Uniform boundedness
From Wikipedia, the free encyclopedia
In mathematics, bounded functions are functions for which there exists a lower bound and an upper bound, in other words, a constant which is larger than the absolute value of any value of this function. If we consider a family of bounded functions, this constant can vary between functions. If it is possible to find one constant which 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.
Contents |
[edit] Definition
[edit] Real line and complex plane
Let
be a family of functions with Y being a set of real (or complex) numbers, then we call 
uniformly bounded if there exists a real number c such that
[edit] Metric space
In general let Y be a metric space with metric d with some 
then
is called uniformly bounded if there exists a real number c such that
[edit] Examples
- The family of functions
defined for real x with n 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 M such that 
for all integers n.
[edit] References
- Ma, Tsoy-Wo (2002). Banach-Hilbert spaces, vector measures, group representations. World Scientific, 620pp. ISBN 9812380388, important to look up the site on its preface.
