4.10 Tensor Vector Valued Multilinear Map Structure

Tensor Vector Valued Multilinear Map Structure is the complete algebraic framework describing a tensor as a map that takes several vector and covector arguments and returns an element of a genuine vector space rather than a bare scalar, combining a slotwise-linear domain structure with a vector codomain into one coherent object. It generalizes the scalar-valued multilinear map structure by allowing the target of the map to itself carry directional, multi-dimensional content, and it is the structure underlying tensor-valued operators, vector-valued curvature and torsion, and any multilinear construction whose result must be combined further with other vectors.

Formal Definition

The Full Structure

A vector-valued multilinear map is a function

where $W$ is a vector space of dimension $m \geq 1$, satisfying slotwise linearity in each of its $r+s$ domain slots exactly as in the scalar case. The structure is fully specified by three pieces of data: the domain structure inherited from $V$ and $V^{*}$, the linearity condition per slot, and the vector codomain $W$, which may or may not be related to $V$.

Isomorphism With a Tensor of Extended Type

Choosing a basis ${f_1, \ldots, f_m}$ of $W$ realizes the structure as an $m$-tuple of scalar-valued tensors of type $(r,s)$, one per basis direction of $W$; equivalently, the space of all such vector-valued multilinear maps is naturally isomorphic to $T^{r}_{s}(V) \otimes W$, embedding the vector-valued case inside the tensor product framework alongside an extra factor for the codomain.

Structural Features

Component Representation With an Extra Codomain Index

Once bases are fixed for both the domain factors and $W$, the structure reduces to a component array carrying one additional index beyond the ordinary $(r,s)$ indices, ranging over the dimension $m$ of the codomain:

where the index $a$, ranging from $1$ to $m$, labels which component of the output in the chosen basis of $W$ is being described, and the semicolon separates this codomain index from the ordinary domain indices.

Transformation Under Two Independent Basis Changes

Because both the domain factors and the codomain carry their own bases, the components transform under two logically separate change-of-basis actions: the ordinary $A^{-1}$ and $A$ factors acting on the domain indices $i_1, \ldots, i_r, j_1, \ldots, j_s$, and an independent matrix acting on the codomain index $a$ whenever the basis of $W$ is changed, with no requirement that these two transformations be related.

Common Instances

The Torsion Tensor

The torsion tensor of an affine connection is a vector-valued bilinear map, taking two tangent vectors and producing a third tangent vector; its codomain is the same space $V$ as its domain factors, a common but not universal feature of vector-valued tensors arising in differential geometry.

Vector-Valued Wedge-Like Products

Certain cross-product-style constructions, generalizing the three-dimensional cross product to other settings, are vector-valued antisymmetric bilinear maps, combining the vector codomain structure with the additional antisymmetry constraint associated with forms.

Operator-Producing Tensors

A vector-valued multilinear map with codomain $W = \operatorname{Hom}(U, V)$ produces, upon evaluation, a linear map from $U$ to $V$ rather than a single vector; since $\operatorname{Hom}(U,V)$ is itself a vector space, this remains within the vector-valued codomain framework even though the individual output elements are operators.

Relation to the Scalar-Valued Structure

Recovery via Codomain Pairing

Pairing a vector-valued map $T$ against a fixed covector $\omega \in W^{}$ produces a scalar-valued multilinear map $\omega \circ T$ on the same domain; ranging $\omega$ over a full basis of $W^{}$ recovers all $m$ scalar component maps and reconstructs the vector-valued structure completely, confirming that the vector-valued case reduces without loss of information to a finite collection of scalar-valued cases.

Summary of Key Points

• A vector-valued multilinear map structure combines the ordinary domain structure of $V$ and $V^{*}$ with a codomain that is a genuine vector space $W$, rather than the base field.
• The space of such maps is isomorphic to $T^{r}_{s}(V) \otimes W$, embedding the vector-valued case within the standard tensor product framework.
• Components carry an additional codomain index beyond the usual domain indices, transforming under an independent change-of-basis action on $W$.
• Familiar examples include the torsion tensor, vector-valued antisymmetric products, and operator-producing tensors valued in a Hom space.
• Pairing against a covector in $W^{*}$ recovers a scalar-valued component map, showing the vector-valued structure decomposes fully into scalar-valued pieces.