multilineares

Multilineares is a term used in mathematics to refer to objects that exhibit multilinearity. In this context, a multilinear map or form is a function that is linear in each argument separately.

Definition and basic idea

Let V1, V2, ..., Vk and W be vector spaces over a field F. A map f: V1 × V2 × ... × Vk → W is multilinear if, for every i from 1 to k, holding all other arguments fixed, the function that varies the i-th argument is linear. That is, for all vi, wi in Vi and scalars α in F, f(v1, ..., αvi + v'i, ..., vk) = α f(v1, ..., vi, ..., vk) + f(v1, ..., v'i, ..., vk).

Examples and special cases

A bilinear form is a multilinear map with k = 2, such as the dot product on R^n or any matrix-induced form. An alternating (or skew-symmetric) multilinear form changes sign under odd permutations of its arguments. The determinant is a classic example of an alternating multilinear form on a vector space.

Relation to tensors

The space of all multilinear maps from V1 × ... × Vk to W is denoted Mult(V1, ..., Vk; W). There is a natural isomorphism Mult(V1, ..., Vk; W) ≅ Hom(V1 ⊗ ... ⊗ Vk, W) via the tensor product, which provides a coordinate-free framework for multilinear objects.

Applications

Multilinear maps appear prominently in multilinear algebra, tensor theory, differential geometry, and the theory of differential forms. They underpin constructions such as tensor products, the wedge product, and various invariants in geometry and physics.

Notes

Symmetry properties (symmetry, invariance under permutations) and alternation play important roles in classifying multilinear forms and in defining associated structures, such as tensors and differential forms.