A universal theorem for stability of ε-isometries of Banach spaces

Let X, Y   be two Banach spaces, and f:X→Yf:X→Y be a standard ε  -isometry for some ε≥0ε≥0. In this paper, we show the following sharp weak stability inequality of f  : for every x⁎∈X⁎x⁎∈X⁎ there exists ϕ∈Y⁎ϕ∈Y⁎ with ‖ϕ‖=‖x⁎‖≡r‖ϕ‖=‖x⁎‖≡r such that|〈x⁎,x〉−〈ϕ,f(x)〉|≤2εrfor allx∈X. It is not only a sharp quantitative extension of Figiel's theorem, but it also unifies, generalizes and improves a series of known results about stability of ε-isometries. For example, if the mapping f   satisfies C(f)≡co¯[f(X)∪−f(X)]=Y, then it is equivalent to the following sharp stability theorem: There is a linear surjective operator T:Y→XT:Y→X of norm one such that ‖Tf(x)−x‖≤2ε‖Tf(x)−x‖≤2ε, for all x∈Xx∈X; When the ε-isometry f   is surjective, it is equivalent to Omladič–Šemrl's theorem: There is a surjective linear isometry U:X→YU:X→Y so that‖f(x)−Ux‖≤2ε,for allx∈X.