14.8.1 Tensor Map Product Codomain Factor Spaces

Tensor Map Product Codomain Factor Spaces is the pair of target vector spaces W1 and W2 whose tensor product W1 tensor W2 forms the codomain of a tensor product of maps, distinguished individually as the two spaces each contributing one factor to that codomain before any tensoring takes place.

Identifying the Codomain Factor Spaces

The Two Spaces Contributing to the Codomain

For a tensor product of maps f tensor g, with

the codomain factor spaces are W1 and W2 individually, the two spaces whose tensor product forms the full codomain W1 tensor W2 of the induced map, with W1 receiving the image of f and W2 receiving the image of g, in the order fixed by the factor space relation.

Codomain Factor Spaces Considered Prior to Tensoring

The codomain factor spaces are meaningful as individual objects before any tensoring occurs, since they are simply the codomains of the two maps f and g taken separately; the tensor product W1 tensor W2 is a derived construction built from these two spaces, and the codomain factor spaces retain their own independent identity as ordinary vector spaces regardless of whether a tensor product is ever formed from them.

Structural Role of the Codomain Factor Spaces

Dimension Contribution

If the codomain factor spaces are finite-dimensional with dimensions p and q respectively, the dimension of the full codomain W1 tensor W2 is the product p q, so each codomain factor space contributes multiplicatively, rather than additively, to the dimension of the tensor product codomain.

Basis Contribution

A basis of the tensor product codomain is built directly from bases of the two codomain factor spaces, by forming every elementary tensor of a basis vector of W1 with a basis vector of W2; consequently, any basis chosen for either codomain factor space propagates directly into the induced basis of the full tensor product codomain, without requiring any further adjustment.

Independence and Interaction of the Codomain Factor Spaces

No Assumed Relationship Between the Two Spaces

The two codomain factor spaces are not required to bear any particular relationship to one another: they may have different dimensions or be defined over different bases, so long as both are vector spaces, or more generally modules, over the same underlying field or ring.

Interaction Only Through the Tensor Product

The codomain factor spaces interact with one another only through the tensor product construction itself; outside of this tensoring operation, elements of W1 and elements of W2 have no direct means of combination, and the codomain factor spaces remain otherwise unconnected as vector spaces.

Codomain Factor Spaces Under Composition and Restriction

Codomain Factor Spaces of a Composite

When two tensor products of maps are composed, the codomain factor spaces of the composite are exactly the codomain factor spaces of the second tensor product of maps in the chain, since composition changes only the domain side of the resulting map as seen from the outside, leaving the identity of the final codomain factor spaces unaffected by any intermediate stage.

Enlargement to a Superspace

If a map g has codomain X strictly larger than the intended codomain factor space W2, the codomain factor space for the tensor product of maps may still be taken as X itself, since the construction is defined relative to the stated codomain rather than relative to the image, and this enlargement does not alter the values produced by the tensor product of maps, only the ambient space in which those values are considered to lie.