12.14.2 Tensor Inclusion Target Space

Tensor Inclusion Target Space is the larger ambient tensor space T into which a source subspace U is embedded by a tensor inclusion operation ι : U → T, considered as the receiving structure that must already exist, with its own linear operations fixed, in order for the inclusion map to be defined at all. Unlike the target domain of a scalar extension, which is newly constructed by the extension operation itself, the target space of an inclusion is typically a pre-existing space — a tensor product, a larger module, or an ambient direct sum — into which the source subspace is mapped without altering the target space's own prior structure.

Role of the Target Space in the Inclusion

A Pre-Existing Receiving Structure

The target space T must already be a well-defined vector space or module over the same base field or ring as the source subspace U, since the inclusion is required to be linear and this presupposes both spaces share compatible scalar operations. T does not depend on U for its existence; it is simply the fixed codomain into which the inclusion happens to map.

The Embedded Image as a Proper Piece of the Target Space

The inclusion identifies U with the subspace ι(U) ⊆ T, which is generally a proper subspace unless ι happens to be surjective as well as injective. The target space typically contains elements outside ι(U) entirely, and part of the value of forming the inclusion is precisely to situate U within this larger context alongside such additional elements.

Typical Forms of the Target Space

Tensor Product Ambient Space

When U = V₁ is included via v₁ ↦ v₁ ⊗ v₂ for a fixed v₂ ∈ V₂, the target space is the full tensor product V₁ ⊗ V₂, which contains not only the embedded copy of V₁ but every other simple tensor v₁′ ⊗ v₂′ for v₂′ not proportional to the fixed v₂, as well as all their linear combinations.

Direct Sum Ambient Space

When T = U ⊕ W for some complementary subspace W, the target space is this entire direct sum, and the inclusion u ↦ (u, 0) situates U as exactly the first summand, leaving W as the complementary region of T untouched by the image of ι.

Extended Scalar Space Viewed as a Target

When the canonical map v ↦ v ⊗ 1 used in scalar extension happens to be injective, the extended domain V ⊗_R S simultaneously plays the role of a target space for an inclusion of the original module V, illustrating that the target space of an inclusion and the target domain of an extension can coincide as the same underlying object viewed from two related perspectives.

Diagram of the Target Space Containing the Embedded Source

Structural Consequences for the Target Space

The Target Space's Own Basis Need Not Align with the Image

If T has a basis chosen independently of U, that basis generally does not consist entirely of vectors inside ι(U); only a specially adapted basis, constructed using the basis of U together with a basis for a chosen complement, aligns cleanly with the image of the inclusion. The target space's structure is therefore not automatically compatible with the inclusion unless such an adapted basis is deliberately chosen.

Dimension Relationship

When both spaces are finite-dimensional, dim(U) ≤ dim(T), with equality if and only if ι is also surjective, in which case the inclusion is an isomorphism and the distinction between source subspace and target space collapses to a relabeling of the same space. Strict inequality is the typical case that motivates calling the map an "inclusion" in the first place, since it reflects U occupying only part of the available structure in T.

Multiple Source Subspaces Sharing One Target Space

A single target space T can simultaneously receive several different inclusions ι₁ : U₁ → T, ι₂ : U₂ → T, from unrelated source subspaces, with their images ι₁(U₁) and ι₂(U₂) possibly overlapping, disjoint except at the origin, or spanning complementary pieces of T. The target space itself places no restriction on how many such inclusions may target it, since it is only the codomain, not a party to the inclusion's defining injectivity condition.

Relation to the Paired Projection

Target Space as the Domain of the Complementary Projection

When an inclusion ι : U → T is paired with a projection π : T → U satisfying π ∘ ι = id_U, the target space T of the inclusion is simultaneously the source space of the projection. This dual role is what allows the two operations to be studied as complementary halves of a single decomposition of T, rather than as independent, unrelated constructions.

Recovering U's Complement Inside T

Once the target space T and the inclusion ι are fixed, and a projection π satisfying the retraction identity is chosen, the kernel of π furnishes an explicit complementary subspace W inside T with T = ι(U) ⊕ W, giving a complete structural account of how the source subspace sits inside its target space.