c © 2003 Universitat de Barcelona Fully absolutely summing and Hilbert-Schmidt multilinear mappings

The space of the fully absolutely (r; r1,..., rn)-summing n-linear mappings between Banach spaces is introduced along with a natural (quasi-)norm on it. If r, rk ∈ [1,+∞], k = 1,..., n, this space is characterized as the topological dual of a space of virtually nuclear mappings. Other examples and properties are considered and a relationship with a topological tensor product is stablished. For Hilbert spaces and r = r1 =... = rn ∈ [2,+∞ [ this space is isomorphic to the space of the Hilbert-Schmidt multilinear mappings. 1