Representation of \(L_ p\)-norms and isometric embedding in \(L_ p\)- spaces
type: Item
instance of: scholarly article
title: Representation of \(L_ p\)-norms and isometric embedding in \(L_ p\)- spaces
zbMATH Open document ID: 0599.46047
DOI: 10.1007/BF02761158
author: Abraham Neyman
published in: Israel Journal of Mathematics
publication date: 1984-01-01T00:00:00Z
review text: For fixed \(1\leq p<\infty\) the \(L_ p\)-semi-norms on \({\mathbb{R}}^ n\) are identified with positive linear functionals on the closed linear subspace of \(C({\mathbb{R}}^ n)\) spanned by the functions \(| <\xi,\cdot >|^ p\), \(\xi \in {\mathbb{R}}^ n\). For every positive linear functional \(\sigma\), on that space, the function \(\phi_{\sigma}:{\mathbb{R}}^ n\to {\mathbb{R}}\) given given by \(\phi_{\sigma}(\xi)=\sigma (| <\xi,\cdot >|^ p)^{1/p}\) is an \(L_ p\)-semi-norm and the mapping \(\sigma \to \phi_{\sigma}\) is 1-1 and onto. The closed linear span of \(| <\xi,\cdot >|^ p\), \(\xi \in {\mathbb{R}}^ n\) is the space of all even continuous functions that are homogeneous of degree p, if p is not an even integer and is the space of all homogeneous polynomials of degree p when p is an even integer. This representation is used to prove that there is no finite list of norm inequalities that characterizes linear isometric embeddability, in any \(L_ p\) unless \(p=2\).
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 47A30
zbMATH DE Number: 3967098
zbMATH Keywords: \(L_ p\)-semi-norms
zbMATH Keywords: positive linear functionals
zbMATH Keywords: space of all homogeneous polynomials
zbMATH Keywords: norm inequalities that characterizes linear isometric embeddability
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3967098