Let X, Y be two Banach spaces, \(f:X\rightarrow Y\) be an \(\varepsilon \)-isometry with \(f(0)=0\) for some \(\varepsilon \ge 0\), and let \(Y_f\equiv \overline{\mathrm{span}}f(X)\). In this paper, we first introduce a notion of \(w^*\)-stability of an \(\varepsilon \)-isometry f. Then we show that stability of f implies its \(w^*\)-stability; the two notions of stability and \(w^*\)-stability coincide whenever X is a dual space and they are not equivalent in general. Making use of a recent sharp weak stability estimate of f, we then improve some known results.

让X，  ÿ是两个Banach空间，\（F：X \ RIGHTARROW Y \）是\（\ varepsilon \） -isometry与\（F（0）= 0 \）对于一些\（\ varepsilon \ GE 0 \ ），然后让\（Y_f \ equiv \ overline {\ mathrm {span}} f（X）\）。在本文中，我们首先介绍的概念\（W ^ * \）的的-稳定性\（\ varepsilon \） -isometry ˚F。然后，我们证明f的稳定性意味着其\（w ^ * \）-稳定性；每当X时，稳定性和\（w ^ * \） -稳定性的两个概念就重合了。是对偶空间，通常它们并不等效。利用最近对f的尖锐弱稳定性估计，我们可以改善一些已知的结果。