Schauder bases and the bounded approximation property in separable Banach spaces
Volume 196 / 2010
Studia Mathematica 196 (2010), 1-12
MSC: Primary 46B15, 46B28, 46G20.
DOI: 10.4064/sm196-1-1
Abstract
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.
