Continuity of coordinate functionals of filter bases in Banach spaces

TL;DR

This paper proves that coordinate functionals linked to analytic filters in Banach spaces are continuous, providing a ZFC solution to Kadets' problem and connecting basis continuity to analytic filters.

Contribution

It establishes the continuity of coordinate functionals under analytic filters in Banach spaces without large-cardinal assumptions, advancing the understanding of basis behavior.

Findings

Coordinate functionals are continuous for analytic filters in Banach spaces.

Provides a ZFC solution to Kadets' problem on statistical convergence.

Shows bases with continuous coordinate functionals relate to analytic filters.

Abstract

We prove that the coordinate functionals associated with filter bases in Banach spaces are continuous as long as the underlying filter is analytic. This removes the large-cardinal hypothesis from the result established by the two last-named authors ([Bull. Lond. Math. Soc. 53 (2021)]) at the expense of reducing the generality from projective to analytic. In particular, we obtain a ZFC solution to Kadets' problem of continuity of coordinate functionals associated with bases with respect to the filter of statistical convergence. Even though the automatic continuity of coordinate functionals beyond the projective class remains a mystery, we prove that a basis with respect to an arbitrary filter that has continuous coordinate functionals is also a basis with respect to a filter that is analytic.