5.5.3 Tensor Elementary Factor Membership

Tensor Elementary Factor Membership is the requirement fixing which space each component of an elementary tensor v ⊗ w must belong to — v ranging over all of V and w ranging over all of W, with no restriction to a subspace, a basis, or any other proper subset — and the consequences that follow from this membership being unrestricted.

The Membership Requirement

Each Factor Ranges Over Its Full Space

For v ⊗ w to be an elementary tensor of V ⊗ W, it suffices that v belongs to V and w belongs to W; no further condition is placed on which vectors of V or W are permitted. This is in contrast to, for instance, requiring v to be a basis vector or requiring w to lie in some fixed subspace — no such restriction is part of the definition of elementary tensor, and the full generality of v and w is what makes the set of elementary tensors far larger than the induced basis.

Membership Is Checked Independently in Each Factor

Whether v ⊗ w is a valid elementary tensor is determined by checking v ∈ V and w ∈ W separately; there is no joint condition linking the two beyond both memberships holding simultaneously. This mirrors the argument-by-argument character of bilinearity itself, where each slot is treated independently rather than as a single joint condition on the pair.

Consequences of Unrestricted Membership

The Full Cartesian Product Is Available as Input

Because v and w range over the entirety of V and W, the canonical bilinear map ⊗: V × W → V ⊗ W has domain the full Cartesian product V × W, with every pair producing a valid elementary tensor; this is what allows the universal property to be stated for bilinear maps defined on the whole of V × W, since restricting membership to a proper subset would leave the universal property's factorization undefined for pairs outside that subset.

Zero Vectors Are Permitted and Produce the Zero Tensor

Since 0 belongs to both V and W, v ⊗ 0 and 0 ⊗ w are elementary tensors by the membership requirement, and both equal the zero element of V ⊗ W by homogeneity: v ⊗ 0 = v ⊗ (0 · w) = 0 · (v ⊗ w) = 0 for any fixed w. Membership does not exclude the zero vector from either factor, so this degenerate case is fully accounted for rather than treated as an exception.

Membership Alone Does Not Determine Decomposition

Because membership places no restriction on v beyond belonging to V, many different elements of V can serve as the first factor of an elementary tensor equal to a given t, once t is known to be decomposable at all; membership fixes only where the factors may be drawn from, not how many pairs (v, w) produce the same elementary tensor t, a question addressed separately under factorization of pure elements.

Relation to Adjacent Structure

Distinct from Factor Order

Factor membership concerns which spaces v and w belong to; it says nothing about the significance of v appearing on the left and w on the right, which is a separate structural question addressed under factor order. Membership would be satisfied identically whether the roles of v and w were swapped, so long as the swapped element is still checked against the correct space.

Foundation for the Product Form

Membership requirements on v and w are the precondition that must hold before the product form — the specific bilinear combination rule producing v ⊗ w — can even be applied; the product form presupposes valid membership and builds the elementary tensor from factors already known to satisfy it.