Suárez Granero, AntonioJiménez Sevilla, María Del MarMontesinos, AlejandroMoreno, José PedroPlichko, Anatolij2023-06-202023-06-2020030039-3223https://hdl.handle.net/20.500.14352/58609We introduce and study the Kunen-Shelah properties KSi, i = 0, 1,..., 7. Let us highlight for a Banach space X some of our results: (1) X ∗ has a w ∗-nonseparable equivalent dual ball iff X has an ω1-polyhedron (i.e., a bounded family {xi}i<ω1 such that xj / ∈ co({xi: i ∈ ω1 \ {j}}) for every j ∈ ω1) iff X has an uncountable bounded almost biorthonal system (UBABS) of type η, for some η ∈ [0, 1), (i.e., a bounded family {(xα, fα)}1≤α<ω1 ⊂ X × X ∗ such that fα(xα) = 1 and |fα(xβ) | ≤ η, if α = β); (2) if X has an uncountable ω-independent system then X has an UBABS of type η for every η ∈ (0, 1); (3) if X has not the property (C) of Corson, then X has an ω1-polyhedron; (4) X has not an ω1-polyhedron iff X has not a convex right-separated ω1-family (i.e., a bounded family {xi}i<ω1 such that xj / ∈ co({xi: j < i < ω1}) for every j ∈ ω1) iff every w ∗-closed convex subset of X ∗ is w ∗-separable iff every convex subset of X ∗ is w ∗-separable iff µ(X) = 1, µ(X) being the Finet-Godefroy index of X (see [1]).engjournal articlehttp://journals.impan.gov.pl/sm/index.htmlhttp://www.impan.pl/restricted access515.1Uncountable basic sequencesBiorthogonal and Markuschevich systemsW-independenceKunen-Shelah properties.Topología1210 Topología