In mathematics, a uniformly bounded family of functionss is a family of bounded functions that can all be bounded by the same constant. This constant is larger than or equal to 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 is either the set of real or complex numbers uniformly bounded such that

Another way of stating this would be the following:

Metric space

be a metric space, then the set

is called uniformly bounded such that

Another way of stating this would be the following:

Examples

• Every uniformly convergent sequence of bounded functions is uniformly bounded.

• defined for real traveling through the integers, is uniformly bounded by 1.

• The family of derivatives is not

References