Abstract In this paper we prove that every infinite-dimensional and separable Banach space ( X , ‖ ⋅ ‖ X ) admits an equivalent norm ‖ ⋅ ‖ X , 1 such that ( X , ‖ ⋅ ‖ X , 1 ) has both the Kadec-Klee and the Opial properties. This result also has a quantitative aspect and when combined with the properties of Schauder bases and the Day norm it constitutes a basic tool in the proof of our main theorem: each infinite-dimensional, reflexive and separable Banach space ( X , ‖ ⋅ ‖ X ) has an equivalent norm ‖ ⋅ ‖ 0 such that ( X , ‖ ⋅ ‖ 0 ) is LUR and contains a diametrically complete set with empty interior.

中文翻译：

---

在自反和可分离的 Banach 空间中存在具有空内部的径向完备集

摘要 在本文中，我们证明了每一个无限维可分的 Banach 空间 ( X , ‖ ⋅ ‖ X ) 都承认一个等价范数 ‖ ⋅ ‖ X , 1 使得 ( X , ‖ ⋅ ‖ X , 1 ) 同时具有 Kadec- Klee 和 Opial 属性。这个结果也有一个定量的方面，当结合 Schauder 基的性质和 Day 范数时，它构成了证明我们主要定理的基本工具：每个无限维、自反和可分离的 Banach 空间 ( X , ‖ ⋅ ‖ X ) 有一个等价范数 ‖ ⋅ ‖ 0 使得 ( X , ‖ ⋅ ‖ 0 ) 是 LUR 并且包含一个内部为空的径向完备集。