Non-Baire hyperplanes in non-separable Baire spaces

AbstractGiven an infinite-dimensional Banach space E, one may ask: Does E have (1) a properly separable quotient, (2) a dense non-Baire hyperplane? [Every closed hyperplane in a Baire space is Baire.] The famous separable quotient problem (1) remains unsolved, and question (2) is question 13.1.1 of P. Pérez Carreras and J. Bonet (“North-Holland Math. Stud.,” Vol. 131, North-Holland, Amsterdam, 1987). In 1966 Wilansky-Klee conjectured the answer to (2) is always “no”; J. Arias de Reyna denied the conjecture (Math. Ann. 249, 1980, 111–114), proving the answer to (2) is “yes” whenever the answer to (1) is “yes,” under assumption of c-additivity (c-A), a condition weaker than Martin's Axiom. M. Valdivia extended the result to E a Baire (2nd category in itself) topological vector space (Collect. Math. 34, 1983, 287–296). In general, (2) is non-trivial only in case E is Baire with E′ ≠ E∗. An extension of Valdivia's extension yields the complete answer for (2) in the general locally convex setting, affirming question 13.1.1 of Pérez Carreras and Bonet (referenced above) in particular: A [not necessarily] Hausdorff locally convex space E has a dense non-Baire hyperplane if and only if [dim(E′) = ∞ and] E′ ≠ E∗. Even for E non-locally convex, (dim(E′) = ∞ and E′ ≠ E∗) suffices, but is not necessary, whereas E′ ≠ E∗ is obviously always necessary in order that E have a dense non-Baire hyperplane. Whether E′ ≠ E∗ suffices in the Hausdorff non-locally convex case, and whether the assumption of c-A can be omitted, we do not know. Contrastingly, any infinite-dimensional E is the union of a sequence of hyperplanes, and if E is Baire, one of the hyperplanes must be dense and Baire; and every hyperplane of E must be unordered Baire-like if E is