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

1.1L1givenchoose

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

2.1A1step 1.1

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].

3.1L2step 2.1∎

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

A locally finite sum of smooth functions is smooth

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources