Abstract: The space of the strictly absolutely (s; r1,,T)-summing multilinear map- pings between Banach spaces is introduced along with a natural (quasi-) norm on it. For Hilbert spaces and sr... r E [2, +00) it is shown that such space is isomorphic to the space of the Hilbert-Schmidt multilinear mappings. If s, r E [1, +oo], k = 1,...,n this space is characterized as the topological dual of a space of quasi-nuclear mappings. Other properties are considered and a relationship with a topological tensor product is stablished Ver menos