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This is a very simple dumb question as I’m completely new to the topic. I was reading Wikipedia’s entry on “Topological Tensor Products”, and there’s one thing I’m confused about.

Let $ A $ and $ B $ be Banach spaces, and let $ A \odot B $ denote the algebraic tensor product of $ A $ and $ B $. A norm $ \rho $ on $ A \odot B $ is said to be a cross-norm if $$ \forall a \in A, ~ \forall b \in B: \quad \rho(a \otimes b) = \| a \| \| b \| \quad \text{and} \quad \rho'(a' \otimes b') = \| a' \| \| b' \|. $$ The entry then says, “there is a largest cross-norm $ \pi $...” and I stopped right there.

First off, how can there be a “largest cross-norm”? If I have two cross-norms $ \rho_{1} $ and $ \rho_{2} $ on $ A \odot B $, aren’t they the same since $$ \forall a \in A, ~ \forall b \in B: \quad {\rho_{1}}(a \otimes b) = \| a \| \| b \| = {\rho_{2}}(a \otimes b), $$ hence $ \rho_{1} = \rho_{2} $? What am I missing here?

Please be nice. :)

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  • $\begingroup$ In what you have written (which is correct) you have only considered elementary tensors; what is missing is that two cross norms need not coincide on a sum $\sum_{i=1}^na_i\otimes b_i$. $\endgroup$ Mar 13, 2014 at 13:37
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    $\begingroup$ Oh I get it, sorry, all the cross norms are the same on elementary tensors, an element $u$ of the algebraic tensor product $A \otimes B$ is a finite linear combination of elementary tensors. $\endgroup$
    – The K
    Mar 14, 2014 at 2:37

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