5.5.1 Tensor Elementary Product Form

Tensor Elementary Product Form is the specific bilinear combination rule that produces an elementary tensor v ⊗ w from a pair of vectors v and w, identified as exactly the canonical map fixed during the tensor product's construction, and distinguished from other ways two vectors might be combined that do not follow this rule.

The Rule Itself

The Canonical Bilinear Map Applied to a Pair

Given v in V satisfying factor membership and w in W satisfying factor membership, the product form assigns the pair (v, w) the value v ⊗ w, defined as the image of (v, w) under the canonical map ⊗: V × W → V ⊗ W fixed at the end of the tensor product construction. There is no alternative rule under this name; "the product form of v ⊗ w" and "the value of the canonical bilinear map at (v, w)" refer to the same thing.

Bilinearity Is the Defining Behavioral Signature

What characterizes the product form, beyond simply being some assignment of a value to each pair, is that it satisfies

together with the analogous identity in the second argument and the two homogeneity identities (cv) ⊗ w = c(v ⊗ w) = v ⊗ (cw); these four identities, inherited directly from bilinear relation imposition during construction, are what make the product form a bilinear rule rather than an arbitrary pairing.

Distinguishing the Product Form From Other Combinations

Not the Same as Any Rule Producing a Scalar

A bilinear form, such as an inner product ⟨v, w⟩, also combines v and w bilinearly but produces an element of the base field rather than an element of V ⊗ W; the product form specifically targets the tensor product space itself and is, in a precise sense, the most general possible bilinear combination of v and w, since every other bilinear combination — including every bilinear form — factors uniquely through it by the universal property.

Not the Same as Direct Sum Pairing

Combining v and w as (v, w) in the direct sum V ⊕ W produces an element of a space of dimension dim(V) + dim(W), with v and w remaining visible as separate, independently addressable components; the product form instead merges v and w into a single element of a space of dimension dim(V) · dim(W), from which the original v and w cannot in general be recovered as separately stored components, only reconstructed through factorization when the element happens to be elementary.

Not an Operation Internal to a Single Factor Space

The product form always combines a vector from V with a vector from W into an element of the new space V ⊗ W; it is not to be confused with an internal operation such as addition within V alone or within W alone, both of which stay within a single one of the two factor spaces rather than producing an element of their tensor product.

Role Within Elementary Tensor Structure

The Mechanism Behind Every Elementary Tensor's Existence

Every elementary tensor exists precisely because the product form assigns it a value; the symbolic form v ⊗ w is the notation for the output of this rule, factor membership fixes the valid inputs to it, and factor order fixes which of the two input positions v and w occupy — the product form itself is the rule connecting valid inputs, in their fixed order, to the resulting elementary tensor.