Editing Atiyah–Singer index theorem (section)

==Analytical index==
As the elliptic differential operator ''D'' has a pseudoinverse, it is a [[Fredholm operator]]. Any Fredholm operator has an ''index'', defined as the difference between the (finite) dimension of the  [[kernel (algebra)|kernel]] of ''D'' (solutions of ''Df'' = 0), and the (finite) dimension of the [[cokernel]] of ''D'' (the constraints on the right-hand-side of an inhomogeneous equation like ''Df'' = ''g'', or equivalently the kernel of the adjoint operator). In other words,
:Index(''D'') = dim Ker(D) − dim Coker(''D'') = dim Ker(D) − dim Ker(''D*'').
This is sometimes called the '''analytical index''' of ''D''.

'''Example:''' Suppose that the manifold is the circle (thought of as '''R'''/'''Z'''), and ''D'' is the operator d/dx − λ for some complex constant λ. (This is the simplest example of an elliptic operator.) Then the kernel is the space of multiples of exp(λ''x'') if λ is an integral multiple of 2π''i'' and is 0 otherwise, and the kernel of the adjoint is a similar space with λ replaced by its complex conjugate.  So ''D'' has index 0. This example shows that the kernel and cokernel of elliptic operators can jump discontinuously as the elliptic operator varies, so there is no nice formula for their dimensions in terms of continuous topological data. However the jumps in the dimensions of the kernel and cokernel are the same, so the index, given by the difference of their dimensions, does indeed vary continuously, and can be given in terms of topological data by the index theorem.