The Segre cone of Banach spaces and multilinear operators

We prove that any pair of reasonable cross norms defined on the tensor product of $n$ Banach spaces induce $(2k)^{n-1}$-Lipschitz equivalent metrics (and thus, a unique topology) on the set $S^k_{X_1,\ldots, X_n}$ of vectors of rank $\leq k$. With this, we define the Segre cone of Banach spaces, $\Sigma_{X_1,\ldots, X_n},$ and state when $S^k_{X_1,\ldots, X_n}$ is closed. We introduce an auxiliary mapping (a $\Sigma$-operator) that allows us to study multilinear mappings with a geometrical point of view. We use the isometric correspondence between multilinear mappings and Lipschitz $\Sigma$-operators, to have a strategy to generalize ideal properties from the linear to the multilinear settitng.