Banach-Steinhaus theorem

Let $X$ be a Banach space and $Y$ a normed space. If a family $ℱ\subset ℬ(X,Y)$ of bounded operators from $X$ to $Y$ satisfies

for each $x\in X$, then

i.e. $ℱ$ is a bounded subset of $ℬ(X,Y)$ with the usual operator norm. In other words, there exists a constant $c$ such that for all $x\in X$ and $T\in ℱ$,

$$\parallel Tx\parallel \le c\parallel x\parallel .$$

Title	Banach-Steinhaus theorem
Canonical name	BanachSteinhausTheorem
Date of creation	2013-03-22 14:48:39
Last modified on	2013-03-22 14:48:39
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	5
Author	Koro (127)
Entry type	Theorem
Classification	msc 46B99
Synonym	Principle of Uniform Boundedness
Synonym	Uniform Boundedness Principle