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We prove that if M is an infinite complete metric space, then the set of strongly norm-attaining Lipschitz functions SNA(M) contains a linear subspace isomorphic to c0. This solves an open question posed by V. Kadets and Ó. Roldán.

Given a Banach space X⁠, we say that a sequence {xn} in the unit ball of X is L-orthogonal if ∥x+xn∥→1+∥x∥ for every x∈X⁠. On the other hand, an element x∗∗ in the bidual sphere is said to be L-orthogonal (to X⁠) if ∥x+x∗∗∥=1+∥x∥ for every x∈X⁠. The aim of this paper is to clarify the relation between L-orthogonal sequences and L-orthogonal elements. Namely, we study whether every L-orthogonal sequence contains L-orthogonal elements in its weak*-closure. We provide an affirmative answer whenever the ambient space has small density character. Nevertheless, we show that, surprisingly, the general answer is independent of the usual axioms of set theory.

A recent result of T. Abrahamsen, P. Hájek and S. Troyanski states that a separable Banach space is almost square if and only if there exists h ∈SX∗∗∗∗such that ||x +h|| =max{||x||, 1}for all x ∈X. The proof passes through a sequential version of being almost square which we call being sequentially almost square. In this article we study these conditions in the nonseparable setting. On one hand, we show that a Banach space Xcontains a copy of c0if and only if there exists an equivalent renorming | ·|on Xfor which there exists h ∈SX∗∗∗∗such that |x +h| =max{|x|, 1}for every x ∈X. On the other hand, although it is unclear whether the aforementioned result of T. Abrahamsen et al. holds in the nonseparable setting, we show that, under the existence of selective ultrafilters, if Xis a sequentially almost square Banach space, then there exists h ∈SX∗∗∗∗such that ||x +h|| =max{||x||, 1}for all x ∈X.

A well-known classical result states that c0is linearly embeddable in a Banach lattice if and only if it is lattice embeddable. Improving results of H.P. Lotz, H.P. Rosenthal and N. Ghoussoub, we prove that C[0, 1] shares this property with c0. Furthermore, we show that any infinite-dimensional closed sublattice of C[0, 1] is either lattice isomorphic to c0 or contains a closed sublattice isomorphic to C[0, 1]. As a consequence, it is proved that for a separable Banach lattice X the following conditions are equivalent: (1) X is linearly embeddable in a Banach lattice if and only if it is lattice embeddable; (2) X is lattice embeddable into C[0, 1].