I aksed this question on Math Stack Exchange 6 days ago, with no answer: https://math.stackexchange.com/q/4875445/1297919

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:

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

There is no subset with at least two elements 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 $\max\{\text{card}\, s:s\in A(u)\}<\infty$?