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## injective tensor norms for real tensors

If $A$ is an element of $\mathbb{R}^n \otimes\mathbb{R}^n \otimes\mathbb{R}^n$, then define its injective tensor norm to be $$\|A\|_{\rm inj} := \max_{x,y,z\in \mathbb{R}^n, \|x\|=\|y\|=\|z\|=1} |\langle A, x\otimes y\otimes z\rangle|.$$ Here the norm on vectors is the usual Euclidean norm.

I have two questions.

1. Does this norm change if x,y,z can be taken to arbitrary complex unit vectors?
2. What if $A$ is symmetric under exchange of the first two positions? That is,
$\langle A, x\otimes y\otimes z\rangle = \langle A, y\otimes x\otimes z\rangle$ for all $x,y,z$.
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I'm having trouble getting the  equation to work. – aram Sep 30 2011 at 23:01
Fixed your LaTeX and added the "banach-spaces" tag – Yemon Choi Sep 30 2011 at 23:08
As clarification: what exactly is the second question? Is it whether the norm changes if $A$ is symmetric in two factors and the vectors are allowed to be complex (as in the first question)? – ARupinski Sep 30 2011 at 23:22
Yes. Sorry for being unclear. – aram Oct 1 2011 at 0:58