Two-norm convergence in the \(L_p\) spaces
type: Item
instance of: scholarly article
title: Two-norm convergence in the \(L_p\) spaces
zbMATH Open document ID: 1246.26009
published in: Real Analysis Exchange
publication date: 2012-05-10T00:00:00Z
review text: The authors gave a representation theorem for two-norm continuous functionals defined on \(L_{p}\), \(1 \leq p \leq \infty\). A sequence \({f_{n}}\) of functions is said to be two-term convergent to \(f\) in \(L_{p}\), \(1 \leq p \leq \infty\) if there is an \(M>0\) such that \(\|f_{n}\|_{p}\leq M\) for all \(n\) and \(\lim_{n\rightarrow \infty} \int^{1}_{0} f_{n}(x)g(x)dx\) exists for every \(g\in L_{q}\) with \(\frac {1}{p}+\frac {1} {q} = 1\). A functional \(F\) defined on \(L_{p}\), \(1 \leq p \leq \infty\) is said to be two-norm continuous in \(L_{p}\) if \(F(f_{n})\rightarrow F(f)\) as \(n\rightarrow \infty\) whenever \({f_{n}}\) is two-norm convergent to \(f\) in \(L_{p}\). The authors proved that if \(F\) is a two-norm continuous linear functional on \(L_{p}\), \(1 \leq p \leq \infty\), then there exists a function \(g\in L_{q}\) with \(\frac {1}{p}+\frac {1} {q} = 1\), such that \(F(f)= \int^{1}_{0} fg\), for every \(f\in L_{p}\).
reviewed by: Jitan Lu
Mathematics Subject Classification ID: 26A39
zbMATH DE Number: 6032473
zbMATH Keywords: two-norm
zbMATH Keywords: representation theorem
zbMATH Keywords: \(L_{p}\), \(1 \leq p \leq \infty\)
MaRDI profile type: Publication
full work available at URL: realanalexch.36.1.0055
OpenAlex ID: W2065635027
DOI: 10.14321/REALANALEXCH.36.1.0055
author: Ch. Rini Indrati
description: scientific article; zbMATH DE number 6032473