PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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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.
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The density line and its positive cone

Statement

The densities $D (V)$ form a one-dimensional real vector space under pointwise operations. Evaluation on any basis is a linear isomorphism $D (V) \to R$ . The nonzero nonnegative densities form a canonical positive ray.

Facts & Assumptions

[F1]

A signed one-density on a real vector space: For an $n$ -dimensional real vector space $V$ , a one-density is a real-valued function $δ$ on ordered bases such that $δ (v A) = ∣ \det A ∣ δ (v) (A \in G L (n, R)) .$ For $n \geq 1$ extend its value by zero to dependent $n$ -tuples. For $n = 0$ it is an arbitrary real scalar on the empty basis (the empty determinant is one). Write $D (V)$ for these densities. Positive means strictly positive on every basis; nonnegative includes zero. Negative scalar multiples remain densities; positivity is extra structure on their one-dimensional real space.

Proof

Given: The objects and hypotheses in the statement above.

1.1

Fix a basis $e$ (the empty basis when $n = 0$ ). Every basis is uniquely $e A$ for $A \in G L (n, R)$ . A density is determined by $c = δ (e)$ since $δ (e A) = c ∣ \det A ∣$ . Conversely this formula, with zero on dependent tuples for $n \geq 1$ , satisfies the required transformation law by multiplicativity of determinants.

2.1

The formula is linear in $c$ , so evaluation and its displayed inverse are linear bijections. Since $∣ \det A ∣ > 0$ , positivity is equivalent to $c > 0$ and nonnegativity to $c \geq 0$ , independently of the chosen basis. Thus the nonzero nonnegative densities are exactly one ray. For $n = 0$ this says precisely that scalars form $R$ .

step 1.1algebra∎

Depends on

A signed one-density on a real vector space

Used by

Density bundle and smooth density fields Definition

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

Lee Proposition 16.35(a,b,d), pp.428–429 (standard reference, not scraped)