called uniformly bounded if there exists an element ${\displaystyle a}$ from ${\displaystyle Y}$ and a real number ${\displaystyle M}$ such that

${\displaystyle d(f_{i}(x),a)\leq M\qquad \forall i\in I\quad \forall x\in X.}$