A uniform boundedness theorem for \(L^\infty(\mu,E)\) (Q1813053)
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scientific article; zbMATH DE number 1841
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | A uniform boundedness theorem for \(L^\infty(\mu,E)\) |
scientific article; zbMATH DE number 1841 |
Statements
A uniform boundedness theorem for \(L^\infty(\mu,E)\) (English)
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25 June 1992
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Let \((\Omega,\Sigma,\mu)\) be a finite, atomless measure space and \(E\) be a normed space. Let \(L^ \infty(\mu,E)\) denote the space of essentially bounded, \(\mu\)-measurable functions from \(\Omega\) into \(E\). It is proved that \(L^ \infty(\mu,E)\) is always barrelled, regardless of the barrelledness of \(E\). We also give similar results for some subspaces of \(L^ \infty(\mu,E)\). These results are proved using a sliding-hump technique combined with a proposition about Banach disks generated by bounded sequences.
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spaces of bounded measurable vector-valued functions
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barrelled spaces
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Banach disks generated by bounded sequences
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