Another proof that \(L^p\)-bounded pointwise convergence implies weak convergence (Q416413)
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scientific article; zbMATH DE number 6032466
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Another proof that \(L^p\)-bounded pointwise convergence implies weak convergence |
scientific article; zbMATH DE number 6032466 |
Statements
Another proof that \(L^p\)-bounded pointwise convergence implies weak convergence (English)
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10 May 2012
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Let \(1<p<+\infty\), \(\Omega\subseteq\mathbb{R}^N\) and \((f_n)_{n\in\mathbb{N}}\) be a norm-bounded sequence in \(L^p(\Omega)\) converging to \(f\) pointwise a.e. (or in measure). It is well known that then \(f_n\rightarrow f\) weakly in \(L^p(\Omega)\). The author gives another proof of this fact using a theorem of Banach and Saks.
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\(L^p\)-spaces
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weak convergence
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pointwise convergence
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convergence in measure
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