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. :)