multilinear form

Noun

 * 1)  Given a vector space V over a field K of scalars, a mapping Vk → K that is linear in each of its arguments;  a similarly multiply linear mapping Mr → R defined for a given module M over some commutative ring R.
 * 2) * 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.
 * 1) * 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, 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 $$M\!f$$.

Usage notes

 * A multilinear form $$V^k\rightarrow K$$ (which has $$k$$ variables) is called a multilinear $$k$$-form.
 * A multilinear $$k$$-form on $$V$$ over $$\mathbb{R}$$ is called a $$k$$-, and the vector space of such forms is usually denoted $$\mathcal{T}^k(V)$$ or $$\mathcal{L}^k(V)$$. (But note that many authors use an opposite convention, writing $$\mathcal{T}^k(V)$$ for the contravariant $$k$$-tensors on $$V$$ and $$\mathcal{T}_k(V)$$ for the covariant ones.)
 * A multilinear form differs from a in that the former maps to a field of scalars, whereas the latter maps, in the general case, to a cross product of vector spaces.

Derived terms

 * , multilinear k-form

Translations

 * French: forme multilinéaire
 * German: Multilinearform
 * Spanish: forma multilineal, función multilineal