Another proof that \(L^p\)-bounded pointwise convergence implies weak convergence
type: Item
instance of: scholarly article
title: Another proof that \(L^p\)-bounded pointwise convergence implies weak convergence
zbMATH Open document ID: 1245.46023
author: Marian Jakszto
published in: Real Analysis Exchange
publication date: 2012-05-10T00:00:00Z
review text: 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.
reviewed by: Hans Weber
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 28A20
zbMATH DE Number: 6032466
zbMATH Keywords: \(L^p\)-spaces
zbMATH Keywords: weak convergence
zbMATH Keywords: pointwise convergence
zbMATH Keywords: convergence in measure
MaRDI profile type: Publication
full work available at URL: realanalexch.36.2.0479
OpenAlex ID: W1484669914
DOI: 10.14321/REALANALEXCH.36.2.0479
description: scientific article; zbMATH DE number 6032466