14.3.5 Tensor Map Product Tensor Space Role

The tensor map product defines how tensors interact in spaces, playing a key role in structuring multilinear transformations and tensor algebra operations.

Tensor Map Product Tensor Space Role is the part played by the tensor product spaces themselves within the construction of a tensor product of maps, namely serving as the ambient stage on which the universal property operates to convert a bilinear map into the induced linear map.

The Tensor Space as Universal Recipient

Role in the Factorization

Given the bilinear map

β (v, w) = f (v) \otimes g (w)

the domain tensor space V1 tensor V2 plays the role of universal recipient of bilinear maps out of V1 times V2, meaning that every bilinear map on this product, including beta, factors uniquely through the canonical map

\otimes : V_{1} \times V_{2} \to V_{1} \otimes V_{2} .

Without this universal property, the domain tensor space would have no privileged role in converting beta into a linear map, and the construction of f tensor g could not proceed.

Role of the Codomain Tensor Space

The codomain tensor space W1 tensor W2 plays a complementary role as the target that beta already maps into, so it need not itself possess any universal property for the construction to work; its role is simply to be the fixed receiving space in which the values f(v) tensor g(w) are already assembled by ordinary tensor multiplication.

Distinguishing the Roles of Domain and Codomain Tensor Spaces

Domain Tensor Space as the Site of Factorization

The domain tensor space is the one space in the entire construction whose defining property, the universal property, is actively invoked, since it is through this property that the map f tensor g comes into existence as a linear map rather than merely as a formula on pairs.

Codomain Tensor Space as a Passive Target

The codomain tensor space, by contrast, plays no active role in the factorization step; it functions purely as the space in which elementary tensors f(v) tensor g(w) are formed using its own internal tensor multiplication, a role shared with every other bilinear map that happens to take values in W1 tensor W2, regardless of whether that map arises from a factor map pair.

Role in Supporting Algebraic Properties

Spanning Role in Establishing Uniqueness

The domain tensor space plays a further role beyond factorization: because it is spanned by elementary tensors, any two linear maps agreeing on elementary tensors must agree everywhere on it. This spanning role is what upgrades the existence of f tensor g, guaranteed by the universal property, into the stronger statement that f tensor g is the unique linear map with the stated elementary output rule.

Role in Supporting Composition

Both tensor spaces play a joint role in supporting the composition identity for tensor products of maps, since the intermediate tensor space W1 tensor W2, appearing as the codomain of f tensor g and the domain of a subsequent tensor product of maps, must carry both roles simultaneously, being the passive target of one construction and the universal recipient underlying the next.

Role in the Matrix Representation

Basis Role of the Tensor Spaces

Once bases are chosen for V1, V2, W1, and W2, the tensor spaces play the role of hosting the induced bases of elementary tensors, and it is with respect to these induced bases that the Kronecker product description of f tensor g becomes valid; changing the bases of the tensor spaces without correspondingly changing the induced elementary tensor basis would invalidate the direct correspondence between the abstract map and its Kronecker product matrix.