Multilinear form

English

Noun

1. Template:Lb Given a vector space V over a field K of scalars, a mapping V ^k → K that is linear in each of its arguments;
   Template:Lb a similarly multiply linear mapping M ^r → R defined for a given module M over some commutative ring R.
   • 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, Template:W 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 $Mf$.

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Usage notes

• A multilinear form $V^k\to K$ (which has $k$ variables) is called a multilinear $k$-form.
• A multilinear $k$-form on $V$ over $ℝ$ is called a (Template:M) $k$-Template:M, and the vector space of such forms is usually denoted ${𝒯}^k(V)$ or ${ℒ}^k(V)$. (But note that many authors use an opposite convention, writing ${𝒯}^k(V)$ for the contravariant $k$-tensors on $V$ and ${𝒯}_k(V)$ for the covariant ones.)
• A multilinear form differs from a Template:L 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.

Synonyms

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Hyponyms

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Derived terms

• Template:L, multilinear k-form

Translations

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See also

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Further reading

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