A note on the stability of nonsurjective ε-isometries of Banach spaces.

Let X, Y be Banach spaces, and let ƒ : X → Y be an ε-isometry with ƒ(0) = 0 for some ε ≥ 0. In this paper, we show that for every x* ∈ X*, there exists φ ∈ Y* with ||φ|| = ||x*|| ≡ r such that |<x*,x>| − <φ,ƒ(x)>| ≤ 3rε, ∨x ∈ X. Moreover, the constant ''3'' is the best one in general. [ABSTRACT FROM AUTHOR]