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locally bounded
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(Definition)
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Suppose that  is a topological space and  a metric space.
In the special case of functions on the complex plane where it is often used, the definition can be given as follows.
Definition 2 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
![$ t \in [0,1]$](http://images.planetmath.org:8080/cache/objects/5752/l2h/img21.png "$ t \in [0,1]$") . 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.
- 1
- John B. Conway. Functions of One Complex Variable I. Springer-Verlag, New York, New York, 1978.
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"locally bounded" is owned by jirka.
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Cross-references: natural numbers, constant functions, counterexample, even, unit disc, region, analytic functions, triangle inequality, open ball, bound, point, unbounded, entire functions, complex plane, bounded, neighbourhood, functions, metric space, topological space
There are 4 references to this entry.
This is version 6 of locally bounded, born on 2004-04-11, modified 2005-03-07.
Object id is 5752, canonical name is LocallyBounded.
Accessed 2775 times total.
Classification:
| AMS MSC: | 54-00 (General topology :: General reference works ) | | | 30A99 (Functions of a complex variable :: General properties :: Miscellaneous) |
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Pending Errata and Addenda
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