mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a uniformly bounded

family Family (from ) is a Social group, group of people related either by consanguinity (by recognized birth) or Affinity (law), affinity (by marriage or other relationship). It forms the basis for social order. Ideally, families offer predictabili ...

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 (its image) is bounded. In other words, there exists a real number M such that :, f(x), \le M for all x in X. A functi ...

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 x 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

\mathbb

is either the set of real

\mathbb

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 for ...

\mathbb

. We call

\mathcal F

uniformly bounded if there exists a real number

M>0

such that :

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

Another way of stating this would be the following: :

\sup\limits_ \sup\limits_ , f_i(x), \le M.

Metric space

In general let

Y

be a

metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...

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 real

x

with

n

traveling through the

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

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

derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...

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