LocallyBounded 2004-04-11 21:27:11 2005-03-07 20:02:59 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions \usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \theoremstyle{theorem} \newtheorem*{thm}{Theorem} \newtheorem*{lemma}{Lemma} \newtheorem*{conj}{Conjecture} \newtheorem*{cor}{Corollary} \newtheorem*{example}{Example} \theoremstyle{definition} \newtheorem*{defn}{Definition} Suppose that $X$ is a topological space and $Y$ a metric space. \begin{defn} A set ${\mathcal{F}}$ of functions $f\colon X \to Y$ is said to be {\em locally bounded} if for every $x \in X$, there exists a neighbourhood $N$ of $x$ such that ${\mathcal{F}}$ is uniformly bounded on $N$. \end{defn} In the special case of functions on the complex plane where it is often used, the definition can be given as follows. \begin{defn} A set ${\mathcal{F}}$ of functions $f\colon G \subset {\mathbb{C}} \to {\mathbb{C}}$ is said to be {\em 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 $f \in {\mathcal{F}}$. \end{defn} 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. \begin{thebibliography}{9} \bibitem{Conway:complexI} John~B. Conway. {\em \PMlinkescapetext{Functions of One Complex Variable I}}. Springer-Verlag, New York, New York, 1978. \end{thebibliography}