Subspaces of \(L_p\) isometric to subspaces of \(\ell_p\)
type: Item
instance of: scholarly article
title: Subspaces of \(L_p\) isometric to subspaces of \(\ell_p\)
zbMATH Open document ID: 0934.46008
DOI: 10.1023/A:1009764511096
author: Hans Jarchow
author: Aleksander Pełczyński
author: Freddy Delbaen
published in: Positivity
publication date: 1999-03-07T00:00:00Z
review text: Three theorems are proven. They are as follows: Theorem A. Let \(0<p<\infty\), \(p\not\in 2\mathbb{N}\). Then a closed linear subspace \(X\) of \(L_p\) is isometrically isomorphic to a subspace of \(\ell_p\) if and only if every, equivalently some, unital subspace which is isometrically isomorphic to \(X\), consists of functions with discrete distribution. Theorem B. Let \(p\in 2\mathbb{N}\). Then every finite-dimensional subspace of \(L_p\) is isometrically isomorphic to a subspace of \(\ell_p\). Moreover, for each \(n\in\mathbb{N}\) there exists an \(N\in\mathbb{N}\), depending on \(n\), \(p\) and \(\mathbb{K}\), such that every \(n\)-dimensional subspace of \(L_p\) is isometrically isomorphic to a subspace of \(\ell^N_p\). Theorem C. Let \(1\leq p<\infty\), \(p\not\in 2\mathbb{N}\), and let \(X\) be a Banach space which is isomorphic either to \(L_p\) or to \(\ell_p\). Then, for each \(k= 2,3,\dots\), there is no finite-dimensional Banach space which is isometrically universal for all \(k\)-dimensional subspaces of \(X\).
Mathematics Subject Classification ID: 46B04
Mathematics Subject Classification ID: 46B25
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 46B07
zbMATH DE Number: 1228749
zbMATH Keywords: isometric subspaces
reviewed by: Joe Howard
MaRDI profile type: Publication
full work available at URL: a:1009764511096
OpenAlex ID: W2598914813
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description: scientific article; zbMATH DE number 1228749