locally bounded
Suppose that is a topological space and a metric space.
Definition.
A set of functions is said to be locally bounded if for every , there exists a neighbourhood of such that is uniformly bounded on .
In the special case of functions on the complex plane where it
is often used, the definition can be given as follows.
Definition.
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.
Another example would be say the set of all analytic functions from
some region to the unit disc
. All those functions are bounded
by 1,
and so we have a uniform bound even over all of .
As a counterexample suppose the we take the constant functions for
all natural numbers . While each of these functions is itself bounded,
we can never find a uniform bound for all such functions.
References
- 1 John B. Conway. . Springer-Verlag, New York, New York, 1978.
Title | locally bounded |
---|---|
Canonical name | LocallyBounded |
Date of creation | 2013-03-22 14:17:47 |
Last modified on | 2013-03-22 14:17:47 |
Owner | jirka (4157) |
Last modified by | jirka (4157) |
Numerical id | 9 |
Author | jirka (4157) |
Entry type | Definition |
Classification | msc 30A99 |
Classification | msc 54-00 |