Uniform boundedness

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

For the result in functional analysis, see Uniform boundedness principle. For the conjecture in number theory and algebraic geometry, see Uniform boundedness conjecture.

Definition

Real line and complex plane

Let

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

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

${\displaystyle |f_{i}(x)|\leq M\qquad \forall i\in I\quad \forall x\in X.}$

Metric space

In general let ${\displaystyle Y}$ be a metric space with metric ${\displaystyle d}$, then the set

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

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

${\displaystyle d(f_{i}(x),a)\leq M\qquad \forall i\in I\quad \forall x\in X.}$

Examples

• Every uniformly convergent sequence of bounded functions is uniformly bounded.
• The family of functions ${\displaystyle f_{n}(x)=\sin nx}$ defined for real ${\displaystyle x}$ with ${\displaystyle n}$ traveling through the integers, is uniformly bounded by 1.
• The family of derivatives of the above family, ${\displaystyle f'_{n}(x)=n\,\cos nx,}$ is not uniformly bounded. Each ${\displaystyle f'_{n}}$ is bounded by ${\displaystyle |n|,}$ but there is no real number ${\displaystyle M}$ such that ${\displaystyle |n|\leq M}$ for all integers ${\displaystyle n.}$

References

• Ma, Tsoy-Wo (2002). Banach-Hilbert spaces, vector measures, group representations. World Scientific. p. 620pp. ISBN 981-238-038-8.