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 or equal to the absolute value of any value of any of the functions in the family.

DefinitionEdit

Real line and complex planeEdit

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 spaceEdit

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

 

ExamplesEdit

  • 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  

ReferencesEdit

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