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Assuming the existence of certain large cardinal numbers, we prove that for every projective filter   F   over the set of natural numbers,   F  ‐bases in Banach spaces have continuous coordinate functionals. In particular, this applies to the filter of statistical convergence, thereby we solve a problem by Kadets (at least under the presence of certain large cardinals). In this setting, we recover also a result of Kochanek ( Studia Math ., 2012) who proved continuity of coordinate functionals for countably generated filters.

Large cardinals and continuity of coordinate functionals of filter bases in Banach spaces