Separated sets and Auerbach systems in Banach spaces

Abstract

The paper elucidates the relationship between the density of a Banach space and possible sizes of Auerbach systems and well-separated subsets of its unit sphere. For example, it is proved that for a large enough space X, the unit sphere S X always contains an uncountable (1+)-separated subset. In order to achieve this, new results concerning the existence of large Auerbach systems are established, that happen to be sharp for the class of WLD spaces. In fact, we offer the first consistent example of a non-separable WLD Banach space that contains no uncountable Auerbach system, as witnessed by a renorming of c 0 (ω 1 ). Moreover, the following optimal results for the classes of, respectively, reflexive and super-reflexive spaces are established: the unit sphere of an infinite-dimensional reflexive space contains a symmetrically (1 + ε)-separated subset of any regular cardinality not exceeding the density of X; should the space X be superreflexive, the unit sphere of X contains such a subset of cardinality equal to the density of X. The said problem is studied for other classes of spaces too, including WLD spaces, RNP spaces, or strictly convex ones.

Embedding Banach spaces into the space of bounded functions with countable support

“…Remark Theorem A implies that the unit sphere of a non‐separable WLD subspace of contains an uncountable symmetrically (1 + )‐separated subset, that is, a set A such that for distinct ; this is because has this property. This observation complements [, Corollary 3.6], where it was proved that WLD spaces of density greater than the continuum contain such sets. It should be noted however that not every renorming of embeds isometrically into ; indeed any strictly convex renorming is a counterexample (such renormings do exist as proved by Day ).…”

Banach spaces in which large subsets of spheres concentrate

“…It has been clear, at least since the paper [32] of Mercourakis and Vassiliadis, that the nature of separation in uncountable subsets of the unit sphere of a nonseparable Banach space could be equally indicative of the global and diverse properties of the space like in the separable case. The question of whether the unit sphere of a nonseparable Banach space must contain an uncountable a (1+)-separated set, and if so, of what cardinality compared to the density of the space, has been studied for various classes of Banach spaces e.g., in [4,21,31,32] and recently culminated in the paper [16], where it is highlighted as a central question. Notably the existence of (1 + ε)-separated sets of the size equal to the density of the space was proved for super-reflexive Banach spaces by Kania and Kochanek in [21] and for C(K) spaces, where K is compact Hausdorff and totally disconnected by Mercourakis and Vassiliadis.…”

Large Banach spaces with no infinite equilateral sets

“…Thus equilateral sets are related to the questions concerning separation of points in the spheres of Banach spaces (see e.g. [8] for references). Recall that a subset Y of a Banach space X is called δ-separated if y − y ′ ≥ δ for all distinct y, y ′ ∈ Y.…”

“…It is called (δ+)-separated if y − y ′ > δ for all distinct y, y ′ ∈ Y. By Remark 3.16 [8] the unit sphere of every renorming of ℓ 1 ([0, 1]) contains a subset Y of cardinality continuum such that y − y ′ ≥ 1 + ε some ε > 0 and for every two distinct y, y ′ ∈ Y.…”

“…A close link between separated subsets in the sphere and Auerbach bases was demonstrated in [8]. In fact Godun's renorming of ℓ 1 ([0, 1]) was designed to prove that the space has no fundamental Auerbach system ( [6]).…”