This is a follow-up question to https://math.stackexchange.com/a/4872343

Let $X$ and $Y$ two Banach spaces and let $X\otimes Y$ their tensor product. Let $A(u)$ be the collection of all finite sets of simple tensors $\{x_1\otimes y_1,\dots ,x_n\otimes y_n\}$ such that:

  1. $u=\sum_{i=1}^n x_i\otimes y_i,$

  2. There is no subset of $\{x_1\otimes y_1,\dots ,x_n\otimes y_n\}$ such that the sum of its elements is a simple tensor.

Is it true that for every $u\in X\otimes Y$ we have $\max\{\text{card}\, s:s\in A(u)\}<\infty$?



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