5.3.4 Tensor Product Quotient Formation

Tensor Product Quotient Formation is the step of the tensor product construction in which the free vector space F(V × W) is divided by the relation subspace R, producing the vector space V ⊗ W := F(V × W)/R together with the canonical surjective linear map that sends each formal symbol to its equivalence class.

Forming the Quotient Vector Space

Equivalence Classes as Elements

Given the subspace R of F(V × W) fixed during bilinear relation imposition, the quotient F(V × W)/R has as its elements the cosets x + R for x in F(V × W), with two elements x and y representing the same coset exactly when x − y lies in R. This is the standard construction of a quotient vector space by a subspace, applied here to the specific free vector space and relation subspace already fixed in the earlier stages.

Inherited Vector Space Operations

Addition and scalar multiplication on the quotient are defined by

and

with well-definedness — independence from the choice of representative x or y within a coset — guaranteed because R is a subspace: if x_1 + R = x_2 + R, then x_1 − x_2 is in R, and adding or scaling this difference keeps it in R since R is closed under both operations.

The Canonical Map

From Symbols to Cosets to Tensor Elements

The quotient projection π: F(V × W) → F(V × W)/R sends x to x + R, and is linear and surjective by the general theory of quotient vector spaces. Composing π with the formal symbol assignment (v, w) ↦ (v, w) gives the canonical map ⊗: V × W → V ⊗ W, (v, w) ↦ (v, w) + R, from which the standard notation v ⊗ w is adopted.

Bilinearity Is Inherited from R, Not Reproven Independently

Because R was generated exactly by the elements expressing failures of additivity and homogeneity in each argument, each generator of R becomes the zero coset in the quotient; this is precisely what makes ⊗ bilinear, and no separate verification beyond checking that these specific generators vanish is needed to establish bilinearity of the canonical map.

Consequences for the Resulting Space

Surjectivity Guarantees Every Element Is a Sum of Decomposables

Since π is surjective and F(V × W) is spanned by the symbols (v, w), every coset in V ⊗ W is the image of some finite sum of symbols, and therefore every element of V ⊗ W is a finite sum of elements of the form v ⊗ w; this is the origin of the fact, used throughout the element and space areas, that decomposable elements span the entire tensor product.

The Quotient Is What Makes the Space Finite-Dimensional

When V and W are finite-dimensional, the vast, generally infinite-dimensional free vector space F(V × W) is collapsed by the quotient down to a space of dimension dim(V) · dim(W); quotient formation is the step responsible for this collapse, since formal symbol creation alone produces no such reduction.