On the entire bounded approximation property
type: Item
instance of: scholarly article
title: On the entire bounded approximation property
zbMATH Open document ID: 0696.46012
DOI: 10.1007/BF00971507
published in: Siberian Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: Let X be a Banach space and \(X^*\) its dual space. A subspace \(M\subset X^*\) is said to be norming if there exists \(c>0\) such that \((\forall x\in X)(\sup_{0\neq f\in M}| f(x)| /\| f\| \geq c\| x\|).\) A subspace \(M\subset X^*\) is said to be quasibasic if there exists a sequence of linear continuous finite-dimensional operators \(v_ n: X\to X\), \(n\in {\mathbb{N}}\) such that: 1) (\(\forall x\in x)(\lim_{n\to \infty}\| v_ n(x)-x\| =0);\) 2) (\(\forall n\in {\mathbb{N}})(Im v^*_ n\subset M)\), where \(v^*_ n\)- operators, conjugate to \(v_ n.\) A Banach space X is said to have the entire bounded approximation property if every norming subspace \(M\subset X^*\) is quasibasic. This property is introduced in connection with investigations of regularizability of unbounded linear operators. The author proved that there exist Banach spaces isomorphic to their biduals which have bases but haven't entire bounded approximation property. This result complements previous results due to I. Singer, V. A. Vinokurov, A. N. Plichko, F. S. Vakher, L. V. Gladun.
Mathematics Subject Classification ID: 46B15
Mathematics Subject Classification ID: 47A50
zbMATH DE Number: 4140521
zbMATH Keywords: quasibasic
zbMATH Keywords: entire bounded approximation property
zbMATH Keywords: norming subspace
zbMATH Keywords: regularizability of unbounded linear operators
author: Mikhail Ostrovskii
MaRDI profile type: Publication
cites work: Q3333584
cites work: Q4136031
cites work: On the subspaces of \(L^p\) \((p > 2)\) spanned by sequences of independent random variables
cites work: On uncomplemented subspaces of \(L_p\), \(1<p<2\)
cites work: Q3920142
description: scientific article; zbMATH DE number 4140521