Extracting long basic sequences from systems of dispersed vectors

TL;DR

This paper investigates Banach spaces with specific geometric properties, demonstrating how dispersed transfinite sequences can be extracted into biorthogonal or weakly null basic sequences, and establishing the Separable Complementation Property under weak conditions.

Contribution

It introduces methods to extract structured sequences from dispersed transfinite sequences in Banach spaces and proves the Separable Complementation Property for spaces with an M-basis under weak geometric assumptions.

Findings

Extraction of biorthogonal sequences from dispersed transfinite sequences

Separable Complementation Property established for spaces with an M-basis

Analogy of Baire Category Theorem for the lattice of closed subspaces

Abstract

We study Banach spaces satisfying some geometric or structural properties involving tightness of transfinite sequences of nested linear subspaces. These properties are much weaker than WCG and closely related to Corson's property (C). Given a transfinite sequence of normalized vectors, which is dispersed or null in some sense, we extract a subsequence which is a biorthogonal sequence, or even a weakly null monotone basic sequence, depending on the setting. The Separable Complementation Property is established for spaces with an M-basis under rather weak geometric properties. We also consider an analogy of the Baire Category Theorem for the lattice of closed linear subspaces.