Ling Xin BAO, Li Xin CHENG, Qing Jin CHENG, Duan Xu DAI
Let X, Y be two real Banach spaces and ε ≥ 0. A map f: X→Y is said to be a standard ε-isometry if |||f(x)-f(y) - x - y||| ≤ ε for all x, y ∈ X and with f(0) = 0. We say that a pair of Banach spaces (X, Y) is stable if there exists γ > 0 such that, for every such ε and every standard ε-isometry f : X → Y , there is a bounded linear operator T : L(f) ≡ span f(X) → X so that Tf(x)-x ≤ γε for all x ∈ X. X(Y) is said to be universally left-stable if (X, Y) is always stable for every Y (X). In this paper, we show that if a dual Banach space X is universally left-stable, then it is isometric to a complemented w*-closed subspace of l∞(Γ) for some set Γ, hence, an injective space; and that a Banach space is universally left-stable if and only if it is a cardinality injective space; and universally left-stability spaces are invariant.
