5.20.2 Tensor Product Linear Map Domain Relation

Tensor Product Linear Map Domain Relation is the fact that the domain of the induced map f ⊗ g : V ⊗ W → V′ ⊗ W′ is exactly the tensor product V ⊗ W of the domains of the individual maps f : V → V′ and g : W → W′, so that the tensor product operation on linear maps always takes as input the tensor product of the original input spaces rather than any other combination of them. This relation, the domain-side counterpart of the codomain relation, fixes precisely what kind of object f ⊗ g is allowed to act on, and it governs how domains combine under composition, restriction to subspaces, and duality.

Statement of the Relation

The Basic Domain Correspondence

Given f : V → V′ and g : W → W′, the induced map has signature

with domain V ⊗ W, the tensor product of the two individual domains V and W. This is forced by the definition of f ⊗ g on simple tensors, (f ⊗ g)(v ⊗ w) = f(v) ⊗ g(w), since the input v ⊗ w must already be an element of V ⊗ W for the expressions f(v) and g(w) to make sense.

Consistency with the Universal Property

The domain relation traces back to how f ⊗ g is constructed: the bilinear map β(v, w) = f(v) ⊗ g(w) has domain V × W, and the universal property of the tensor product converts this bilinear map, uniquely, into a linear map whose domain is the tensor product V ⊗ W of exactly those two spaces — no other domain is produced by the construction.

Domain Behavior Under Special Cases

Identity on One Factor

When f = id_V, the domain relation specializes to id_V ⊗ g : V ⊗ W → V ⊗ W′, with domain V ⊗ W reflecting that the first factor's space, V, is unchanged as an input space while only the second factor's domain space, W, is subject to g.

Domain Behavior Under Composition

If f₁ : V₁ → V₂ and f₂ : V₂ → V₃ are composable, then for any g : W → W′,

with domain V₁ ⊗ W correctly reflecting the domain V₁ of the first map in the composed chain, matching the identity (f₂ ∘ f₁) ⊗ g = (f₂ ⊗ id) ∘ (f₁ ⊗ g).

Diagram of the Domain Correspondence

Restriction of the Domain

Restricting to a Subspace of the Domain

If V₀ ⊆ V is a subspace, the domain relation restricts coherently: V₀ ⊗ W is naturally identified with a subspace of V ⊗ W, and (f ⊗ g)|_{V₀ ⊗ W} = (f|_{V₀}) ⊗ g, so the domain relation is compatible with restricting the tensor product map to a tensor product of subspaces of the original domain.

Kernel Considerations Tied to the Domain

Because the domain of f ⊗ g is V ⊗ W in full, the kernel of f ⊗ g is a subspace of this entire domain and, for finite-dimensional spaces, equals ker(f) ⊗ W + V ⊗ ker(g), a description only meaningful because the domain relation guarantees f ⊗ g is defined on the whole of V ⊗ W rather than some smaller or unrelated space.

Interaction with Duality

Domain Relation for Dual Maps

For the transpose map f* : V′* → V* of f : V → V′, the domain relation for f* ⊗ g* reads f* ⊗ g* : V′* ⊗ W′* → V* ⊗ W*, with domain V′* ⊗ W′* built from the domains of the individual transposes, illustrating that the domain relation applies uniformly regardless of whether the maps involved are "original" maps or their duals.

Significance of the Domain Relation

Guaranteeing Type-Correctness of Inputs

The domain relation guarantees, in advance and without needing to examine specific elements, exactly what kind of input a tensor product map f ⊗ g accepts, namely elements of V ⊗ W, which is essential for correctly composing tensor product maps in longer chains of transformations.

Foundation for Functorial Reasoning

Together with the codomain relation, the domain relation establishes that ⊗ acts functorially on pairs of linear maps, with both the input and output spaces of f ⊗ g determined entirely and predictably by the input and output spaces of f and g, supporting systematic reasoning about tensor constructions built from chains of linear maps.