 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.