Assume that X and Y are real Banach spaces with the same finite dimension. In this paper we show that if a standard coarse isometry f: X → Y satisfies an integral convergence condition or weak stability on a basis, then there exists a surjective linear isometry U: X → Y such that ‖f (x) − Ux‖ = o(‖x‖) as ‖x‖ → ∞. This is a generalization about the result of Lindenstrauss and Szankowski on the same finite dimensional Banach spaces without the assumption of surjectivity. As a consequence, we also obtain a stability result for ε-isometries which was established by Dilworth.

中文翻译：

---

有限维巴拿赫空间之间的粗等距

假设X和Y是具有相同有限维数的实 Banach 空间。在本文中，我们证明，如果一个标准的粗等距f : X → Y在基础上满足积分收敛条件或弱稳定性，则存在一个满射线性等距U : X → Y使得 ‖ f ( x ) − Ux ‖ = o (‖ x ‖) as ‖ x‖ → ∞。这是对 Lindenstrauss 和 Szankowski 在相同有限维 Banach 空间上的结果的推广，没有假设满射性。因此，我们还获得了Dilworth 建立的ε等距的稳定性结果。