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Banach-Steinhaus theorem
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(Theorem)
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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 $T\in \mathcal{F}$ $$\|Tx\|\leq c\|x\|.$$
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"Banach-Steinhaus theorem" is owned by Koro.
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| Other names: |
Principle of Uniform Boundedness, Uniform Boundedness Principle |
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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 8412 times total.
Classification:
| AMS MSC: | 46B99 (Functional analysis :: Normed linear spaces and Banach spaces; Banach lattices :: Miscellaneous) |
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Pending Errata and Addenda
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