In
mathematics, a uniformly boundedfamily of
functions 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 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