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

1.1

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.

L1given

2.1

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 $M$ .

L1L2step 1.1∎

Depends on

Used by

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Sources

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