14.1 Tensor Map Product Scope

Tensor Map Product Scope is the overall boundary of the subject area concerned with forming, combining, and analyzing tensor products of linear maps, encompassing the conditions under which such products can be constructed, the algebraic properties they satisfy, and the range of settings, from finite-dimensional vector spaces to more general modules, in which the construction remains meaningful.

Definition

The scope of tensor map products covers every linear map obtainable as:

for linear maps $$f:V\to W$$ and $$g:X\to Y$$, together with the identities, extensions, and specializations that arise from combining, composing, or restricting such products.

Boundaries of the Scope

Included Topics

The scope includes the basic construction of a tensor product map from two factor maps, as delineated by the construction scope; the algebraic identities satisfied by such products, including their behavior under composition and their interaction with the identity map; and the extension of the construction to finite products of more than two factors.

Excluded Topics

The scope excludes constructions that are not expressible as tensor products of linear maps acting on tensor product spaces, such as general bilinear or multilinear maps that do not factor through independent linear maps on each argument, and excludes questions specific to tensor contraction, which, while related, concerns a distinct operation reducing tensor order rather than combining maps.

Structural Properties Within Scope

Compatibility With Composition

Within the scope, tensor products of maps satisfy a compatibility identity with composition of linear maps:

allowing composed tensor product maps to be built up from composed factor maps in either order.

Identity Preservation

The scope covers the fact that tensoring two identity maps produces the identity map on the tensor product space, providing a basic reference case against which more general tensor product maps can be compared.

Diagram

Relation to the Rest of Tensor Algebras

Tensor map product scope sits alongside the study of tensor contractions as one of the two principal operations examined within tensor algebras: contraction reduces the order of a single tensor by pairing indices, while the tensor product of maps combines separate linear maps into a single map on a larger tensor product space, and together these two operations, along with their respective notations, verification procedures, and boundary conditions, provide the core operational toolkit for manipulating tensors and the maps that act on them.