A Banach space with $L$-orthogonal sequences but without $L$-orthogonal elements

TL;DR

This paper constructs a Banach space that has sequences with a specific orthogonality property but lacks individual orthogonal elements, showing this is independent of ZFC, and introduces $Q$-measures generalizing $Q$-points.

Contribution

It demonstrates the independence of the existence of such Banach spaces from ZFC and generalizes classical ultrafilter concepts to $Q$-measures.

Findings

Existence of the Banach space is independent of ZFC.

Introduces $Q$-measures generalizing $Q$-point ultrafilters.

Extends results by Miller and Bartoszynski.

Abstract

We prove that the existence of Banach spaces with $L$-orthogonal sequences but without $L$-orthogonal elements is independent of the standard foundation of Mathematics, ZFC. This provides a definitive answer to \cite[Question~1.1]{AvilesMartinezRueda}. Generalizing classical $Q$-point ultrafilters, we introduce the notion of $Q$-measures and provide several results generalizing former theorems by Miller \cite{Miller} and Bartoszynski \cite{Bartoszynski} for $Q$-point ultrafilters.