14.7.1 Tensor Map Product Domain Factor Spaces

Tensor Map Product Domain Factor Spaces is the pair of source vector spaces V1 and V2 whose tensor product V1 tensor V2 forms the domain of a tensor product of maps, distinguished individually as the two spaces each contributing one factor to that domain before any tensoring takes place.

Identifying the Domain Factor Spaces

The Two Spaces Contributing to the Domain

For a tensor product of maps f tensor g, with

the domain factor spaces are V1 and V2 individually, the two spaces whose tensor product forms the full domain V1 tensor V2 of the induced map, with V1 supplying the first factor and V2 supplying the second, in the order fixed by the factor space relation.

Domain Factor Spaces Considered Prior to Tensoring

The domain factor spaces are meaningful as individual objects before any tensoring occurs, since they are simply the domains of the two maps f and g taken separately; the tensor product V1 tensor V2 is a derived construction built from these two spaces, and the domain 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 Domain Factor Spaces

Dimension Contribution

If the domain factor spaces are finite-dimensional with dimensions m and n respectively, the dimension of the full domain V1 tensor V2 is the product m n, so each domain factor space contributes multiplicatively, rather than additively, to the dimension of the tensor product domain.

Basis Contribution

A basis of the tensor product domain is built directly from bases of the two domain factor spaces, by forming every elementary tensor of a basis vector of V1 with a basis vector of V2; consequently, any basis chosen for either domain factor space propagates directly into the induced basis of the full tensor product domain, without requiring any further adjustment.

Independence and Interaction of the Domain Factor Spaces

No Assumed Relationship Between the Two Spaces

The two domain factor spaces are not required to bear any particular relationship to one another: they may have different dimensions, be defined over different bases, or in the module-theoretic generalization even carry different additional structure, so long as both are modules over the same underlying ring of scalars.

Interaction Only Through the Tensor Product

The domain factor spaces interact with one another only through the tensor product construction itself, which introduces the bilinear relations characteristic of tensor products; outside of this tensoring operation, elements of V1 and elements of V2 have no direct means of combination, and the domain factor spaces remain otherwise unconnected as vector spaces.

Domain Factor Spaces Under Composition and Restriction

Domain Factor Spaces of a Composite

When two tensor products of maps are composed, the domain factor spaces of the composite are exactly the domain factor spaces of the first tensor product of maps in the chain, since composition changes only the codomain side of the construction, leaving the identity of the original domain factor spaces V1 and V2 unaffected.

Restriction to Subspaces of the Domain Factor Spaces

If a subspace U of V1 is chosen, the tensor product U tensor V2 forms a subspace of the full domain V1 tensor V2, and restricting the tensor product of maps to this subspace corresponds to restricting f itself to U while leaving g and its domain factor space V2 unchanged, illustrating how modifications to one domain factor space propagate predictably into the structure of the full tensor product domain.