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.
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
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. ISBN981-238-038-8.
December 14, 2023
uniform, boundedness, result, functional, analysis, principle, conjectures, number, theory, algebraic, geometry, conjecture, disambiguation, mathematics, uniformly, bounded, family, functions, family, bounded, functions, that, bounded, same, constant, this, co. For the result in functional analysis see Uniform boundedness principle For the conjectures in number theory and algebraic geometry see Uniform boundedness conjecture disambiguation 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 Contents 1 Definition 1 1 Real line and complex plane 1 2 Metric space 2 Examples 3 ReferencesDefinition editReal line and complex plane edit Let F f i X K i I displaystyle mathcal F f i X to K i in I nbsp be a family of functions indexed by I displaystyle I nbsp where X displaystyle X nbsp is an arbitrary set and K displaystyle K nbsp is the set of real or complex numbers We call F displaystyle mathcal F nbsp uniformly bounded if there exists a real number M displaystyle M nbsp such that f i x M i I x X displaystyle f i x leq M qquad forall i in I quad forall x in X nbsp Metric space edit In general let Y displaystyle Y nbsp be a metric space with metric d displaystyle d nbsp then the set F f i X Y i I displaystyle mathcal F f i X to Y i in I nbsp is called uniformly bounded if there exists an element a displaystyle a nbsp from Y displaystyle Y nbsp and a real number M displaystyle M nbsp such that d f i x a M i I x X displaystyle d f i x a leq M qquad forall i in I quad forall x in X nbsp Examples editEvery uniformly convergent sequence of bounded functions is uniformly bounded The family of functions f n x sin n x displaystyle f n x sin nx nbsp defined for real x displaystyle x nbsp with n displaystyle n nbsp traveling through the integers is uniformly bounded by 1 The family of derivatives of the above family f n x n cos n x displaystyle f n x n cos nx nbsp is not uniformly bounded Each f n displaystyle f n nbsp is bounded by n displaystyle n nbsp but there is no real number M displaystyle M nbsp such that n M displaystyle n leq M nbsp for all integers n displaystyle n nbsp References editMa Tsoy Wo 2002 Banach Hilbert spaces vector measures group representations World Scientific p 620pp ISBN 981 238 038 8 Retrieved from https en wikipedia org w index php title Uniform boundedness amp oldid 1115707897, wikipedia, wiki, book, books, library,