We know what a partition of unity is, but not all partitions of unity are very useful. For instance, the single function defined by for all points is a partition of unity all on its own — its support is itself, which is clearly a locally finite cover of , and it adds up to the constant unit function. But we can’t really do anything with it.
What we need is a partition of unity subordinate to an open cover. That is, given a collection of open sets that cover , we want a partition of unity such that for every there is some so that is supported in . In particular, we can let be the collection of coordinate patches in a smooth atlas, so each of the functions “lives in” a single local coordinate system.
But do any such things exist? Remember, except for the trivial example above I haven’t actually given any examples of a smooth partition of unity at all. The example last time was differentiable, and even twice-differentiable, but not smooth. So this is a nice concept, but it might well be vacuous.
Still, all is not lost: I say that given any open cover of a smooth manifold, there is a countable smooth partition of unity subordinate to that cover. In particular, given any smooth structure on a manifold we can always find a partition of unity with each function supported completely within a single coordinate patch. The proof of this fact, however, is one of the few really annoying, fiddly, technical bits in differential geometry. It will take a few days of doing, and I fully understand if you’d rather just skip it. All you really need to know is: whenever we need a partition of unity to break global things defined over our entire manifold up into nice chunks that fit into coordinate patches, we can do it.
However, I should point this out: analytic manifolds are not nearly so forgiving. The basic (but sketchy) idea is that in order to construct our partitions of unity we’ll need to create “bump” functions sort of like the one we did last time, but ones that are smooth instead of just twice-differentiable. This means using a piecewise definition, just like last time, and at the edge of a piece we’ll have points such that in any neighborhood of that point we need two different definitions of the function. But if the function is supposed to be analytic, then the definition that works on one side should keep working on the other side, and so we can’t make the bump functions we need.
This is a big reason why people stop at smooth manifolds rather than working with analytic ones, despite the fact that analytic functions are arguably “nicer”. Unfortunately, this also means that not everything we do carries over quite so easily to complex manifolds — based on complex vector spaces — which must always be analytic.