Erratum to: Every Banach space admits a homogenous measure of non-compactness not equivalent to the Hausdorff measure

Abstract

1, Theorem 4.4] states that every infinite dimensional Banach space admits a homogenous measure of noncompactness not equivalent to the Hausdorff measure. Howevere, there is a gap in the proof. In fact, we found that [1, Lemma 4.3] is not true. In this erratum, we give a corrected proof of [1, Theorem 4.4].For a Banach space X, let C (X) (resp. B(X), K (X)) be the collection of all non-empty bounded closed convex (resp. nonempty bounded, nonempty convex compact) sets of X endowed with the Hausdorff metric. If there is no confusion, we simply denote them by C , B and K , respectively. We use Ω to denote the closed unit ball B X * of the dual X * , and C b (Ω), the Banach space of all real-valued bounded norm-continuous functions on Ω endowed with the sup-norm. Let J :With the symbols as above, we first recall the main results presented in [2], which will be used in the proof. Theorem 1. (1) (See [2, Theorem 2.3i)]) Given a Banach space X, the collection C consisting of all nonempty closed bounded convex sets of X endowed with the set addition A ⊕ B = A + B, the usual scalar multiplication of sets λC = {λc : c ∈ C}, and the norm ||| · ||| defined by |||C||| = sup c∈C ∥c∥ is a complete normed convex cone. (2) (See [2, Theorem 2.3ii)]) If we endow with the Hausdorff metric d H on C , i.e., d H (A, B) = max { sup a∈A inf b∈B ∥a − b∥, sup b∈B inf a∈A ∥b − a∥ }