9.8.1 Tensor Basis Tensor Product Form

The Tensor Basis Tensor Product Form constructs multilinear relationships through basis elements, enabling algebraic manipulation of tensor spaces.

Tensor Basis Tensor Product Form is the explicit expression of a composite tensor basis element as a multilinear map built directly from the individual actions of its factors, defined so that evaluating the composite on a full list of arguments equals the product of each factor evaluated on its own matching argument; it gives the precise functional meaning of a symbol such as e_i ⊗ e^j, specifying not just how the symbol is built from factors but exactly what output it produces when supplied with arguments.

The Defining Rule of the Product Form

Evaluating a Composite on a Full Argument List

Given a composite basis tensor e_i ⊗ e^j, accepting one covector argument and one vector argument, the tensor product form specifies its value on a pair of arguments (α, v) as the product of e_i evaluated on α and e^j evaluated on v.

(e_{i} \otimes e^{j}) (α, v) = e_{i} (α) \times e^{j} (v)

Each Factor Consumes Only Its Own Matching Argument

The tensor product form assigns each argument in the list to exactly one factor, according to that factor's position in the ordered tuple defining the composite; no factor evaluates any argument other than the one supplied to its own designated slot.

Multilinearity of the Product Form

Linearity Holds in Every Slot Separately

Because each factor is itself linear in its single argument, the tensor product form of a composite basis tensor is guaranteed to be linear in each of its arguments taken separately, with all other arguments held fixed, which is exactly the defining multilinearity required of any tensor.

(e_{i} \otimes e^{j}) (α + β, v) = (e_{i} \otimes e^{j}) (α, v) + (e_{i} \otimes e^{j}) (β, v)

The Product Form Extends Uniformly to Any Number of Factors

For a basis element composed from any number of factors, the same rule applies: evaluating the composite on a full argument list produces the product of each individual factor's evaluation on its own matching argument, regardless of how many factors are involved.

Using the Product Form to Verify the Pairing Condition

Reconfirming the Kronecker Delta Pairing

The tensor product form makes it straightforward to verify that a composite basis element behaves correctly under the assignment mechanism of a tensor coordinate basis system, since substituting basis or dual-basis elements for the arguments reduces the product form directly to a product of Kronecker deltas.

(e_{i} \otimes e^{j}) (e^{k}, e_{l}) = δ_{i}^{k} δ_{l}^{j}

Verifying Multilinear Expansion of General Tensors

The product form also underlies checking that the expansion role of a tensor basis is consistent: expanding an arbitrary tensor as a weighted sum of composite basis elements and evaluating the sum on a chosen argument list reduces, by the tensor product form, to a computable sum of products, matching the component array obtained through ordinary component assignment.

Diagram of the Product Form

Consequences of the Product Form

It Gives Every Composite Basis Element a Computable Value

Because the product form specifies exactly how to evaluate a composite basis element on any argument list, no composite basis tensor remains an abstract, uninterpreted symbol; every one comes equipped with a concrete rule for producing a number from any admissible set of arguments.

It Reduces Verification of Tensor Identities to Ordinary Arithmetic

Any identity claimed about composite basis tensors can be checked directly by substituting the tensor product form and reducing to arithmetic on the individual factor evaluations, replacing what might otherwise require abstract argument with a concrete, step-by-step calculation.