PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-01

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Tensor product of multilinear tensors is associative and bilinear

Statement

On a finite-dimensional real vector space, the tensor product of multilinear tensors is associative and bilinear in each factor.

Facts & Assumptions

Given: Tensors $R, S, T$ on the same finite-dimensional real vector space and scalars $a, b$ .

[F1]

The tensor product is defined by multiplying the two factor values on concatenated arguments (The tensor product of multilinear tensors).

Proof

technique · direct

1.1F1givenalgebra

Fix a list of arguments of the right total type. By [F1], $((R \otimes S) \otimes T)$ evaluates on that list as the product of the three separate values $R (\dots) S (\dots) T (\dots)$ .

1.2F1givenalgebra

The same formula [F1] gives exactly the same scalar for $R \otimes (S \otimes T)$ on the same list of arguments. Hence $(R \otimes S) \otimes T = R \otimes (S \otimes T)$ .

1.3F1givenalgebra

Again by [F1], $((a R + b S) \otimes T) (\dots) = (a R + b S) (\dots) T (\dots) = a (R \otimes T) (\dots) + b (S \otimes T) (\dots)$ , and the same computation in the second slot gives bilinearity there as well.

2.1step 1.2step 1.3∎

Therefore the tensor product is associative and bilinear in each factor.

Depends on

The tensor product of multilinear tensors

Dependency tree · 2 levels

Sources

Will J. Merry, Differential Geometry