4.1.1 Tensor Multilinear Map Domain Structure

The domain structure of a tensor multilinear map defines how multiple vector spaces interact through bilinear and multilinear operations in algebra.

Tensor Multilinear Map Domain Structure is the precise description of the Cartesian product space on which a tensor, viewed as a multilinear map, is defined, together with the rules that determine which factors of that product are dual spaces and which are the original vector space, and in what order the arguments must appear for the map's values to match the tensor's components.

The Domain as a Product of Spaces

Composition of the Domain

For a tensor of type (p, q) over a vector space V with dual V*, the domain of its associated multilinear map is the (p + q)-fold Cartesian product

\times_{k = 1}^{p} V^{*} \times \times_{l = 1}^{q} V

with the convention that the p covector slots are listed before the q vector slots. This ordered structure is not arbitrary: it fixes which index positions of the tensor's components are contravariant (upper) and which are covariant (lower).

Slot Typing

Each factor of the domain carries a fixed type, either V* or V, and this typing cannot be altered without changing the tensor itself. A slot of type V* accepts a covector and, when the tensor is written in coordinates, corresponds to an upper (contravariant) index; a slot of type V accepts a vector and corresponds to a lower (covariant) index. The domain structure is therefore the coordinate-free record of the tensor's valence pattern.

Codomain and the Scalar Field

Why the Codomain Is the Base Field

The domain structure only becomes a genuine multilinear map when paired with a codomain, which for a real tensor is the base field ℝ. The map

T : \times_{k = 1}^{p} V^{*} \times \times_{l = 1}^{q} V \to ℝ

is uniquely determined once its values on all tuples of basis covectors and basis vectors are known, because the domain is spanned, in the multilinear sense, by such tuples.

Domain Dimension Bookkeeping

If V is finite-dimensional with dimension n, each factor in the domain, whether V or V*, also has dimension n. The number of independent evaluation points needed to determine T fully is n^(p+q), matching the dimension of the tensor product space V*⊗p ⊗ V⊗q that T is naturally identified with.

Structural Constraints Imposed on the Domain

Order and Index Correspondence

Because multilinear maps are not required to be symmetric between slots of different types, the order in which factors appear in the domain fixes a bijection between domain slots and tensor index positions. Swapping the order of two slots of the same type (two V factors or two V* factors) corresponds to a relabeling of indices, while swapping a V factor with a V* factor changes the type of tensor being represented entirely, since it would require accepting a vector where a covector was expected.

Restriction to Subdomains

A partial evaluation of T, where some arguments are fixed and others left open, restricts the domain to the product of the remaining factors and yields a new multilinear map of correspondingly lower valence. This is the domain-level description of what happens when a tensor is contracted with, or partially saturated by, fixed vectors or covectors.

Interaction with Linear Maps

Domain Transformation Under Pullback

Given a linear map f: V → W, the dual map f*: W* → V* acts on the domain factors of type V* that belong to purely covariant tensors on W, replacing each W* factor with a V* factor. This is precisely why a purely covariant tensor's multilinear-map domain transforms contravariantly: the domain factors of type W* are relabeled to V* factors by precomposing each argument with f before evaluation.

Domain Transformation Under Pushforward

When f is invertible, the domain factors of type V belonging to a tensor can likewise be transformed by (f^-1)* acting on the covector slots, or by f acting directly on the vector slots, allowing the full multilinear map domain of a mixed tensor to be consistently relocated from V to W.

Domain Structure and Basis Representation

Basis-Induced Domain Grid

Choosing a basis for V and the dual basis for V* effectively discretizes the domain into an n^(p+q) grid of basis-tuples, and the tensor's components are the values of the multilinear map on this grid. Any two tuples that differ only by linear combination within a single slot are related by the multilinearity axiom, so the domain structure together with the basis choice fully encodes how components transform under a change of basis.