4.7 Tensor Multilinear Domain Structure

Tensor Multilinear Domain Structure is the overall organization of the input side of a tensor's defining map: the specific arrangement of vector spaces, their variance types, and their combination via the Cartesian product that together form the set on which a multilinear map is defined. It is the umbrella structure encompassing the argument slot structure, the domain product, and the variance labeling of each factor, and it is what must be specified before any statement about multilinearity can even be made.

Formal Definition

Composing the Domain from Its Factors

The multilinear domain structure of a tensor of type $(r, s)$ built from a vector space $V$ is the Cartesian product

This structure is more than a bare set: it carries the ordering of its factors, the variance (dual or primal) assigned to each factor, and the dimension inherited from $V$, all of which together determine what kinds of multilinear maps can be defined on it and how those maps transform under change of basis.

Distinguishing Domain Structure from the Map Itself

The domain structure is prior to, and independent of, any particular multilinear map defined on it. Many different tensors, with entirely different output values, can share the exact same multilinear domain structure; the domain structure only fixes the "shape" of admissible inputs, not the rule assigning outputs to them.

Components of the Domain Structure

Factor Spaces and Their Variance

Each factor in the product is either the base space $V$ (a covariant factor) or its dual $V^{*}$ (a contravariant factor). This variance labeling is what allows the domain structure to encode the type $(r,s)$ of any tensor defined on it, and it governs how each factor individually transforms under a change of basis of $V$.

Order of Factors

The factors appear in a fixed sequence, and this sequence matters whenever the multilinear map defined on the domain lacks symmetry: reordering the factors of the domain structure corresponds to reordering the argument slots of any map defined on it, generally yielding a structurally different, though related, multilinear map.

Product, Not Sum

The domain structure is built using the Cartesian product of its factors, not their direct sum, since multilinear maps are only required to be linear one factor at a time. This choice of product over sum is the structural reason multilinear maps can exhibit the characteristic degree-$k$ homogeneity under simultaneous scaling of all arguments, rather than the degree-one homogeneity of an ordinary linear map on a direct sum.

Role in the Broader Tensor Formalism

Foundation for the Universal Property

The tensor product space is constructed as the universal object receiving multilinear maps out of a given domain structure: every multilinear map on the domain structure factors uniquely through the corresponding tensor product. The domain structure is therefore the input specification that the universal property of the tensor product is stated relative to.

Consistency Requirement for Change of Basis

Because every factor of the domain structure has a fixed variance, a coordinated change of basis of $V$ induces a predictable, opposite transformation on the dual factors and a matching transformation on the primal factors. The internal consistency of the domain structure, specifically that dual and primal factors transform inversely to one another, is what guarantees that any multilinear map defined on it transforms as a genuine tensor under change of basis, rather than as an arbitrary basis-dependent array of numbers.

Basis for Constructing Related Structures

The domain structure also underlies the definitions of symmetric and antisymmetric tensor spaces, since symmetrization and antisymmetrization are operations that act on maps defined over a domain structure by exploiting permutations among factors sharing the same variance; a domain structure with mixed variance naturally supports independent symmetrization within its dual block and within its primal block.

Summary of Key Points

• The multilinear domain structure specifies the ordered, variance-labeled Cartesian product of factor spaces on which a tensor's defining map is built.
• It is prior to and independent of any specific multilinear map, fixing only the shape of admissible inputs.
• Variance labeling of each factor determines the tensor type $(r,s)$ of any map defined on the structure.
• Using the Cartesian product rather than the direct sum is what produces the characteristic degree-$k$ homogeneity of multilinear maps.
• The domain structure underlies the universal property of the tensor product and guarantees consistent transformation behavior under change of basis.