5.3.1 Tensor Product Input Space Selection

Tensor Product Input Space Selection is the preliminary step of the tensor product construction in which the two vector spaces V and W to be combined are fixed, together with the requirement that both be vector spaces over the same base field F, before any formal symbol is created or any relation is imposed.

What Must Be Fixed

Two Vector Spaces Over a Common Field

The construction requires both V and W to be vector spaces over the same field F; there is no tensor product defined between a real vector space and a complex one, or between a vector space and a set lacking vector space structure altogether, since the relations imposed later — homogeneity with respect to scalar multiplication by elements of F — presuppose a single field acting on both factors. Selecting V and W over mismatched fields is not a case the construction handles by some fallback; it is simply outside what the construction is defined for.

Order Matters at This Stage

V and W are selected as an ordered pair, giving rise to V ⊗ W, distinct in general presentation from W ⊗ V, even though the two are later shown to be canonically isomorphic. Input space selection fixes this order, since the free vector space F(V × W) created in the next step is built on the specific Cartesian product V × W and not on W × V.

No Restriction to Finite Dimension

Nothing in input space selection requires V or W to be finite-dimensional; the construction proceeds identically for infinite-dimensional spaces, though results relying on a finite basis — such as the dimension formula dim(V) · dim(W) — apply only once finite-dimensionality is separately assumed at a later stage.

Why This Step Is Separated

Distinguishing the Choice of Inputs from the Construction Applied to Them

Selecting V and W is a choice external to the construction machinery itself: the same construction procedure — formal symbol creation, quotient formation — applies uniformly no matter which two vector spaces over F are chosen. Isolating the selection step keeps clear that properties of the resulting tensor product depending on the specific choice of V and W, such as its dimension, are consequences of that choice, not features built into the construction procedure.

Downstream Dependence on This Choice

Every subsequent step depends on V and W exactly as selected here: formal symbol creation builds symbols from pairs drawn from this specific V × W, and the relations imposed during quotient formation reference the addition and scalar multiplication of these specific spaces. Changing the selected input spaces changes the entire downstream construction, even though the procedure connecting selection to result stays fixed.

Relation to Adjacent Steps

Precedes Formal Symbol Creation

Input space selection is logically and procedurally prior to formal symbol creation: symbols (v, w) cannot be formed until V and W are fixed, since the set of symbols to be created is exactly the Cartesian product V × W determined by this selection.

Does Not Overlap with Quotient Formation

Input space selection fixes which spaces are being combined; it says nothing about which relations will later identify certain formal symbols with combinations of others. That question belongs entirely to the later step of imposing bilinear relations, and is independent of how V and W themselves were chosen.