|
|
|
|
locally bounded
|
(Definition)
|
|
|
Suppose that $X$ is a topological space and $Y$ a metric space.
Definition 1 A set ${\mathcal{F}}$ of functions $f\colon X \to Y$ is said to be locally bounded if for every $x \in X$ , there exists a neighbourhood $N$ of $x$ such that ${\mathcal{F}}$ is uniformly bounded on $N$ .
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 ${\mathcal{F}}$ of functions $f\colon G \subset {\mathbb{C}} \to {\mathbb{C}}$ is said to be locally bounded if for every $a \in G$ there exist constants $\delta > 0$ and $M > 0$ such that for all $z \in G$ such that $\lvert z-a \rvert < \delta$ , $\lvert f(z) \rvert < M$ for all $f \in {\mathcal{F}}$ .
As an example we can look at the set ${\mathcal{F}}$ of entire functions where $f(z) = z^2 + t$ for any $t \in [0,1]$ . Obviously each such $f$ is unbounded itself, however if we take a small neighbourhood around any point we can bound all $f \in {\mathcal{F}}$ . Say on an open ball $B(z_0,1)$ we can show by triangle inequality that $\lvert f(z) \rvert \leq (\lvert z_0 \rvert +1)^2 + 1$ for all $z \in B(z_0,1)$ . So this set of functions is locally bounded.
Another example would be say the set of all analytic functions from some region $G$ to the unit disc. All those functions are bounded by 1, and so we have a uniform bound even over all of $G$ .
As a counterexample suppose the we take the constant functions $f_n(z) = n$ for all natural numbers $n$ . 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.
|
"locally bounded" is owned by jirka.
|
|
(view preamble | get metadata)
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 5 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 3337 times total.
Classification:
| AMS MSC: | 54-00 (General topology :: General reference works ) | | | 30A99 (Functions of a complex variable :: General properties :: Miscellaneous) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
