# Banach-Steinhaus theorem

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 $T\in\mathcal{F}$,

 $\|Tx\|\leq c\|x\|.$
Title Banach-Steinhaus theorem BanachSteinhausTheorem 2013-03-22 14:48:39 2013-03-22 14:48:39 Koro (127) Koro (127) 5 Koro (127) Theorem msc 46B99 Principle of Uniform Boundedness Uniform Boundedness Principle