BoundedFunction 2003-07-06 11:48:55 2004-12-11 11:07:47 Definition supremum norm sup norm sup-norm uniform norm bounded function unbounded function % 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 \newcommand{\sR}[0]{\mathbb{R}} \newcommand{\sC}[0]{\mathbb{C}} \newcommand{\sN}[0]{\mathbb{N}} \newcommand{\sZ}[0]{\mathbb{Z}} % The below lines should work as the command % \renewcommand{\bibname}{References} % without creating havoc when rendering an entry in % the page-image mode. \makeatletter \@ifundefined{bibname}{}{\renewcommand{\bibname}{References}} \makeatother \newcommand*{\norm}[1]{\lVert #1 \rVert} \newcommand*{\abs}[1]{| #1 |} {\bf Definition} Suppose $X$ is a nonempty set. Then a function $f:X\to \sC$ is a \emph{\PMlinkescapetext{bounded function}} if there exist a $C<\infty$ such that $|f(x)|<C$ for all $x\in X$. The set of all bounded functions on $X$ is usually denoted by $B(X)$ (\cite{aliprantis}, pp. 61). Under standard point-wise addition and point-wise multiplication by a scalar, $B(X)$ is a complex vector space. If $f\in B(X)$, then the \emph{sup-norm}, or \emph{uniform norm}, of $f$ is defined as $$ ||f||_\infty = \sup_{x\in X} |f(x)|. $$ It is straightforward to check that $||\cdot||_\infty$ makes $B(X)$ into a normed vector space, i.e., to check that $||\cdot||_\infty$ satisfies the assumptions for a norm. \subsubsection{Example} Suppose $X$ is a compact topological space. Further, let $C(X)$ be the set of continuous complex-valued functions on $X$ (with the same vector space structure as $B(X)$). Then $C(X)$ is a vector subspace of $B(X)$. \begin{thebibliography}{9} \bibitem{aliprantis} C.D. Aliprantis, O. Burkinshaw, \emph{Principles of Real Analysis}, 2nd ed., Academic Press, 1990. \end{thebibliography}