5.4.1 Tensor Product Space Carrier Structure

Tensor Product Space Carrier Structure is the underlying set of V ⊗ W, considered apart from the addition and scalar action defined on it: the set of cosets x + R of the free vector space F(V × W) modulo the relation subspace R, which becomes a vector space only once those two operations are layered on top of it.

The Carrier Set Itself

Elements as Equivalence Classes

The carrier set of V ⊗ W consists of the equivalence classes of F(V × W) under the relation x ~ y if and only if x − y is in R; each class is a coset x + R, and the carrier set is simply the collection of all such cosets, with no addition or scalar multiplication yet specified as part of its description as a set. This is the same underlying set that carries the addition structure and the scalar action structure, but considered here prior to and independently of either.

Distinguishing the Carrier from the Operations Placed on It

A vector space, in the general algebraic sense, is a carrier set together with an addition operation and a scalar action satisfying the vector space axioms; the carrier structure names the "together with" part's first component alone. The same carrier set of cosets could in principle support other, different operations, though only the specific addition and scalar action already fixed are the ones that make it agree with the tensor product as characterized by the universal property.

How the Carrier Set Is Populated

Every Coset Traces Back to a Finite Sum of Symbols

Since F(V × W) is spanned by the formal symbols (v, w), every coset in the carrier set is x + R for some finite linear combination x = ∑ c_i (v_i, w_i); after the scalar coefficients are absorbed into either factor using the relations in R, every element of the carrier set is realized as ∑ (v_i ⊗ w_i), a finite sum of decomposable elements, matching the description given once the resulting space is discussed as a whole.

Distinct Cosets Correspond to Distinct Tensor Elements

Two finite sums of symbols determine the same element of the carrier set exactly when their difference lies in R; this is what makes the carrier set neither larger nor smaller than it needs to be — every genuinely distinct tensor product element corresponds to exactly one coset, and no two algebraically different elements are ever conflated into a single coset by the quotienting already performed.

Role Among the Space's Structural Components

The Set on Which Addition and Scalar Action Are Layered

The carrier structure is logically prior to both the addition structure and the scalar action structure: those two operations are defined as specific functions from pairs of carrier elements, or from a scalar and a carrier element, back into the carrier set, and neither operation can be described without first fixing what the carrier set consists of.

Why the Carrier Is Isolated as Its Own Topic

Separating the carrier set from the operations placed on it keeps clear which facts about V ⊗ W are facts about which cosets exist — for instance, that every coset is representable as a finite sum of decomposable elements — as opposed to facts about how those cosets combine under addition or scale under the field action, which are the concerns of the separately treated addition and scalar action structures.