5.3.5 Tensor Product Resulting Space

Tensor Product Resulting Space is the final output of the tensor product construction pipeline, the vector space V ⊗ W = F(V × W)/R obtained once input space selection, formal symbol creation, bilinear relation imposition, and quotient formation have all been carried out, considered here specifically as the endpoint of that construction rather than as an object whose further internal structure is being explored.

What Emerges at the End of the Pipeline

A Vector Space, by Inheritance from the Quotient

V ⊗ W, being defined as a quotient of the vector space F(V × W) by the subspace R, is a vector space immediately upon quotient formation, with no further step needed to establish this; addition and scalar multiplication on V ⊗ W are exactly the operations inherited from the quotient construction, already shown well-defined during quotient formation.

A Fixed Bilinear Map Accompanying the Space

The resulting space arrives together with the canonical map ⊗: V × W → V ⊗ W, (v, w) ↦ v ⊗ w, which is bilinear precisely because the generators of R were chosen during bilinear relation imposition to force exactly that property. The resulting space is never considered apart from this accompanying map; the pair (V ⊗ W, ⊗) together is what the construction produces, not the space alone.

Every Element Is a Finite Sum of Decomposables

As a direct carryover from the surjectivity of the quotient projection, every element of the resulting space is expressible as a finite sum ∑ v_i ⊗ w_i; this fact is established at the moment the space is formed and does not require any additional argument once the earlier stages are complete.

What Is Deliberately Not Addressed Here

Dimension and Basis

That dim(V ⊗ W) = dim(V) · dim(W) in the finite-dimensional case, and that bases of V and W induce a basis of V ⊗ W, are facts about the resulting space that require an additional argument beyond the construction itself; they belong to the separate treatment of the tensor product's basis and space structure, not to the description of the resulting space as the construction's immediate output.

Decomposability of Individual Elements

Whether a particular element of the resulting space is decomposable, and what its tensor rank is, are questions about individual elements that go beyond what the construction pipeline itself settles; the pipeline guarantees every element is some finite sum of decomposables, but says nothing about how short that sum can be made for a given element.

Role as the Pipeline's Terminus

Where Construction Ends and Structural Study Begins

The resulting space marks the boundary between the constructive stages of the tensor product — the mechanical steps of selecting inputs, forming symbols, imposing relations, and quotienting — and the subsequent study of that space's properties, such as its dimension, its basis, the behavior of induced maps upon it, and the classification of its elements, each of which starts from the resulting space as already given and asks further questions about it.