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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 $u\ne 0$?

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

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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.

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