To prove the APS theorem it is illuminating to embed the index problem in a one-parameter family. Namely the Sobolev spaces extend to weighted spaces, x sHĮn(X ’1 L), for s Є R, where x Є C°°(X) is a defining function for the boundary. Then

The eigenvalues of 9o,e form a discrete subset spec(9o,£;) C R, unbounded above and below, so inds(9į) is defined for — s Є R \ spec(Əo,£). It is convenient to extend the definition of the index, even to the case that the operator is not Fredholm by setting

The parameter, 5, can be absorbed into the operator by observing that the weighting factor, xs, can be treated as ‘rescaling’ (in the sense of Chapter 8) of the coefficient bundle E to a bundle E(s).