injective tensor norms for real tensors - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T21:36:00Z http://mathoverflow.net/feeds/question/76894 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/76894/injective-tensor-norms-for-real-tensors injective tensor norms for real tensors aram 2011-09-30T22:59:24Z 2011-10-02T17:43:53Z <p>If $A$ is an element of $\mathbb{R}^n \otimes\mathbb{R}^n \otimes\mathbb{R}^n$, then define its <a href="http://en.wikipedia.org/wiki/Topological_tensor_product" rel="nofollow">injective tensor norm</a> to be <code>$$\|A\|_{\rm inj} := \max_{x,y,z\in \mathbb{R}^n, \|x\|=\|y\|=\|z\|=1} |\langle A, x\otimes y\otimes z\rangle|.$$</code> Here the norm on vectors is the usual Euclidean norm.</p> <p>I have two questions.</p> <ol> <li>Does this norm change if x,y,z can be taken to arbitrary <i>complex</i> unit vectors?</li> <li>What if $A$ is symmetric under exchange of the first two positions? That is,<br> $\langle A, x\otimes y\otimes z\rangle = \langle A, y\otimes x\otimes z\rangle$ for all $x,y,z$.</li> </ol> http://mathoverflow.net/questions/76894/injective-tensor-norms-for-real-tensors/76902#76902 Answer by aram for injective tensor norms for real tensors aram 2011-10-01T01:02:47Z 2011-10-01T05:47:50Z <p>I just found <a href="http://dmle.cindoc.csic.es/pdf/RRACEFN_2000_94_04_05.pdf" rel="nofollow">this paper</a>, which gives an example in which the real and complex version of the norm <i>are</i> different. The tensor in this example is also symmetric, which provides an example for part 2 as well.</p>