  We develop the duality theory between ideals of multilinear operators and
tensor norms that arises from the geometric approach of $\Sigma$-operators. To
this end, we introduce and develop the notions of $\Sigma$-ideals of
multilinear operators and $\Sigma$-tensor norms. We establish the foundations
of this theory by proving a representation theorem for maximal $\Sigma$-ideals
of multilinear operators by finitely generated $\Sigma$-tensor norms and a
duality theorem for $\Sigma$-tensor norms. For these notions we also develop
basic theory and present concrete examples.

Functional Analysis