12.14 Tensor Inclusion Operation

Tensor Inclusion Operation is the operation that embeds a tensor from a smaller or simpler space into a larger ambient tensor space by a linear map ι : U → T that is injective, thereby realizing U as a genuine subspace of T without loss of information. It is the structural counterpart to the tensor projection operation: where a projection collapses a larger space down onto a smaller piece, an inclusion transports a smaller space faithfully up into a larger one, and the two operations are frequently paired so that a projection followed by its associated inclusion reproduces the identity on the subspace being handled.

Defining Properties

Injectivity

An inclusion must never send two distinct tensors of U to the same tensor of T; distinct source elements always have distinct images. This is what allows U to be identified, without ambiguity or collapse, with its image ι(U) ⊆ T.

Linearity

As with the projection operation, the inclusion is required to be linear, so that its image ι(U) is genuinely a linear subspace of T rather than an arbitrary subset, and so that the algebraic structure of U — its sums and scalar multiples — is carried over faithfully.

Image Is a Genuine Subspace

Since ι is linear and injective, ι(U) is a subspace of T isomorphic to U via ι itself. Every computation performed inside U has a directly corresponding computation performed inside ι(U) ⊆ T, with no distortion introduced by the embedding.

Typical Realizations of Tensor Inclusion

Inclusion of a Tensor Product Factor

Given a fixed nonzero vector v₂ ∈ V₂, the map v₁ ↦ v₁ ⊗ v₂ embeds V₁ into V₁ ⊗ V₂. This inclusion depends on the choice of v₂; different choices of v₂ produce different, generally non-identical, embedded copies of V₁ inside the tensor product.

Inclusion of a Direct Summand

When T = U ⊕ W is a fixed direct-sum decomposition, the map ι : U → T sending u ↦ (u, 0) is an inclusion, and it is paired canonically with the projection π : T → U sending (u, w) ↦ u, satisfying π ∘ ι = id_U.

Inclusion via Scalar Extension

The canonical map v ↦ v ⊗ 1 used in scalar extension, sending a source domain V into an extended domain V ⊗_R S, is itself a tensor inclusion operation whenever R → S is injective and V is a free R-module, since in that case the map is injective and linear, embedding the source domain faithfully inside the target domain.

Diagram of an Inclusion into an Ambient Space

Relationship Between Inclusion and Projection

Retraction Identity

When a projection π : T → U and an inclusion ι : U → T are paired, the composite π ∘ ι equals the identity on U, meaning that including a tensor and then projecting it back always recovers the original tensor exactly. This is the precise sense in which the two operations are inverse to one another on the subspace U, even though ι ∘ π is generally not the identity on the larger space T.

Constructing a Projection from an Inclusion and a Complement

Given an inclusion ι : U → T whose image has a chosen complementary subspace W with T = ι(U) ⊕ W, a corresponding projection π : T → U is constructed by decomposing any t ∈ T as t = ι(u) + w and setting π(t) = u. This shows inclusion and projection operations are not independent notions but are generated jointly from a single choice of direct-sum decomposition.

Idempotent Composite in the Other Order

The composite ι ∘ π : T → T is generally not the identity but is idempotent, (ι∘π)∘(ι∘π) = ι∘π, since it equals the projection operation onto ι(U) viewed as an endomorphism of T. This links the inclusion operation directly back to the idempotency property central to the tensor projection operation.

Consequences of Requiring Injectivity

Failure of Injectivity Breaks Faithful Embedding

If a candidate inclusion map failed to be injective, distinct tensors of U would collapse to a single tensor in T, and no well-defined "undoing" projection could recover the original distinct elements. Injectivity is therefore not an optional refinement but the essential property that justifies calling the operation an inclusion at all, rather than merely a linear map.

Compatibility with Further Extensions

Because an inclusion operation preserves all linear relations of U, any further linear or multilinear construction performed on ι(U) inside T corresponds exactly, via ι⁻¹ restricted to the image, to the same construction performed directly on U. This compatibility is what allows tensor computations to be legitimately transferred back and forth between a subspace and its ambient space through the inclusion operation.