# Uniform boundedness

In mathematics, a bounded function is a function for which there exists a lower bound and an upper bound, in other words, a constant that is larger than the absolute value of any value of this function. If we consider a family of bounded functions, this constant can vary across functions in the family. If it is possible to find one constant that bounds all functions, this family of functions is **uniformly bounded**.

The uniform boundedness principle in functional analysis provides sufficient conditions for uniform boundedness of a family of operators.

## 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

- Every uniformly convergent sequence of bounded functions is uniformly bounded.
- The family of functions defined for real with traveling through the integers, is uniformly bounded by 1.
- The family of derivatives of the above family, is
*not*uniformly bounded. Each is bounded by but there is no real number such that for all integers

## References

- Ma, Tsoy-Wo (2002).
*Banach-Hilbert spaces, vector measures, group representations*. World Scientific. p. 620pp. ISBN 981-238-038-8.