  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.
