Schauder bases and the bounded approximation property in separable Banach spaces

Jorge Mujica ¹   ; Daniela M. Vieira ²

¹ IMECC-UNICAMP Caixa Postal 6065 13083-970 Campinas, SP, Brazil
² IMECC-UNICAMP Caixa Postal 6065 13083-970 Campinas, SP, Brazil and Instituto de Matemática e Estatística Universidade de São Paulo Caixa Postal 66281 05315-970 São Paulo, SP, Brazil

Studia Mathematica, Tome 196 (2010) no. 1, pp. 1-12

Résumé

Let $E$ be a separable Banach space with the $\lambda$-bounded approximation property. We show that for each $\epsilon >0$ there is a Banach space $F$ with a Schauder basis such that $E$ is isometrically isomorphic to a $1$-complemented subspace of $F$ and, moreover, the sequence $(T_{n})$ of canonical projections in $F$ has the properties $$ \sup_{n \in \mathbb{N}} \|T_{n}\| \le \lambda+ \epsilon \quad\hbox{and}\quad \limsup_{n \rightarrow \infty} \|T_{n}\| \le \lambda. $$ This is a sharp quantitative version of a classical result obtained independently by Pełczyński and by Johnson, Rosenthal and Zippin.

Détail

DOI : 10.4064/sm196-1-1

Keywords: separable banach space lambda bounded approximation property each epsilon there banach space schauder basis isometrically isomorphic complemented subspace moreover sequence canonical projections has properties sup mathbb lambda epsilon quad hbox quad limsup rightarrow infty lambda sharp quantitative version classical result obtained independently czy ski johnson rosenthal zippin