# 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 $$u\ne 0$$?

• I am not sure I understand what is meant by the set $S_k$. Can you add a description? Commented Mar 19 at 22:48
• @WillieWong Clarified, Thanks! Commented Mar 20 at 9:57
• What is $S_k$? It's still not clear. Commented Mar 26 at 11:20
• In Q2, you mean $\inf_๐ผ๐(๐^\alpha)=0$ for some nonzero element of $๐\otimes ๐$, right? Otherwise it's trivial.
– Uagi
Commented Mar 26 at 12:15
• @Uagi yes of course, I corrected Commented Mar 26 at 14:34

Take $$X = Y= M_2$$ and consider the tensor $$e_1 \otimes e_1 + e_2 \otimes e_2$$. You get the same result for any choice of orthonormal basis of $$\mathbb{C}^2$$, so there can already be two expressions for a tensor that share no common simple tensors.