PointwiseConvergence 2002-12-11 12:11:56 2002-12-11 12:11:56 Definition pointwise % 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 Let $X$ be any set, and let $Y$ be a topological space. A sequence $f_1,f_2,\dots$ of functions mapping $X$ to $Y$ is said to be \emph{pointwise convergent} (or simply convergent) to another function $f$, if the sequence $f_n(x)$ converges to $f(x)$ for each $x$ in $X$. This is usually denoted by $f_n\rightarrow f$.