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In a recent note with J. Swaczyna (arXiv:2005.04873), we proved that assuming the existence of certain large cardinals (for example, the existence of a super-compact cardinal) that filter bases whose underlying filter is a projective subset of the Cantor set, have continuous coordinate functionals. The role of large cardinals is to make the heuristic proof I outlined in my other answer work.

The filter of statistical convergence is actually $F_{\sigma \delta}$ (hence Borel, hence projective), so in a theory stronger than ZFC, the answer to your question is affirmative. We still expect that, at least for Borel filters, the question about continuity of coordinate functionals should have a positive answer in ZFC. However, our proof method does not give a chance to invoke Schoenfield's absoluteness theorem in that context.

Edit 26.04.2022. The problem has been recently solved in ZFC. arxiv.org/abs/2203.15123.

In a recent note with J. Swaczyna (arXiv:2005.04873), we proved that assuming the existence of certain large cardinals (for example, the existence of a super-compact cardinal) that filter bases whose underlying filter is a projective subset of the Cantor set, have continuous coordinate functionals. The role of large cardinals is to make the heuristic proof I outlined in my other answer work.

The filter of statistical convergence is actually $F_{\sigma \delta}$ (hence Borel, hence projective), so in a theory stronger than ZFC, the answer to your question is affirmative. We still expect that, at least for Borel filters, the question about continuity of coordinate functionals should have a positive answer in ZFC. However, our proof method does not give a chance to invoke Schoenfield's absoluteness theorem in that context.

In a recent note with J. Swaczyna (arXiv:2005.04873), we proved that assuming the existence of certain large cardinals (for example, the existence of a super-compact cardinal) that filter bases whose underlying filter is a projective subset of the Cantor set, have continuous coordinate functionals. The role of large cardinals is to make the heuristic proof I outlined in my other answer work.

The filter of statistical convergence is actually $F_{\sigma \delta}$ (hence Borel, hence projective), so in a theory stronger than ZFC, the answer to your question is affirmative. We will upload the paper to the arXiv in the near future.

In a recent note with J. Swaczyna (arXiv:2005.04873), we proved that assuming the existence of certain large cardinals (for example, the existence of a super-compact cardinal) that filter bases whose underlying filter is a projective subset of the Cantor set, have continuous coordinate functionals. The role of large cardinals is to make the heuristic proof I outlined in my other answer work.

The filter of statistical convergence is actually $F_{\sigma \delta}$ (hence Borel, hence projective), so in a theory stronger than ZFC, the answer to your question is affirmative. We still expect that, at least for Borel filters, the question about continuity of coordinate functionals should have a positive answer in ZFC. However, our proof method does not give a chance to invoke Schoenfield's absoluteness theorem in that context.

In a recent note with J. Swaczyna (arXiv:2005.04873), we proved that assuming the existence of certain large cardinals (for example, the existence of a super-compact cardinal) that filter bases whose underlying filter is a projective subset of the Cantor set, have continuous coordinate functionals. To role of the use of large cardinals is to make the heuristic proof I outlined in my other answer work.

The filter of statistical convergence is actually $F_{\sigma \delta}$ (hence Borel, hence projective), so in a theory stronger than ZFC, the answer to your question is affirmative. We will upload the paper to the arXiv in the near future.

In a recent note with J. Swaczyna (arXiv:2005.04873), we proved that assuming the existence of certain large cardinals (for example, the existence of a super-compact cardinal) that filter bases whose underlying filter is a projective subset of the Cantor set, have continuous coordinate functionals. The role of large cardinals is to make the heuristic proof I outlined in my other answer work.

The filter of statistical convergence is actually $F_{\sigma \delta}$ (hence Borel, hence projective), so in a theory stronger than ZFC, the answer to your question is affirmative. We will upload the paper to the arXiv in the near future.