Uniform boundedness


In mathematics, a uniformly bounded family of functions is a family of bounded functions that can all be bounded by the same constant. This constant is larger than the absolute value of any value of any of the functions in the family.

Definition

Real line and complex plane

Let
be a family of functions indexed by, where is an arbitrary set and is the set of real or complex numbers. We call uniformly bounded if there exists a real number such that

Metric space

In general let be a metric space with metric, then the set
is called uniformly bounded if there exists an element from and a real number such that

Examples