Authors: Ling Xin Bao Li Xin Cheng Qing Jin Cheng Duan Xu Dai
Publish Date: 2013/10/15
Volume: 29, Issue: 11, Pages: 2037-2046
Abstract
Let X Y be two real Banach spaces and ɛ ≥ 0 A map f X → Y is said to be a standard ɛisometry if ‖fx − fy‖ − ‖x − y‖ ≤ ɛ for all x y ∈ X and with f0 = 0 We say that a pair of Banach spaces X Y is stable if there exists γ 0 such that for every such ɛ and every standard ɛisometry f X → Y there is a bounded linear operator TLf equiv overline span fX to X so that ‖Tfxtx‖ ≤ γɛ for all x ∈ X XY is said to be universally leftstable if X Y is always stable for every Y X In this paper we show that if a dual Banach space X is universally leftstable then it is isometric to a complemented wclosed subspace of ℓ ∞Γ for some set Γ hence an injective space and that a Banach space is universally leftstable if and only if it is a cardinality injective space and universally leftstability spaces are invariant
Keywords: