Unconditional almost squareness and applications to spaces of Lipschitz functions

Abstract

Abstract. We introduce an unconditional concept of almost squareness in order to provide a partial negative answer to the problem of existence of any dual almost square Banach space. We also take advantage of this notion to provide some criterion of non-duality of some subspaces of scalar as well as vector valued Lipschitz functions.

On the structure of spaces of vector-valued Lipschitz functions

“…This would have two implications. On the one hand, given a Banach space X such that F(M ) or X * has (AP), lip τ (M, X) = lip τ (M ) ⊗ ε X would be an unconditionally almost square Banach space [11] and, consequently, it would not be a dual Banach space, extending Proposition 2.11. On the other hand, such a metric space M would satisfy the thesis of Proposition 4.1.…”

“…Now, by Theorem 2.8, we have that lip ω, * (B X * , Y ) is a subspace of K(Y * , lip ω, * (B X * )) which clearly contains lip ω, * (B X * ) ⊗ Y . Proposition 2.7 in [11] provides unconditional almost squareness of lip ω, * (B X * , Y ). Finally, the non-duality of this space follows from Theorem 2.5 in [11].…”

“…Proposition 2.7 in [11] provides unconditional almost squareness of lip ω, * (B X * , Y ). Finally, the non-duality of this space follows from Theorem 2.5 in [11].…”

“…Finally, we will take advantage of the notion of unconditional almost squareness, introduced in [11], in order to prove non duality of the space lip ω, * (B X * , Y ) under above hypotheses. According to [11], a Banach space X is said to be unconditionally almost square (UASQ) if, for each ε > 0, there exists a subset {x γ } γ∈Γ ⊆ S X such that It is known that there is not any dual UASQ Banach space [11, Theorem 2.5].…”

“…First we prove that lip ω, * (B X * ) is UASQ. By [11,Proposition 3.3], it suffices to show that there exists a point x * 0 ∈ B X * and sequences r n of positive numbers and f n ∈ lip ω, * (B X * ) such that f n = 0, f n (x * 0 ) = 0 and f n vanishes out of B(x * 0 , r n ). Let x * 0 ∈ S X * be a continuity point of the identity I : (B X * , w * ) → (B X * , · ), that is, x * 0 has relative weakstar neighbourhoods of arbitrarily small diameter.…”