5.9.3 Tensor Product Basis Element Construction

Constructing tensor product basis elements involves combining basis vectors from multiple spaces to form a new basis for the tensor product space.

Tensor Product Basis Element Construction is the formal procedure by which a single basis tensor of a tensor product space is built as a genuine, well-defined element of that space, starting from a chosen basis vector in each factor, and the accompanying argument establishing that the constructed elements are linearly independent and together form a basis rather than merely a spanning or redundant family.

Constructing a Single Basis Element

Given vector spaces V1, V2, …, Vn over a field F with chosen bases, and a specific index tuple (k1, …, kn), the corresponding basis element of the tensor product is constructed as the image, under the canonical multilinear map

τ : V_{1} \times \dots \times V_{n} \to V_{1} \otimes \dots \otimes V_{n}

of the tuple of basis vectors (e^(1)_k1, …, e^(n)_kn). Concretely, in the standard quotient construction of the tensor product, this is the equivalence class of the formal symbol (e^(1)_k1, …, e^(n)_kn) inside the quotient of the free vector space on all tuples by the subspace enforcing multilinearity.

Well-Definedness of the Construction

Before the constructed object can be called a basis element, it must be shown to be a single, unambiguous element of the tensor product space, independent of any incidental choices made along the way.

Independence from the Underlying Construction Model

The tensor product can be built by several different but isomorphic methods — the free-vector-space quotient, or as the target of the universal property characterized abstractly. Basis element construction is designed to be insensitive to which model is used: in every construction, the image of (e^(1)_k1, …, e^(n)_kn) under the canonical multilinear map is, up to the canonical isomorphism identifying the different models, the same element.

Immediate Consequence of Multilinearity

Because the canonical map τ is multilinear by the very definition of the tensor product, the constructed basis element depends only on the chosen basis vectors themselves — not on how those vectors happen to be expressed as combinations of other vectors — so the construction is unambiguous the moment the factor bases are fixed.

Establishing Linear Independence

The heart of basis element construction is not the act of forming the tensors but the proof that distinct index tuples produce linearly independent elements, since without this the constructed family would only span, not form a basis.

The Dual Functional Argument

For each factor space Vi with dual basis f^(i)_1, …, f^(i)_di (satisfying f^(i)_j(e^(i)_k) equal to one when j = k and zero otherwise), one constructs a multilinear functional

ϕ (x_{1}, \dots, x_{n}) = f_{k_{1}}^{(1)} (x_{1}) \dots f_{k_{n}}^{(n)} (x_{n})

whose linear extension to the tensor product, guaranteed by the universal property, takes the value one on the basis element indexed by (k1, …, kn) and the value zero on every basis element indexed by a different tuple. The existence of such a "detector" functional for every index tuple is precisely what rules out any nontrivial linear relation among the constructed basis elements.

From Independence to Basis Status

Combined with the earlier spanning fact — that the finitely many constructed elements already span the tensor product — the linear independence established via dual functionals completes the proof that the constructed family is a genuine basis: spanning plus independence.

Construction for Multiple Factors

The construction generalizes uniformly from two factors to any finite number, with the same argument structure applying at every stage.

Iterated Construction

For three or more factors, the basis element for tuple (k1, k2, k3) can equivalently be constructed in two stages — first forming e^(1)_k1 ⊗ e^(2)_k2 as a basis element of the two-factor tensor product V1 ⊗ V2, then tensoring the result with e^(3)_k3 — relying on the associativity isomorphism of the tensor product to confirm that this iterated construction agrees, up to canonical identification, with the direct n-factor construction.

Uniform Treatment

Because the associativity isomorphism is itself canonical, basis element construction does not depend on the order in which factors are grouped during an iterated build-up, so results proved by induction on the number of factors transfer without modification to the general n-factor statement.

Illustrative Diagram

The diagram shows two chosen factor basis vectors passing through the canonical multilinear map τ to produce a single, well-defined basis element of the tensor product space.