  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.

Functional Analysis