5.3.2 Tensor Product Formal Symbol Creation

Tensor Product Formal Symbol Creation is the initial step of the tensor product construction in which every ordered pair (v, w), with v in V and w in W, is treated as an independent formal symbol serving as a basis vector of a free vector space, with no algebraic relation yet connecting these symbols to one another or to the vector space structure already present on V and W.

What a Formal Symbol Is

An Uninterpreted Label, Not Yet an Operation

At this stage, (v, w) is a label attached to the pair (v, w), not a computation or an application of any operation; it carries no meaning beyond naming which pair of vectors it corresponds to. In particular, (v, w) is not yet asserted to depend linearly on v or on w — that dependence is introduced only later, through the relations imposed during quotient formation.

One Symbol per Pair, Without Exception

A distinct formal symbol is created for every pair (v, w) in the Cartesian product V × W, including pairs that are related in an obvious way once vector space structure is taken into account, such as (v_1 + v_2, w) versus (v_1, w) and (v_2, w). At the symbol-creation stage, these are three unrelated symbols; nothing here identifies (v_1 + v_2, w) with any combination of the other two.

Why Creation Precedes Relation

Isolating the Raw Material from the Rules Imposed on It

Separating symbol creation from the imposition of relations makes explicit that the free vector space F(V × W) is a genuinely different, much larger object than the tensor product it will become: F(V × W) has one basis vector for every pair in V × W, and its dimension, when V and W are finite-dimensional, is the cardinality of V × W as a set of basis labels — infinite in general — rather than the finite product dim(V) · dim(W) that the tensor product will end up having once relations are imposed.

A Prerequisite for Well-Defined Quotient Formation

Quotient formation, the step that imposes the bilinearity relations, requires a vector space to quotient in the first place; formal symbol creation supplies that vector space by fixing, once and for all, what the "unrelated" starting objects are before any relation identifies some of them with combinations of others.

Relation to the Final Notation

From (v, w) to v ⊗ w

The symbol (v, w) created at this stage is not the same notation as the eventual v ⊗ w; the tensor symbol ⊗ is introduced only after the quotient is taken, to denote the image of (v, w) under the quotient map. Writing v ⊗ w presupposes that the relations of bilinearity are already in force, whereas (v, w) at the symbol-creation stage presupposes nothing beyond membership in V × W.

What Carries Over and What Does Not

The pairing of a specific v with a specific w carries over unchanged from the formal symbol to the final tensor element; what does not carry over is the assumption of independence between distinct symbols, which is precisely what the relations imposed during quotient formation remove.