mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a uniformly bounded

family Family (from la, familia) is a group of people related either by consanguinity (by recognized birth) or affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its members and of society. Idea ...

of functions is a family of

bounded function In mathematics, a function ''f'' defined on some set ''X'' with real or complex values is called bounded if the set of its values is bounded. In other words, there exists a real number ''M'' such that :, f(x), \le M for all ''x'' in ''X''. ...

s that can all be bounded by the same constant. This constant is larger than or equal to the

absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...

of any value of any of the functions in the family.

Definition

Real line and complex plane

Let :

\mathcal F=\

be a family of functions indexed by

I

, where

X

is an arbitrary set and

K

is the set of

real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...

s. We call

\mathcal F

uniformly bounded if there exists a real number

M

such that :

, f_i(x), \le M \qquad \forall i \in I \quad \forall x \in X.

Metric space

In general let

Y

be a

metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...

with metric

d

, then the set :

\mathcal F=\

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

* Every uniformly convergent sequence of bounded functions is uniformly bounded. * The family of functions

f_n(x)=\sin nx

defined for

x

with

n

traveling through the

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

s, is uniformly bounded by 1. * The family of

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

s 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

*{{cite book , last = Ma , first = Tsoy-Wo , title = Banach-Hilbert spaces, vector measures, group representations , publisher = World Scientific , date = 2002 , isbn = 981-238-038-8 , page = 620pp Mathematical analysis