TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-30

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A locally finite sum of smooth functions is smooth

Statement

Let $M$ be a smooth manifold and let $(f_{i})_{i \in I}$ be smooth functions $f_{i} : M \to R$ whose supports form a locally finite family. Then the pointwise sum $f : = \sum_{i \in I} f_{i}$ is well defined and smooth.

Facts & Assumptions

Given: A family $(f_{i})_{i \in I}$ of smooth real-valued functions on $M$ whose supports are locally finite.

[L1]

If the supports are locally finite, then the cozero sets are locally finite (Locally finite supports have locally finite cozero sets).

[L2]

Smoothness is local on the source (Smoothness is local on the source).

[A1]

A finite sum of smooth real-valued functions on a smooth manifold is smooth.

Proof

technique · direct

1.1

Fix $p \in M$ ; by [L1], there is an open neighbourhood $U$ of $p$ meeting only finitely many cozero sets, say those with indices $i_{1}, \dots, i_{m}$ .

L1givenchoose

2.1

On $U$ , the pointwise sum equals the finite sum $f_{i_{1}} + \dots + f_{i_{m}}$ , so it is well defined and smooth by [A1].

A1step 1.1

3.1

Because every point has such a neighbourhood, [L2] implies that $f$ is smooth on all of $M$ .

L2step 2.1∎

Depends on

Used by

Dependency tree · two levels

9 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)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)