14.1.1 Tensor Map Product Construction Scope

Tensor Map Product Construction Scope is the precise delineation of which linear maps, and over which vector spaces, are admissible as factors when forming a tensor product of maps, specifying the domain and codomain requirements that each individual map must satisfy before their tensor product can be constructed as a single well-defined linear map on the tensor product of the underlying spaces.

Definition

Given linear maps $$f:V\to W$$ and $$g:X\to Y$$, the construction scope specifies that their tensor product is defined as a linear map:

acting on simple tensors by:

Requirements Defining the Scope

Independence of Domains

The construction scope requires only that $$f$$ and $$g$$ be defined on their own, possibly entirely unrelated, vector spaces $$V$$ and $$X$$; no relationship between these domains, such as equality or inclusion, is needed for the tensor product of the two maps to be well defined.

Linearity of Each Factor

Each map entering the construction must individually be linear on its own domain; the tensor product construction does not extend to maps that fail linearity, since the well-definedness of $$f\otimes g$$ on simple tensors, and its extension by linearity to the full tensor product space, relies on each factor respecting vector addition and scalar multiplication.

Codomain Independence

Similarly, the codomains $$W$$ and $$Y$$ need not bear any particular relationship to one another; the construction scope places no constraint linking the codomain of one factor to the codomain of the other.

What Falls Outside the Scope

Non-Linear Maps

A map that fails to be linear on its stated domain lies outside the construction scope entirely, since the defining formula on simple tensors would not extend consistently to a well-defined map on the full tensor product space.

Maps With Mismatched Simple Tensor Structure

An attempt to apply the tensor product construction to an object that is not itself expressible via the universal property of the tensor product, such as an arbitrary bilinear expression not arising from two independent linear maps, falls outside the scope of this specific construction, though it may still be addressed by other constructions in tensor algebra.

Diagram

Placement Within Tensor Products of Maps

The construction scope forms the foundational entry point for the broader study of tensor products of maps, establishing the exact preconditions, independent linear maps on their own respective domains, that must be met before any further properties of the tensor product map, such as its behavior under composition or its interaction with contraction, can be meaningfully investigated.