 in Lp plus norm of u in Lp. But this is one, this is zero. So we know that this is bounded by a constant c. Well, let me take in fact here L plus one p's and I would have here w one p, but this doesn't change the argument. So again, we have a bound by a constant. But now I know that because of the compactness of the embeddings of some of the spaces, I know that the sequence u n viewed as a sequence in w k p is compact. That is, there is a convergent subsequence. Subsequence, I will still denote this by u n and this converges strongly into some u infinity in w Lp. So that what I meant is L. Right, but our operator is a bounded one, so we can apply L to, so L u n converges to L u infinity. This is zero, so u infinity is again in the kernel of L, which means that the sphere, the unit sphere in this space L is compact and you know very well that if you take the unit sphere in a vector space and it happens to be compact, then this vector space is finite dimensional. So this is how the proof works and what you see, what we used in the proof was soberly embedding theorems and elliptic estimates and that's what is essential about elliptic operators. Okay, are there any questions to that? Yes? We didn't need to, it was just convenient for me to start because we can, in the setting I described, we can apply L to functions in W Lp only on K plus Lp. So here I took K equals zero or K equals one to be more precise, but sort of as I described this, the differentiability order here must be bigger than L. In fact, this is really not important. We could have started here with any differentiability degree, for instance, zero, but you would need to know how to deal in this case. By the way, this elliptic estimate is still valid in this case when K is sort of negative, but you need to know how to define the norm for negative Ks. It can be done. So one more useful generalization of the notions we consider so far is the notion of an elliptic complex. So what we have is, we have a complex and that is we consider sections of some bundle, say E1 into maps, say a map L1 from sections of E1 into sections of E2, say L2 in gamma E3, and so on. So let me take a finite complex for simplicity, that is gamma EK, so here is LK minus one, then zero. So assume this is, let it be two star as a complex. In the usual sense, that is when we take L1 and L2, we have zero, well for any sequence of two maps, you know. So it assumes that this is a complex, then this is called elliptic, so two stars as elliptic, if the sequence of symbols, that is what we have is pi star E1, pi star E2, pi star E3, and so on, so pi star EK, and we have maps here, so sigma L1, sigma L2, sigma L3, and so on. If this sequence of symbols is exact. Right, in particular for a very short complex, if you have just pi star E to pi star F, so the exactness of this sequence means exactly that the symbol of the operator L is invertible, that is the operator L itself is elliptic. Now since we have a complex, we can define its homology groups, so we have Hj of, well let me skip the notation, so this is just the kernel of say Lj divided by the image of Lj minus one, so this is J's homology of two star. Now what I will also need is the notion of the formal adjoint, so Lj star, this is a formal adjoint of Lj, which means simply that if I take L2 scalar product of Lj S with any section T in L2, this is simply S Lj star T, again in L2, and so you can prove that for any differential operator there is a formal adjoint operator and this is again a differential operator of the same order, and this notion at hand, we can define the corresponding Laplacian, delta J is Lj star Lj plus Lj minus one star Lj minus one. So that's if we are here in the complex, what we do, we go by L2 in here, say, and then we take the adjoint in here, and also in the other direction, so we go first here and then here, and the sum of these operators is just delta J, and an easy exercise is to show that this is an elliptic operator, whenever our complex two star is an elliptic complex. So this is a simple exercise in Dini algebra, so let me denote by Hj, the kernel of delta J, yeah, mm-hmm, thank you. So for J minus one, you have to go first in this direction, then take Lj, J minus one. So these are called harmonic sections, and the main theorem is if M is compact, our complex is elliptic, then the following holds. So first of all, the space of harmonic sections is finite dimensional. Secondly, S is a harmonic section, if and only if Lj of S is zero, and Lj minus one, star of S is zero, and the set claim and the main one is that we have a natural map from the space of harmonic maps into the J's cohomology group of the complex, since whenever we have a harmonic section, it is sort of closed, so we can take its cohomology class, so S is mapped simply to the cohomology class of S, and the claim is that this is an isomorphism. So the proof of this theorem is not really hard. I gave a proof in my notes, so if you're interested, take a look, but this is really sort of an elementary computation, computation was what we have done so far. Let me actually give you an interpretation of this fact. So what we have done is the following. So let me say maybe linear gauge theory. What we have is we have a manifold C, which is, let us take this to be C infinity Ej. This is our manifold. So the notation suggests that I refer to what we had in the very first lecture, and we have a group G, which acts on this manifold. This is now C infinity Ej minus one, and the action is whenever you have S and G, you can send this to S plus Lj minus one of G. So here clearly S is an element in here, and G is an element in here. So we have an infinite dimensional manifold and an infinite dimensional group, and the group acts on this manifold. So we can write, we can ask, and we also have a map Lj from C into V, C into V, and V is just taken to be C infinity Ej plus one, viewed as a trivial G representation. In particular, any point here is a fixed point of the action, but I can take, say, the origin here, and I can consider the modular space. So I can take Lj inverse of zero, and I can divide this by the gauge group, and what you immediately realize what we have here is a J's cohomology group of the complex just by definition, and this is our modular space. Now, because everything is linear, we know that this is a vector space, so the only invariance that we have is the dimension of this space, and so I could define Pj, be the dimension of Hj. So let me consider one, perhaps, more concrete example where we take the drum complex. So here is an example. So what we have, we have a manifold, say compact manifold oriented, so oriented, Romanian, and we have the drum complex that is in omega zero M and omega one M and so on. Now, what we know is that this is an elliptic complex, and so by what we have done so far, we know that the J's cohomology group of this complex is finite dimensional, and as we have defined, Pj is the dimension of the J's cohomology group. This is our invariant. So of course, we know that this complex, so the cohomology groups here, Hj, are just the RAM cohomology groups, and Pj is just the J's petty number. But just by the way, an interesting invariant of a topological manifold, as you all surely know, and this is what happens more or less generally if you have a linear elliptic complex, the invariants that you can get are topological invariants of your manifold. So this is not quite what we are looking for in gauge theory. We are looking for somewhat more interesting invariants, but this requires a non-linear version of the theory. So what we will do in the next lecture will be an abstract setup for non-linear frithole maps, and then we will go to the cyberquit in gauge theory. Okay, so that's all I wanted to tell you today. Are there any questions? All right, then. See you tomorrow.