Editing Local boundedness (section)

== Locally bounded family ==

A [[Set (mathematics)|set]] (also called a [[Indexed family|family]]) ''U'' of real-valued or complex-valued functions defined on some topological space <math>X</math> is called '''locally bounded''' if for any <math>x_0 \in X</math> there exists a [[Neighborhood (topology)|neighborhood]] <math>A</math> of <math>x_0</math> and a positive number <math>M > 0</math> such that
<math display=block>|f(x)| \leq M</math> 
for all <math>x \in A</math> and <math>f \in U.</math> In other words, all the functions in the family must be locally bounded, and around each point they need to be bounded by the same constant.

This definition can also be extended to the case when the functions in the family ''U''  take values in some metric space, by again replacing the absolute value with the distance function.