文摘
Let X and Y be two Banach spaces, and f: X → Y be a standard ε-isometry for some ε ≥ 0. In this paper, by using a recent theorem established by Cheng et al. (2013–2015), we show a sufficient condition guaranteeing the following sharp stability inequality of f: There is a surjective linear operator T: Y → X of norm one so that $$\left\| {Tf(x) - x} \right\| \leqslant 2\varepsilon , for all x \in X.$$As its application, we prove the following statements are equivalent for a standard ε-isometry f: X → Y:(i)lim inft→∞ dist(ty, f(X))/|t| < 1/2, for all y ∈ SY;