5.1.5 Tensor Product Map Scope

The tensor product map scope defines how tensor products operate between vector spaces, establishing their structure and mapping properties within linear algebra.

Tensor Product Map Scope is the delineation of what falls under the study of linear maps built from, or acting between, tensor product spaces, separating the induced map f ⊗ g obtained from linear maps on the factors, and its functorial behavior, from the bilinear map ⊗ that produces individual elements and from the universal-property factorization of arbitrary bilinear maps through the tensor product.

What Lies Inside the Scope

The Induced Map on Tensor Products

Given linear maps f: V → V' and g: W → W', the scope covers the uniquely determined linear map

f \otimes g : V \otimes W \to V^{'} \otimes W^{'}

satisfying (f ⊗ g)(v ⊗ w) = f(v) ⊗ g(w) on decomposable elements, extended linearly to all of V ⊗ W; existence and uniqueness of this induced map follow from the universal property applied to the bilinear map (v, w) ↦ f(v) ⊗ g(w).

Functoriality

The scope includes the compatibility of this construction with composition and identity: (f_1 ∘ f_2) ⊗ (g_1 ∘ g_2) = (f_1 ⊗ g_1) ∘ (f_2 ⊗ g_2) and id_V ⊗ id_W = id_{V ⊗ W}. These identities are what make ⊗ behave as a functor of two linear-map arguments, so that composing maps before or after tensoring gives the same result.

Special Cases: One Factor Fixed

When one of the two maps is the identity, the induced map f ⊗ id_W: V ⊗ W → V' ⊗ W acts only on the first tensor factor; this partial application is within the scope as the mechanism by which a linear map on one factor of a tensor product is extended to act on the whole product while leaving the other factor untouched, which underlies operations such as contraction against a single slot of a tensor.

What Lies Outside the Scope

The Bilinear Map Defining Individual Elements

The map ⊗: V × W → V ⊗ W sending (v, w) to v ⊗ w is the bilinear map used to construct the tensor product space and its elements; it is not itself an instance of the induced-map construction, since it has no pair of prior linear maps f and g behind it, and belongs instead to the construction and element scopes.

General Bilinear Maps Out of a Tensor Product

An arbitrary linear map T: V ⊗ W → U that does not arise as f ⊗ g for any pair of linear maps on the factors is outside this scope; not every linear map on V ⊗ W decomposes this way, and the general factorization of bilinear maps through the tensor product is treated under the universality scope rather than here.

Multiplication and Contraction Operations

Products or contractions that combine two tensors into a third, such as forming v ⊗ w from two vectors or contracting an index between two tensor factors, use the induced-map machinery as an ingredient but are themselves separate operations belonging to their own topics within tensor algebra.

Why Maps Between Tensor Products Are Scoped Separately

Isolating the Functorial Behavior of the Tensor Product

Scoping the induced map on its own keeps the functoriality of ⊗ — its predictable behavior under composition and identity — available as a self-contained fact, usable whenever linear maps on tensor factors need to be extended to the tensor product, without re-deriving it from the universal property each time.