12.3.2 Tensor Addition Same Space Requirement

Tensor Addition Same Space Requirement is the specific prerequisite within the tensor addition operation demanding that both tensors being summed be built over the identical underlying vector space, so that their components refer to the same family of bases and the same basis-change matrices, without which the component-wise sum defining addition has no consistent transformation law to satisfy.

Foundational Setting

Why This Requirement Is Distinct from Type Matching

Tensor addition requires both matching variance type and this same-space requirement, but the two conditions serve different purposes: type matching ensures the two tensors have the same arrangement of upper and lower indices, while the same-space requirement ensures those indices are drawn from, and transform according to, one common vector space rather than two unrelated ones.

What "Same Underlying Vector Space" Means

Two tensors satisfy this requirement when their components are defined relative to bases of one shared vector space $$V$$, so that a single basis-change matrix $$A$$ governs how both tensors' components transform under any change of basis considered.

Why the Requirement Is Necessary

No Shared Transformation Law Without It

If two tensors were defined over genuinely different vector spaces, there would be no single, common basis-change matrix relating their components, and the component-wise sum, even if numerically computable by simply adding corresponding entries, would fail to transform as a tensor under any meaningful change of basis, since there would be no unambiguous basis change to test it against.

An Illustrative Failure

Consider two spaces $$V$$ and $$W$$ of the same dimension but with no fixed relationship between them. Components $$v^i$$ from $$V$$ and $$u^i$$ from $$W$$ might happen to have matching dimension and even matching numerical values in some chosen bases, but a change of basis on $$V$$ alone would transform $$v^i$$ while leaving $$u^i$$ unaffected, breaking any hope of a consistent combined transformation law for their sum.

Distinguishing Genuinely Different Spaces from Isomorphic Ones

Isomorphism Alone Is Not Sufficient

Two vector spaces of the same dimension are always isomorphic to one another, but the mere existence of some isomorphism between them does not, by itself, satisfy the same-space requirement, since addition needs the two tensors to already be expressed relative to one common space and its basis-change structure, not merely to belong to spaces that could in principle be identified through an arbitrary chosen map.

Explicit Identification as a Separate Step

If two tensors genuinely belong to different but related spaces, an explicit linear map can be used, through the mapping area's pushforward or pullback procedures, to transport one tensor into the other's space first, satisfying the same-space requirement only after this transport step, rather than attempting to add the two tensors directly beforehand.

Practical Recognition of the Requirement

A Common Source of Confusion

A frequent point of confusion arises when two tensors happen to have matching dimension and matching variance type but originate from conceptually distinct vector spaces, such as a tangent space at one point of a curved space and a tangent space at a different point; despite superficial numerical compatibility, these tensors do not satisfy the same-space requirement and cannot be added directly without first transporting one of them into the other's space.

Checking the Requirement in Practice

Confirming this requirement means confirming that both tensors are expressed relative to bases belonging to one identified, common vector space, a check that must be made explicitly whenever the two tensors under consideration arise from different geometric or algebraic contexts rather than being introduced together as elements of a single space from the outset.

Summary of Key Traits

Defining Characteristics

• The same-space requirement demands that both tensors involved in addition be defined relative to one common underlying vector space.
• This requirement is distinct from matching variance type, addressing the shared basis-change structure rather than the arrangement of indices.
• Without this requirement, no single, well-defined transformation law would govern the component-wise sum under a change of basis.
• Tensors from genuinely different, even if isomorphic or same-dimensional, spaces must first be transported into a common space, using pushforward or pullback, before addition can be validly applied.