# Largest/smallest cross-norm: A simple question about cross-norms on tensor products of Banach spaces.

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?

• 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 $\sum_{i=1}^na_i\otimes b_i$. Mar 13, 2014 at 13:37
• 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. Mar 14, 2014 at 2:37