Learn and talk about Uniform boundedness, Mathematical analysis

digplanet beta 1: Athena

Share digplanet:

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.

1 Definition
- 1.1 Real line and complex plane
- 1.2 Metric space
2 Examples
3 References

Definition [edit]

Real line and complex plane [edit]

Let

$\mathcal F=\{f_i: X \to K, i \in I\}$

be a family of functions indexed by $I$ , where $X$ is an arbitrary set and $K$ is the set of real or complex numbers. We call $\mathcal F$ uniformly bounded if there exists a real number $M$ such that

$|f_i(x)|\leq M \qquad \forall i \in I \quad \forall x \in X.$

Metric space [edit]

In general let $Y$ be a metric space with metric $d$ , then the set

$\mathcal F=\{f_i: X \to Y, i\in I\}$

is called uniformly bounded if there exists an element $a$ from $Y$ and a real number $M$ such that

$d(f_i(x), a) \leq M \qquad \forall i \in I \quad \forall x \in X.$

Examples [edit]

Every uniformly convergent sequence of bounded functions is uniformly bounded.

The family of functions $f_n(x)=\sin nx\,$ defined for real $x$ with $n$ traveling through the integers, is uniformly bounded by 1.

The family of derivatives of the above family, $f'_n(x)=n\, \cos nx,$ is not uniformly bounded. Each $f'_n\,$ is bounded by $|n|,\,$ but there is no real number $M$ such that $|n|\le M$ for all integers $n.$

References [edit]

Ma, Tsoy-Wo (2002). Banach-Hilbert spaces, vector measures, group representations. World Scientific. p. 620pp. ISBN 981-238-038-8, important to look up the site on its preface.

Original courtesy of Wikipedia: http://en.wikipedia.org/wiki/Uniform_boundedness — Please support Wikipedia.
A portion of the proceeds from advertising on Digplanet goes to supporting Wikipedia.

235 videos foundNext >

Mod-01 Lec-33 Baires Category & Uniform Boundedness Theorems

Functional Analysis by Prof. P.D. Srivastava, Department of Mathematics, IIT Kharagpur. For more details on NPTEL visit http://nptel.iitm.ac.in.

The Uniform Boundedness Principle - Dr Joel Feinstein

This is a lecture from Dr Feinstein's 4th-year module G14FUN Functional Analysis. See also Dr Feinstein's blog at http://explainingmaths.wordpress.com/ and, ...

Uniform convergence and pointwise convergence - Dr Joel Feinstein

This video is a combination of the three screencasts from Chapter 9 of the second year module G12MAN Mathematical Analysis, lectured by Dr J. Feinstein (Nott...