# 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

- 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.

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