On stability of nonlinear non-surjective ε-isometries of Banach spaces

AbstractLet X, Y be two Banach spaces, ε⩾0, and let f:X→Y be an ε-isometry with f(0)=0. In this paper, we show first that for every x⁎∈X⁎, there exists ϕ∈Y⁎ with ‖ϕ‖=‖x⁎‖≡r such that|〈ϕ,f(x)〉−〈x⁎,x〉|⩽4εr,for all x∈X. Making use of it, we prove that if Y is reflexive and if E⊂Y [the annihilator of the subspace F⊂Y⁎ consisting of all functionals bounded on co¯(f(X),−f(X))] is α-complemented in Y, then there is a bounded linear operator T:Y→X with ‖T‖⩽α such that‖Tf(x)−x‖⩽4ε,for all x∈X. If, in addition, Y is Gateaux smooth, strictly convex and admitting the Kadec–Klee property (in particular, locally uniformly convex), then we have the following sharp estimate‖Tf(x)−x‖⩽2ε,for all x∈X