Abstract In the first part of our note we prove that every Weakly Lindelof Determined (WLD) (in particular, every reflexive) non-separable Banach X space contains two dense linear subspaces Y and Z that are not densely isomorphic. This means that there are no further dense linear subspaces Y 0 and Z 0 of Y and Z which are linearly isomorphic. Our main result ( Theorem B ) concerns the existence of biorthogonal systems in normed spaces. In particular, we prove under the Continuum Hypothesis (CH) that there exists a dense linear subspace of l 2 ( ω 1 ) (or more generally every WLD space of density ω 1 ) which contains no uncountable biorthogonal system. This result lies between two fundamental results concerning biorthogonal systems, namely the construction of Kunen (under CH) of a non-separable Banach space which contains no uncountable biorthogonal system, and the construction of Todorcevic (under Martin Maximum) of an uncountable biorthogonal system in every non-separable Banach space.

中文翻译：

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在稠密同构赋范空间上

摘要 在我们笔记的第一部分中，我们证明了每个弱林德洛夫确定 (WLD)（特别是每个自反）不可分 Banach X 空间包含两个非密集同构的密集线性子空间 Y 和 Z。这意味着不再有线性同构的 Y 和 Z 的稠密线性子空间 Y 0 和 Z 0 。我们的主要结果（定理 B）涉及规范空间中双正交系统的存在。特别地，我们在连续统假设 (CH) 下证明存在一个稠密线性子空间 l 2 ( ω 1 )（或更一般地说，每个密度为 ω 1 的 WLD 空间）不包含不可数的双正交系统。这个结果介于关于双正交系统的两个基本结果之间，