Minimal Volume‐Product in Banach Spaces with a 1‐Unconditional Basis
Minimal Volume‐Product in Banach Spaces with a 1‐Unconditional Basis
复制标题
具有 1-无条件基础的 Banach 空间中的最小体积积
DOI：
10.1112/jlms/s2-36.1.126
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发表时间：
1987
影响因子：
1.2
通讯作者：
S. Reisner
中科院分区：
文献类型：
--
作者：
S. Reisner
定理1的“仅当”部分，即HL空间满足（1）和（2）的事实是简单的，如前所述，可以在例如[7]中找到。证明满足（1）或（2）的具有1-uc基的空间必定是HL空间构成了本文的其余部分。证据有两个主要成分，在第2节和第3节中描述。两个引理的证明被推迟到第四节，以便不打断证明的主线。在这一节中，我们仔细检查[7]中不等式（1.2）的证明（第一节中的符号（1.2）代表'（2）\类似的符号在后续中使用）。有两个不等式出现在这个证明的过程中，原来是关键点;实现平等的两个迫使空间是HL空间。因此，我们必须对[7]中的证明作一个简要的回顾，并找出这些关键点。注意，我们关于单位向量基是E中的1-uc基的协议保证了B（E）和B（E*）都包含ε（f？）并包含在£（/£）中。文[7]中的证明是从公式开始的
The'only if part of Theorem 1, that is, the fact that HL spaces satisfy (1) and (2) is simple and, as mentioned before, can be found, for example, in [7]. The proof that spaces with 1-uc basis which satisfy (1) or (2) must be HL spaces constitutes the rest of this paper. There are two main ingredients of the proof, which are described in Sections 2 and 3. The proof of two lemmas is postponed to Section 4 in order not to interrupt the main line of proof.In this section we make a close check of the proof in [7] of our inequality (1.2)(the notation (1.2) stands for'(2) in Section 1\similar notation is used in the sequel). There are two inequalities which appear in the course of this proof that turn out to be the critical points; fulfilment of equality in both compels the space to be an HL space. We must therefore make a brief review of the proof in [7] and locate these critical points. Note that our agreement that the unit vector basis is a 1-uc basis in E guarantees that both B (E) and B (E*) contain£(/?) and are contained in£(/£). The proof in [7] starts from the formula