4.10.2 Tensor Vector Valued Codomain Space

Tensor Vector Valued Codomain Space is the specific vector space $W$ chosen as the target of a vector-valued multilinear map, considered in its own right as a mathematical object with its own dimension, basis, and possible relationship to the domain factors of the map, independent of any particular tensor defined to land inside it. It is the "output universe" fixed in advance of specifying any actual multilinear map, and its own internal properties, dimension, whether it coincides with $V$, and what additional structure it carries, shape everything that can subsequently be said about tensors valued in it.

Formal Definition

The Codomain Space as a Prior Choice

Before any multilinear map $T : V_1 \times \cdots \times V_k \to W$ can be specified, the codomain space $W$ must itself be fixed as a vector space over the same base field $F$ as the domain factors. This space carries its own dimension $m = \dim W$, and, once a basis is chosen, its own set of coordinates, entirely independent of whatever basis is separately chosen for the domain factors $V_1, \ldots, V_k$.

Relationship to the Domain

A codomain space may coincide with one of the domain factors, as in the torsion tensor where $W = V$; it may be the dual $V^{*}$; it may be built from $V$ via an auxiliary construction such as $\operatorname{End}(V)$ or $\Lambda^{p}(V)$; or it may be an entirely unrelated vector space with no canonical connection to the domain factors at all. This relationship, or lack of one, is a defining feature of the codomain space distinct from the multilinear map itself.

Structural Properties of the Codomain Space

Dimension and Its Effect on Component Count

The dimension $m$ of the codomain space directly multiplies the total number of independent scalar components needed to describe any tensor valued in it: a type $(r,s)$ tensor over an $n$-dimensional $V$, valued in an $m$-dimensional $W$, has $m \cdot n^{r+s}$ independent components, since each of the $n^{r+s}$ domain component slots must additionally be resolved along each of the $m$ codomain directions.

Independent Basis and Transformation Behavior

Because the codomain space's basis is chosen independently of the domain factors' bases, a change of basis in $W$ affects only the extra codomain index of the tensor's components, leaving the transformation behavior of the ordinary domain indices under a change of basis in $V$ entirely unaffected; the two transformation laws act on disjoint sets of indices and can be applied in either order without interference.

Dual and Endomorphism Codomain Spaces

When the codomain space is chosen to be $W = V^{}$, tensors valued in it can be re-expressed, via the canonical pairing, as scalar-valued multilinear maps of one higher covariant order, effectively absorbing the extra codomain index into an additional ordinary domain index. When $W = \operatorname{End}(V)$, the codomain space itself is built from two copies of $V$ (one covariant, one contravariant, via $\operatorname{End}(V) \cong V \otimes V^{}$), giving the codomain a compound internal structure distinct from a bare, unrelated vector space.

Choosing the Codomain Space in Practice

Codomain Determined by the Geometric Meaning Sought

The choice of codomain space is typically dictated by what kind of object the tensor is meant to represent: if the desired output is itself a directional quantity in the same space as the inputs, $W = V$ is the natural choice; if the desired output is a transformation to be applied to further vectors, $W = \operatorname{End}(V)$ or another Hom space is appropriate; if the desired output is an entirely separate physical quantity with its own units and structure, an unrelated codomain space may be required.

Codomain Space Consistency Across a Tensor Field

When tensors valued in $W$ are assembled into a tensor field over a manifold, the codomain space compatibility requirement described elsewhere demands that the codomain spaces at different points, if $W$ is itself built from the tangent space (such as $W = TM$ at each point), fit together into a smoothly varying bundle rather than remaining a single, fixed abstract space throughout.

Summary of Key Points

• The vector valued codomain space is the target vector space $W$ fixed prior to defining any specific multilinear map into it.
• $W$ may coincide with a domain factor, be built from it via a construction such as duality or endomorphisms, or be entirely unrelated to the domain.
• The dimension of $W$ multiplies the total component count of any tensor valued in it, adding an extra, independently transforming index.
• Choosing $W = V^{*}$ allows reinterpretation as a scalar-valued tensor of higher covariant order via the canonical pairing.
• The choice of codomain space is guided by the geometric or algebraic meaning the tensor's output is intended to carry.