  A tensor norm $\beta= (\beta_{n})_{n=1}^{\infty}$ is smooth if the natural
correspondence $(E_{1} \otimes\cdots\otimes E_{n}
\otimes\mathbb{K},\beta_{n+1}) \longleftrightarrow (E_{1} \otimes\cdots\otimes
E_{n} ,\beta_{n})$ is always an isometric isomorphism. In this paper we study
the representation of multi-ideals and of ideals of multilinear forms by smooth
tensor norms.
