PropositionStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-31

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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.

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Smooth sections form a module over smooth functions

Statement

If $E \to M$ is a smooth vector bundle, then the smooth sections of $E$ form a module over $C^{\infty} (M)$ under pointwise addition and scalar multiplication.

Facts & Assumptions

Given: A smooth vector bundle $E \to M$ .

[L1]

A section is smooth exactly when its local frame components are smooth (Smoothness of a section is equivalent to smooth local components).

[L2]

Sums and products of smooth scalar functions are smooth (Sums and scalar multiples of totally differentiable maps are totally differentiable with the expected derivatives).

Proof

technique · direct

1.1

Let $σ, τ$ be smooth sections and let $f \in C^{\infty} (M)$ . On a local frame, write $σ = \sum a_{i} s_{i}$ and $τ = \sum b_{i} s_{i}$ with smooth components $a_{i}, b_{i}$ . Then $σ + τ = \sum (a_{i} + b_{i}) s_{i}$ and $f σ = \sum (f a_{i}) s_{i}$ .

L1given

2.1

By [L2], the component functions $a_{i} + b_{i}$ and $f a_{i}$ are smooth, so [L1] shows that $σ + τ$ and $f σ$ are again smooth sections. The module axioms hold fibrewise because each fibre is a vector space.

L1L2step 1.1∎

Depends on

Used by

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