Meaning of multilinear form | Babel Free

Definitions

1. Given a vector space V over a field K of scalars, a mapping Vᵏ → K that is linear in each of its arguments;
2. Given a vector space V over a field K of scalars, a mapping Vᵏ → K that is linear in each of its arguments; (more generally) a similarly multiply linear mapping Mʳ → R defined for a given module M over some commutative ring R.
   broadly
3. a similarly multiply linear mapping Mʳ → R defined for a given module M over some commutative ring R.
   broadly

Examples

“1985, Jack Peetre, Paracommutators and Minimal Spaces, S. C. Power (editor) Operators and Function Theory, Kluwer Academic (D. Reidel), page 163, 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.”

“1994, Hessam Khoshnevisan, Mohamad Afshar, Mechanical Elimination of Commutative Redundancy, Baudouin Le Charlier (editor), Static Analysis: 1st International Static Analysis Symposium, Proceedings, Volume 1, Springer, LNCS 864, page 454, 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.”

“Since the coordinates of a vector change in transforming to a new basis, the same is true of the coefficients of a multilinear form (since the form itself is to remain invariant).”

CEFR level