Banach spaces from a construction scheme

TL;DR

This paper constructs specific Banach spaces demonstrating the existence of uncountable biorthogonal and basic sequences with particular properties, using recursive amalgamation methods to show limitations in uncountable structures.

Contribution

It introduces new Banach space constructions with controlled uncountable biorthogonal and basic sequences, advancing understanding of their limitations and structure.

Findings

Constructed a Banach space with an uncountable ε-biorthogonal system but no smaller τ-biorthogonal system.

Built a Banach space with an uncountable K-basic sequence but no larger K'-basic sequence.

Both spaces are created via recursive amalgamation using a single construction scheme.

Abstract

We construct a Banach space $\mathcal X_\varepsilon$ with an uncountable $\varepsilon$-biorthogonal system but no uncountable $\tau$-biorthogonal system for $\tau<\varepsilon$. 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 $K'$-basic sequence, for $1\leq K'<K$. A common feature of these examples is that they are both constructed by recursive amalgamations using a single construction scheme.