I just spent the last few days working through the proof of the Hodge theorem in Wells Differential Analysis on Complex Manifolds. There were a few things which confused me right at the start, and which could have been easily cleared up if I had just understood which constructions were canonical and which were not. The point of this blog entry is to record that data. I’ll also add a little extra background, so people other than me have a chance of understanding it. Since I am very much not an analyst, there is a high risk of errors here; I would appreciate anyone who points them out to me. Also, to keep life simple, will be an integer and will be .
Let be a smooth -dimensional Riemmannian manifold and a complex vector bundle equipped with a positive definite Hermitian form.
Things that are true, important, and canonical
For every integer , there is a topological vector space . There are inclusions . These are compact maps, meaning that if is a bounded sequence in then has a convergent subsequence in . The smooth functions, , have compatible inclusions into all of the ‘s, with dense image.
For , a sequence of smooth functions is Cauchy in if, for any differential operator , of order , the sequence is Cauchy in the topology. So and, more generally, can be thought of all the completion of in this variant of the topology.
We have containments , compatible with the injections and . In other words, if we have a sequence in which is -Cauchy, then it approaches a limit in the -topology and, if we have a sequence in approaching a limit in the -topology, then it is -Cauchy.
There is a sesquilinear pairing with respect to which and are adjoint. This exhibits as the conjugate-linear-dual of . (And, I think, vice versa, although Wells doesn’t say this.) When , this is the standard pairing on . The fact that the dual of the smaller space is the larger space is strange. My current intuition is that, although is an injection, it schrunches down, so that it is easier for sequences to converge in the image. Thus, many linear functions which are continuous on are no longer continuous as functions on with the topology restricted from . I still find this hard to think about, though.
An order differential operator will extends to a continuous map to . The above pairing will induce an adjoint map . In the cases I care about, is also a differential operator, but I think this is in general not true.
has the structure of a normed vector space, which is canonical in the sense that the norm is well defined up to a constant distortion factor.
The following is true, but should be ignored
The topology on arises from a Hilbert space structure. Nothing is ever adjoint with respect to this inner product. Orthogonality in this inner product is not a canonical property. Since it is a Hilbert space, is isomorphic to its own dual and, hence, ; this isomorphism is never used. It would have been better had I not known this structure existed.
Below the fold, for the curious, I will repeat Wells’ definitions in the order I would have given them.
Sobolev Structures on
Terry Tao has a very comprehensive blog post on this, so I’ll be brief.
Let be the standard norm on functions on . For , define a norm by
where are positive reals whose exact values don’t matter. (This is the first evidence that the Hilbert space structure on doesn’t matter.)
Now, let be the Fourier transform of and let be the frequency variable dual to . Then we have the convenient formula
Up to a constant distortion, this is the same as
We use this last formula to define the topology for negative .
There are now two ways we can think of , at least for .
From the inside: is the completion of in the above norms. (The subscript means compact support.) One can easily see that is in , so one can also think of it as the corresponding completion of . The injection is by noticing that any Cauchy sequence is also Cauchy.
From the outside: is the subspace of where the integral converges. By a nontrivial lemma, this is actually a subspace of so, for large enough, we can think of it as a subspace of these somewhat smoother functions. The inclusion is given by noticing that this condition on the integral becomes weaker as becomes more negative.
The pairing between and is extended from the standard pairing on : . That this extends continuously to follows from Cauchy-Schwartz.
Sobolev spaces on compact manifolds
The above definition exhibited as a Hilbert space: our norm obeys the parallelogram law. I’m not sure whether or not there is any benefit to knowing this. But, when we get to manifolds, there definitely isn’t.
Let be a compact Riemmannian manifold and a Hermitian vector bundle. For , define the topology on by saying that is Cauchy if, for every differential operator of order , the sequence is Cauchy.
Choose a finite open cover , so that embeds into and such that is trivial on ; choose a trivialization of on , and choose a partition of unity subordinate to . Then any section gives a collection of functions on ; the components of .
Lemma: A sequence is -Cauchy if and only if all the sequences , for all and , are -Cauchy.
This lemma is not completely trivial: I goofed up the first two times I tried to write a proof. Hint: There is some such that is covered by the sets .
This lets us define -Cauchy sequences for negative . We can then define as the completion of in this topology. The fact that this topology can be given by a norm is somewhat important; the fact that this norm can be chosen to obey the paralellogram law is completely unimportant.
When , we can think of as a subspace of , given by those functions whose finite smooth cutoffs have sufficiently slow growing Fourier transforms. The fact that the precise details of the smooth cutoff do not effect the high-frequency behavior of the Fourier transform is really cool and nonobvious, and is perhaps most easily seen from the above lemma.
And, finally, the pairing between and is still given by the continuous extension of . One shows that the integral converges by, once again, passing to local coordinates and using Cauchy-Schwartz.