DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-09-01

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A type $(r, s)$ tensor on a finite-dimensional vector space

Definition

Let $V$ be a finite-dimensional real vector space, and let $r, s \geq 0$ . A type $(r, s)$ tensor on $V$ is a multilinear map

$T : (V^{*})^{r} \times V^{s} \to R .$

With this convention, vectors are type $(1, 0)$ tensors and covectors are type $(0, 1)$ tensors.

Depends on

Used by

Alternating k-covectors Definition
The contraction of a mixed tensor Definition
The permutation action on covariant tensors Definition
The pullback of a covariant tensor by a linear map Definition
The tensor product of multilinear tensors Definition
The type (r,s) tensor bundle Definition
A bilinear form as a type (0,2) tensor Example
An endomorphism as a type (1,1) tensor Example
A tensor is not determined by its values on diagonal tuples without symmetry False statement

Dependency tree · two levels

11 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

John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)

A type (r,s)(r,s) tensor on a finite-dimensional vector space