We consider the following problem (which is a generalisation of a folklore result Proposition 1 below): given a continuous linear operator \(T:X\rightarrow Y\), where Y is a Banach space with a (long) sub-symmetric basis, under which conditions can we find a continuous linear operator \(S:X\rightarrow Y\) such that \(S(B_X)\) contains the basis of Y. As a tool we also consider a non-separable version of Theorem 2 below: Given an infinite subset \(A\subset X^*\), under which conditions can we find a biorthogonal system in \(X\times A\) of cardinality \({{\,\mathrm{card}\,}}A\)?

中文翻译：

---

关于双正交系统的注释

我们考虑以下问题（以下是民俗结果命题1的推广）：给定连续线性算子\（T：X \ rightarrow Y \），其中Y是具有（long）次对称基础的Banach空间，在这种情况下，我们可以找到一个连续的线性算子\（S：X \ rightarrow Y \）使得\（S（B_X）\）包含Y的基。作为工具，我们还考虑以下定理2的不可分版本：给定无限子集\（A \ subset X ^ * \），在这种情况下，我们可以在\（X \ times A \）中找到双正交系统基数\（{{\，\ mathrm {card} \，}} A \）吗？