multilinear form

Noun

multilinear form (plural multilinear forms)

1. (multilinear algebra) Given a vector space V over a field K of scalars, a mapping Vᵏ → K that is linear in each of its arguments;
   • Finally, in the short Lecture 5 we make some remarks on multilinear forms over Hilbert spaces, a theory which is still in a rather embryonic state, motivated by the observation that paracommutators (and Hankel operators too) really should be viewed as forms, not operators.1985, Jack Peetre, Paracommutators and Minimal Spaces, S. C. Power (editor) Operators and Function Theory, Kluwer Academic (D. Reidel), page 163
   • A multilinear form is said to be degenerate if all its function variables are identical. Thus a degenerate m-multilinear form can more concisely be written as M!f.1994, Hessam Khoshnevisan, Mohamad Afshar, Mechanical Elimination of Commutative Redundancy, Baudouin Le Charlier (editor), Static Analysis: 1st International Static Analysis Symposium, Proceedings...
2. (broadly, multilinear algebra) a similarly multiply linear mapping Mʳ → R defined for a given module M over some commutative ring R.