2021/04/12 by Antonio Avilés, Avilés, Antonio, Gonzalo Martínez-Cervantes +3 · 1 citation
Computer Science · Mathematics · #46B04 (Primary) #46B20 #46B26 #54A20 (Secondary) #Advanced Banach Space Theory #Digital Image Processing Techniques #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2104.05535
openalex publication_date 2021/04/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a Banach space X, we say that a sequence \xn\ in the unit ball of X is L-orthogonal if \Vert x+xn\Vert→ 1+\Vert x\Vert 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+\Vert x\Vert for every x∈ X. A result of V. Kadets, V. Shepelska and D. Werner asserts that a Banach space contains an isomorphic copy of ℓ1 if and only if there exists an equivalent renorming with an L-orthogonal sequence, whereas a result of G. Godefroy claims that containing an isomorphic copy of ℓ1 is equivalent to the existence of an equivalent renorming with L-orthogonals in the bidual. 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. We also prove that, even though the set of L-orthogonals is not a vector space, this set contains infinite-dimensional Banach spaces when the surrounding space is separable.