We provide a partial answer to the question of Vladimir Kadets whether given an ℱ-basis of a Banach space X, with respect to some filter ℱ ⊂ 𝒫(ℕ), the coordinate functionals are continuous. The answer is positive if the character of ℱ is less than 𝔭. In this case every ℱ-basis is an M-basis with brackets which are determined by an element of ℱ.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm211-3-7,
author = {Tomasz Kochanek},
title = {F-bases with brackets and with individual brackets in Banach spaces},
journal = {Studia Mathematica},
volume = {209},
year = {2012},
pages = {259-268},
zbl = {1273.46004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm211-3-7}
}
Tomasz Kochanek. ℱ-bases with brackets and with individual brackets in Banach spaces. Studia Mathematica, Tome 209 (2012) pp. 259-268. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm211-3-7/