In 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 ...

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

s 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.

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 set ...

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 In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily s ...

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