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

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A locally finite positive smooth family normalizes to a partition of unity

Statement

Let $(g_{i})_{i \in I}$ be a locally finite family of smooth functions $g_{i} : M \to [0, \infty)$ on a smooth manifold $M$ , and suppose that for every $p \in M$ at least one $g_{i} (p)$ is strictly positive. Put $G : = \sum_{i} g_{i}$ . Then $G$ is a positive smooth function, each $ϕ_{i} : = g_{i} / G$ is smooth, and $(ϕ_{i})_{i \in I}$ is a partition of unity subordinate to $(supp (g_{i}))_{i \in I}$ .

Facts & Assumptions

Given: A locally finite family $(g_{i})_{i \in I}$ of nonnegative smooth functions on $M$ that is pointwise positive.

[L1]

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

[L2]

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

[A1]

A positive smooth real-valued function has a smooth reciprocal.

Proof

technique · direct

1.1

By [L1], the sum $G : = \sum_{i} g_{i}$ is smooth; because the $g_{i}$ are nonnegative and some $g_{i} (p)$ is positive at each point, one has $G (p) > 0$ for all $p \in M$ .

L1given

2.1

By [A1], the reciprocal $1 / G$ is smooth, so each $ϕ_{i} = g_{i} (1 / G)$ is smooth and nonnegative; also $supp (ϕ_{i}) \subseteq supp (g_{i})$ .

A1step 1.1

3.1

The family $(ϕ_{i})$ is locally finite by [L2], and $\sum_{i} ϕ_{i} = (1 / G) \sum_{i} g_{i} = 1$ pointwise. Therefore $(ϕ_{i})$ is a partition of unity subordinate to $(supp (g_{i}))$ .

L2step 1.1step 2.1∎

Depends on

Used by

Smooth partitions of unity exist on manifolds Theorem

Dependency tree · two levels

5 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)