= Construction of a vector bundle connection by a partition of unity
A local trivialization of a smooth <vector bundle> gives a local connection by differentiating its component functions. Weight these connections by a subordinate locally finite <partition of unity> and extend the weighted terms by zero. The weights sum to one, so the connection Leibniz rule survives. This proves existence on a paracompact smooth manifold. Averaging connections is valid with weights summing to one because the collection is the <affine space of vector-bundle connections>.
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