5.3 Tensor Product Construction

Tensor Product Construction is the sequence of steps by which the tensor product V ⊗ W is produced as a concrete vector space, proceeding from selecting the two input spaces, through creating formal symbols and imposing bilinear relations upon them, to forming the resulting quotient space and fixing the notation used to denote its elements.

The Pipeline as a Whole

Ordered Steps, Each Feeding the Next

The construction proceeds through five stages in a fixed order, each depending on the output of the one before it: input space selection fixes V and W; formal symbol creation builds a free vector space with one basis vector for every pair in V × W; bilinear relation imposition identifies a subspace of relations that must be treated as zero; quotient formation divides the free vector space by that subspace; and the resulting space is the tensor product itself, denoted using the notation fixed for its elements. No stage can be carried out before the ones preceding it are complete, since each stage's input is precisely the output of the last.

Why the Pipeline Is Broken Into Separate Stages

Separating the construction into distinct stages isolates which facts about V ⊗ W depend on which choice: the dimension of the result depends on the input spaces selected, the well-definedness of ⊗ as a bilinear map depends on exactly which relations were imposed, and the coordinate description of an element depends on the basis chosen only after the resulting space already exists. Collapsing the stages into a single description would obscure which of these facts follow from which part of the construction.

Summary of Each Stage

Input Space Selection

Fixes the ordered pair of vector spaces V and W over a common field F, with no restriction to finite dimension, and determines the Cartesian product V × W on which every later stage is built.

Formal Symbol Creation

Treats every pair (v, w) in V × W as an independent, uninterpreted basis vector of a free vector space F(V × W), with no relation yet connecting distinct symbols to each other.

Bilinear Relation Imposition

Identifies the subspace R of F(V × W) generated by the four families of relations that express additivity and homogeneity in each argument separately, capturing exactly the algebraic behavior a bilinear map must satisfy.

Quotient Formation

Divides F(V × W) by R to produce V ⊗ W := F(V × W)/R, and defines the canonical bilinear map ⊗: V × W → V ⊗ W as the composite of symbol creation with the quotient projection.

Resulting Space and Notation

Fixes v ⊗ w as the standard notation for the image of (v, w) under this composite map, and establishes that every element of V ⊗ W is expressible as a finite sum of such decomposable elements.

What the Pipeline Establishes

A Concrete Model Satisfying the Universal Property

Carrying out all five stages produces one specific vector space, together with one specific bilinear map into it, that can be shown to satisfy the universal property characterizing the tensor product; the pipeline is what supplies existence for that property, while the separate universal property area supplies the argument that this specific model is unique up to canonical isomorphism among all models satisfying the same characterization.