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

1.1

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 $R (\dots) S (\dots) T (\dots)$ .

F1givenalgebra

1.2

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

F1givenalgebra

1.3

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.

F1givenalgebra

2.1

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

step 1.2step 1.3∎

Depends on

The tensor product of multilinear tensors

Used by

Tensor product and contraction in a basis Example
Tensor products and contractions of smooth tensor fields are smooth Proposition

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Will J. Merry, Differential Geometry (standard reference, not scraped)