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 or equal to the absolute value of any value of any of the functions in the family.
Definition
Real line and complex plane
Let
![{\displaystyle {\mathcal {F}}=\{f_{i}:X\to K,i\in I\}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
be a family of functions indexed by
, where
is an arbitrary set and
is the set of real or complex numbers. We call
uniformly bounded if there exists a real number
such that
![{\displaystyle |f_{i}(x)|\leq M\qquad \forall i\in I\quad \forall x\in X.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Metric space
In general let
be a metric space with metric
, then the set
![{\displaystyle {\mathcal {F}}=\{f_{i}:X\to Y,i\in I\}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
is called uniformly bounded if there exists an element
from
and a real number
such that
![{\displaystyle d(f_{i}(x),a)\leq M\qquad \forall i\in I\quad \forall x\in X.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Examples
- Every uniformly convergent sequence of bounded functions is uniformly bounded.
- The family of functions
defined for real
with
traveling through the integers, is uniformly bounded by 1. - The family of derivatives of the above family,
is not uniformly bounded. Each
is bounded by
but there is no real number
such that
for all integers ![{\displaystyle n.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
References
- Ma, Tsoy-Wo (2002). Banach–Hilbert spaces, vector measures, group representations. World Scientific. p. 620pp. ISBN 981-238-038-8.