Śliwa’s strongly normal subsequence question

Śliwa’s question asks whether every normal sequence in a dual Banach space contains a strongly normal subsequence. A survey identifies it as Problem 7 and records positive answers for important classes, but gives no date for the original question.

Known results

• Every infinite-dimensional WCG Banach space has the required subsequence property.
• Every Banach space with density character $d(X)<\mathfrak b$ has a separable quotient, yielding another positive range.

September 2026 construction

A September 2026 arXiv preprint by Jerzy Kąkol claims a ZFC negative result for cardinalities at least the continuum, while the case $\aleph_1$ is independent of ZFC and an intermediate range remains open. Thus it gives substantial progress, not a full classification.

Current status (as of September 2026): Positive cases are known, while the new construction claims failure at cardinalities at least the continuum; the $\aleph_1$ case is claimed independent of ZFC and an intermediate range remains open.