Combination of simple tensors - II

This is a follow-up question to Combination of simple tensors.

I am interested in devising an alternative norm (I mean, other than the usual $\pi$ or $\epsilon$ norms) in the tensor product of two Banach spaces.

Let $X$ and $Y$ be two Banach spaces and let $X\otimes Y$ their tensor product. Let $u\in X\otimes Y$ and $A(u)$ be the collection of all finite sets of simple tensors of type $S^\alpha=\{x_1^\alpha\otimes y_1^\alpha,\dots ,x_n^\alpha\otimes y_n^\alpha\}$ (where $\alpha$ belongs to a suitable, generally uncountable set of indices $I$) such that:

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

2. For every $\alpha\in I$, there is no subset with at least two elements of $\{x_1^\alpha\otimes y_1^\alpha,\dots ,x_n^\alpha\otimes y_n^\alpha\}$ such that the sum of its elements is a simple tensor.

Q1: Is it possible that $\cap_{\alpha\in I} S^\alpha=\emptyset$?

Q2: Is it possible that $\inf_{\alpha\in I}m(S^\alpha)= 0$, where $m(S^\alpha)$ is the largest product of type $||x_i^\alpha||_{X}||y_i^\alpha||_{Y}$ in each set $S^\alpha$ and assuming $u\ne 0$?