locally bounded

Suppose that $X$ is a topological space and $Y$ a metric space.

Definition.

A set $ℱ$ of functions $f:X\to Y$ is said to be locally bounded if for every $x\in X$, there exists a neighbourhood $N$ of $x$ such that $ℱ$ 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.

A set $ℱ$ of functions $f:G\subset ℂ\to ℂ$ 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 , for all $f\in ℱ$.

As an example we can look at the set $ℱ$ 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 $f\in ℱ$. Say on an open ball $B(z_0,1)$ we can show by triangle inequality that $|f(z)|\le {(|z_0|+1)}^2+1$ 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.

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