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

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Locally finite linear combinations of sections are smooth

Statement

Let $(σ_{i})_{i \in I}$ be smooth sections of a vector bundle $E \to M$ and let $(f_{i})_{i \in I}$ be smooth real-valued functions on $M$ . If the family of supports $(supp (f_{i} σ_{i}))_{i \in I}$ is locally finite, then the pointwise sum

$\sum_{i \in I} f_{i} σ_{i}$

defines a smooth section of $E$ .

Facts & Assumptions

Given: Smooth sections $σ_{i}$ , smooth functions $f_{i}$ , and a locally finite family of supports $supp (f_{i} σ_{i})$ .

[L1]

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

[L2]

A locally finite sum of smooth scalar functions is smooth (A locally finite sum of smooth functions is smooth).

Proof

technique · direct

1.1L1given

Fix a local frame $(s_{1}, \dots, s_{r})$ on an open set $U$ . Write $σ_{i} ∣_{U} = \sum_{j} a_{i j} s_{j}$ with smooth coefficient functions $a_{i j}$ . Then on $U$ the formal sum has components $\sum_{i} f_{i} a_{i j}$ . Because the supports of $f_{i} σ_{i}$ are locally finite, only finitely many terms are nonzero near each point.

2.1L1L2step 1.1∎

Each component $\sum_{i} f_{i} a_{i j}$ is therefore a locally finite sum of smooth scalar functions, so it is smooth by [L2]. Applying [L1] again shows that $\sum_{i} f_{i} σ_{i}$ is a smooth section on $U$ , and hence on all of $M$ .

Depends on

Used by

Every smooth vector bundle admits a connection Theorem

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)