In the special case of functions on the complex plane where it is often used, the definition can be given as follows.
A set of functions is said to be locally bounded if for every there exist constants and such that for all such that , for all .
As an example we can look at the set of entire functions where for any . Obviously each such is unbounded itself, however if we take a small neighbourhood around any point we can bound all . Say on an open ball we can show by triangle inequality that for all . So this set of functions is locally bounded.
- 1 John B. Conway. . Springer-Verlag, New York, New York, 1978.
|Date of creation||2013-03-22 14:17:47|
|Last modified on||2013-03-22 14:17:47|
|Last modified by||jirka (4157)|