12.14.5 Tensor Inclusion Structure Preservation

Tensor Inclusion Structure Preservation is the collection of guarantees that every algebraic feature carried by a source subspace U — its linear relations, any attached multilinear forms, distinguished elements, and compatible symmetries — survives intact and undistorted inside its embedded image ι(U) within the target space T, as a direct consequence of the inclusion map satisfying the embedding map role. Where the embedding map role states the two defining conditions of linearity and injectivity, structure preservation catalogs the downstream consequences of those conditions for every piece of structure U might carry.

Preservation of Linear Relations

Sums and Scalar Multiples

Every equation of the form u = au₁ + bu₂ holding in U transports directly to ι(u) = a·ι(u₁) + b·ι(u₂) holding in T, by linearity of ι. No linear relation among elements of U is created, destroyed, or altered by passing to the embedded image.

Dependence and Independence

If a family {uₖ} in U is linearly dependent, satisfying Σ cₖuₖ = 0 with not all cₖ zero, the same relation Σ cₖι(uₖ) = 0 holds among the images, by linearity. Conversely, injectivity of ι guarantees that a linearly independent family in U remains linearly independent after embedding, so the dependence structure of U is preserved exactly, in both directions, inside ι(U).

Preservation of Attached Multilinear Structure

Bilinear Forms

If U carries a bilinear form β and T carries a compatible bilinear form β′ such that this identity holds for all u, u′ ∈ U, then β is preserved exactly by the inclusion: any computation of β(u, u′) inside U can equivalently be performed as β′ applied to the embedded images inside T, with no discrepancy in the result.

Algebra Multiplication

If U is itself a subalgebra of a larger algebra structure on T, meaning ι(u)·ι(u′) = ι(u·u′) for the multiplications on T and U respectively, the inclusion preserves the entire multiplicative structure of U, not merely its additive vector space structure. This is a strictly stronger preservation guarantee than linearity alone provides, and it must be checked separately whenever an algebra structure is present.

Distinguished Tensors

A distinguished element of U, such as an identity element or a fixed structure tensor, maps under ι to a corresponding distinguished element of ι(U), and any defining property of that element expressed via U's own operations transfers to the corresponding property of its image, expressed via T's operations restricted to ι(U).

Diagram of Structure Transported Intact

Limits of Structure Preservation

Structure External to U Is Not Automatically Preserved

Structure preservation guarantees that features internal to U transport faithfully, but it says nothing about how ι(U) interacts with elements of T outside the image. For instance, U being preserved as a subspace does not by itself imply anything about whether ι(U) is invariant under some operator defined on all of T, since such invariance is a statement about the ambient space's structure, not about U's own internal structure.

Preservation Requires the Structure to Be Expressible via Internal Operations

If a property of U is defined using operations not available purely from U's own vector space structure — for example, a property referring to a specific ambient embedding already assumed — that property is not guaranteed to be preserved automatically, since the preservation guarantee only covers structure statable in terms of U's intrinsic addition, scalar multiplication, and any explicitly compatible multilinear operations.

Compatibility Conditions Must Be Verified, Not Assumed

Preservation of a bilinear form or algebra multiplication requires the explicit compatibility identity relating U's operations to T's operations, as shown above; this identity is an additional hypothesis that must be checked for the specific structure in question and is not automatically implied merely by ι satisfying the embedding map role for the underlying vector space structure alone.

Consequences for Working with Embedded Subspaces

Computations May Be Performed in Either Space

Because structure preservation guarantees an exact correspondence, any computation involving only elements of U and its own internal operations may be carried out either directly in U or, equivalently, inside T after embedding via ι, with identical results translated by ι and ι⁻¹ restricted to the image. This equivalence is what justifies treating U as literally a subspace of T in subsequent reasoning, rather than as merely related to a subspace of T.

Foundation for Compatible Decompositions

Structure preservation under inclusion is what makes it meaningful to speak of a tensor space T decomposing into pieces, each carrying its own internal structure inherited faithfully from a corresponding source subspace, forming the basis for building up complex tensor spaces from simpler, well-understood components via a sequence of structure-preserving inclusions.