Uniform boundedness

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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[edit]

Real line and complex plane[edit]

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[edit]

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[edit]

  • 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[edit]

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