Banach spaces from a construction scheme

Abstract

We construct a Banach space ${\mathcal{X}}_{\epsilon }$ with an uncountable $\epsilon$-biorthogonal system but no uncountable $\tau$-biorthogonal system for $\tau <\epsilon$. In particular the space have no uncountable biorthogonal system. We also construct a Banach space ${\mathcal{X}}_K$ with an uncountable $K$-basic sequence but no uncountable $K^{\prime }$-basic sequence, for $1\le K^{\prime }<K$. A common feature of these examples is that they are both constructed by recursive amalgamations using a single construction scheme.

arXiv comment10 pages, fix errors and included referee suggestions. To appear in J. Math. Anal. Appl