𝜔-independence in nonseparable Banach spaces

On the Kunen–Shelah properties in Banach spaces

“…The implication KS 4 ⇒ KS 3 was proved in [3]. It also follows from Proposition 3.2 and from Proposition 7.3 and a result of Sersouri [12]. 4 ; we will prove that X ∈ KS 4 .…”

“…If X has a τ -polyhedron {x α : α < τ }, it is clear that {C α : α < τ }, where C α = co({x β : α < β < τ }), is a contractive τ -onion. So, the property KS 5 implies KS 4 , whence by Proposition 3.2 we get KS 5 ⇒ KS 3 , a result proved by Sersouri in [12].…”

“…A family {x i } i∈I in a Banach space X is said to be ω-independent if for every sequence (i n ) n≥1 ⊂ I of distinct indices, and every sequence (λ n ) n≥1 ⊂ R, the series ∞ n=1 λ n x i n converges (in norm) to 0 iff λ n = 0 for every n ≥ 1 (see [6], [12]). A Banach space X is said to have the Kunen-Shelah property KS 3 if X has no uncountable ω-independent family.…”

“…Using the proof of [6,Th. 3], as in [12], we can construct inductively a sequence {ε n } n≥1 of signs, a sequence {λ n r } n≥1, 1≤r≤k(n) of real numbers and a sequence {γ n r } n≥1, 1≤r≤k(n) of ordinals such that:…”

ON $\omega$-INDEPENDENCE AND THE KUNEN–SHELAH PROPERTY

“…Spaces with the above property share, in a rather striking way, some of the features of separable spaces (see [3,4,8]). The Shelah space was constructed assuming the diamond principle for ω 1 and solved a problem by Davis and Johnson [9].…”