T_fIsADistributionOfZerothOrder 2003-07-09 05:18:27 2003-12-20 05:24:38 Proof every locally integrable function is a distribution 0 % 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 |} \newcommand{\cD}[0]{\mathcal{D}} To check that $T_f$ is a \PMlinkname{distribution of zeroth order}{Distribution4}, we shall use condition (3) on \PMlinkname{this page}{Distribution4}. First, it is clear that $T_f$ is a linear mapping. To see that $T_f$ is continuous, suppose $K$ is a compact set in $U$ and $u\in \cD_K$, i.e., $u$ is a smooth function with support in $K$. We then have \begin{eqnarray*} |T_f(u)| &=& |\int_K f(x) u(x) dx |\\ &\le& \int_K |f(x)| \,\, |u(x)| dx \\ &\le& \int_K |f(x)| dx \, ||u||_\infty. \end{eqnarray*} Since $f$ is locally integrable, it follows that $C=\int_K |f(x)| dx$ is finite, so $$ |T_f(u)| \le C ||u||_\infty.$$ Thus $f$ is a distribution of zeroth order (\cite{lang}, pp. 381). $\Box$ \begin{thebibliography}{9} \bibitem{lang} S. Lang, \emph{Analysis II}, Addison-Wesley Publishing Company Inc., 1969. \end{thebibliography}