BanachSteinhausTheorem 2004-11-12 07:16:42 2006-08-09 18:43:18 Theorem % 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} \usepackage{mathrsfs} % 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{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Per}{\operatorname{Per}} Let $X$ be a Banach space and $Y$ a normed space. If a family $\mathcal{F}\subset \mathscr{B}(X,Y)$ of bounded operators from $X$ to $Y$ satisfies $$\sup\{\|T(x)\|: T\in \mathcal{F}\}<\infty$$ for each $x\in X$, then $$\sup\{\|T\|: T\in \mathcal{F}\}<\infty,$$ i.e. $\mathcal{F}$ is a bounded subset of $\mathscr{B}(X,Y)$ with the usual operator norm. In other words, there exists a constant $c$ such that for all $x\in X$ and $T\in \mathcal{F}$, $$\|Tx\|\leq c\|x\|.$$