A note on the Banach-Steinhaus theorem
type: Item
instance of: scholarly article
title: A note on the Banach-Steinhaus theorem
zbMATH Open document ID: 0776.46007
published in: Real Analysis Exchange
publication date: 1992-09-26T00:00:00Z
review text: Let \((T_ i)\) be a family of bounded linear operators between Banach spaces such that \(\sup | T_ i|=\infty\). By the Banach-Steinhaus theorem, the set \(E=\{x:\) \(\sup | T_ i x|<\infty\}\) is `thin', namely a countable union of nowhere dense sets. Here it is shown that \(E\) is also thin in a different sense: it is a countable union of so-called porous sets.
Mathematics Subject Classification ID: 46A30
Mathematics Subject Classification ID: 26A21
zbMATH DE Number: 54518
zbMATH Keywords: open mapping theorem
zbMATH Keywords: thin
zbMATH Keywords: family of bounded linear operators between Banach spaces
zbMATH Keywords: Banach-Steinhaus theorem
zbMATH Keywords: countable union of nowhere dense sets
zbMATH Keywords: countable union of porous sets
author: Victor Olevskii
reviewed by: Dirk Werner
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 54518