Cantor–Bernstein theorems for certain symmetric bases in Banach spaces

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• 作者：Marcos J. González
• 关键词：Banach spaces ; Pełczyński’s decomposition method ; Cantor–Bernstein theorems ; Symmetric bases ; Banach sequence spaces ; Orlicz and Lorentz functions ; Geometrically convex ; Geometrically concave
• 刊名：Archiv der Mathematik
• 出版年：2015
• 出版时间：November 2015
• 年：2015
• 卷：105
• 期：5
• 页码：425-433
• 全文大小：474 KB
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• 作者单位：Marcos J. González (1)
  
  1. Departamento de Matemáticas, Universidad Simón Bolívar, Caracas, 1080-A, Venezuela
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Birkh盲user Basel
• ISSN：1420-8938

文摘

We prove the following Cantor–Bernstein type theorem, which applies well to the class of symmetric sequence spaces studied earlier by Altshuler, Casazza, and Lin: Let X and Y be Banach spaces having symmetric bases (x _n ) and (y _n ), respectively. If each of the bases (x _n ) and (y _n ) is equivalent to a basic sequence generated by one vector of the other, then the spaces X and Y are isomorphic. As a consequence, we obtain the strong equivalence that two Lorentz sequence spaces have the same linear dimension if and only if they are isomorphic.