Analytic index

Summary

The analytic index of an elliptic operator $P$ on a closed manifold is $\dim \ker P - \dim \operatorname{coker} P$. Both dimensions are finite because elliptic operators on closed manifolds are Fredholm operators between suitable Sobolev spaces. So the analytic index is a Fredholm index. The Atiyah–Singer index theorem computes it as the topological index.