PlanetMath: Banach-Steinhaus theorem

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Banach-Steinhaus theorem

(Theorem)

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

$\displaystyle \sup\{\Vert T(x)\Vert: T\in \mathcal{F}\}<\infty$

for each $x\in X$ , then

$\displaystyle \sup\{\Vert T\Vert: T\in \mathcal{F}\}<\infty,$

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

such that for all $x\in X$ and $T\in \mathcal{F}$ ,

$\displaystyle \Vert Tx\Vert\leq c\Vert x\Vert.$

"Banach-Steinhaus theorem" is owned by Koro.

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Other names:

Principle of Uniform Boundedness, Uniform Boundedness Principle

Attachments:: proof of Banach-Steinhaus theorem (Proof) by Koro
pointwise limit of bounded operators is bounded (Corollary) by asteroid

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Cross-references: operator norm, subset, bounded, bounded operators, normed space, Banach space
There are 2 references to this entry.

This is version 2 of Banach-Steinhaus theorem, born on 2004-11-12, modified 2006-08-09.
Object id is 6469, canonical name is BanachSteinhausTheorem.
Accessed 6273 times total.

Classification:

46B99 (Functional analysis :: Normed linear spaces and Banach spaces; Banach lattices :: Miscellaneous)

Pending Errata and Addenda

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