12.14.1 Tensor Inclusion Source Subspace

Tensor Inclusion Source Subspace is the smaller tensor space U that is being embedded by a tensor inclusion operation ι : U → T, considered together with its own internal linear structure prior to and independent of the embedding. It plays the same defining role for the inclusion operation that the source domain plays for scalar extension and the source space plays for projection: it is the given input data, fixed in advance, whose faithful transport into a larger ambient space T is the entire purpose of the inclusion.

Identity of the Source Subspace

A Space in Its Own Right, Prior to Embedding

U is defined and equipped with addition and scalar multiplication independently of any ambient space T it might later be included into. The source subspace's own structure — its dimension, its basis, any multilinear forms it carries — is settled entirely by U alone, before the inclusion map ι is even specified.

Distinguishing the Abstract Space from Its Embedded Image

The term "source subspace" refers to U itself, as an abstract object, and is to be distinguished from ι(U), the embedded copy of U sitting concretely inside T. Although ι restricted to U is a linear isomorphism onto ι(U), the two are conceptually distinct: U can be discussed, and can serve as the source subspace for several different inclusions into several different ambient spaces, without reference to any one of those ambient spaces in particular.

Structural Requirements on the Source Subspace

Module or Vector Space Structure

For the inclusion operation to be defined as a linear map, U must already be a well-formed vector space or module over the same base field or ring as the target space T. Without this, the linearity condition required of ι — ι(au + bu′) = a·ι(u) + b·ι(u′) — could not even be stated, since it presupposes that U already supports the operations au + bu′.

Basis Data Carried Forward

If U is finite-dimensional with basis {fₖ}, the source subspace's basis maps under ι to a linearly independent set {ι(fₖ)} in T, which then serves as a basis for the embedded image ι(U). The source subspace's basis is thus the template from which the embedded image's basis is generated, exactly analogous to how a source domain's basis generates the extended domain's basis under scalar extension.

Any Attached Multilinear Structure

If U carries a bilinear form, an algebra multiplication, or a distinguished tensor, that structure is expressed purely in terms of operations internal to U. For the inclusion to transport this structure faithfully, it must be defined using only these internal operations, so that ι applied to the structure agrees with the corresponding structure realized directly inside ι(U) ⊆ T.

Diagram of the Source Subspace Prior to Inclusion

Consequences of the Source Subspace's Independence

Reusability Across Multiple Inclusions

Because U is defined independently of any particular ambient space, the same source subspace can be the domain of several distinct inclusion operations ι₁ : U → T₁, ι₂ : U → T₂, into unrelated target spaces. Nothing about U's own definition needs to change to support this reuse, since the source subspace carries no built-in reference to any specific ambient space.

Isomorphism Class Determines Compatible Inclusions

Two source subspaces that are isomorphic as abstract vector spaces admit inclusions into the same ambient space with the same resulting image structure, up to relabeling by the isomorphism. The source subspace's relevant data for inclusion purposes is therefore its isomorphism class together with any additionally specified multilinear structure, not any particular concrete realization of its elements.

Necessity for Verifying Injectivity

Checking that a candidate map ι is injective requires knowing precisely when two elements u, u′ ∈ U are considered equal, which is determined entirely by the internal structure of the source subspace. An ambiguously specified source subspace — for instance, one where it is unclear whether two expressions denote the same element — makes it impossible to verify the injectivity that the inclusion operation requires.

Relation to the Complementary Roles in Tensor Operations

Mirrors the Source Domain of Extension

The source subspace plays a structurally identical role to the source domain of a tensor domain extension operation: both are the fixed, prior-given input object whose faithful transport into a larger structure is the point of the operation, differing only in that extension changes the base ring while inclusion changes the ambient space over a fixed base ring.

Complements the Target Component Under a Paired Projection

When an inclusion ι : U → T is paired with a projection π : T → U satisfying π ∘ ι = id_U, the source subspace U is exactly recovered as the target component of π applied to elements of ι(U). This closes the loop between the source subspace, the inclusion operation, and the paired projection operation, showing all three concepts describe complementary aspects of a single underlying decomposition of T.