## ABSTRACT

The basic analytic tool developed below to carry out the proof of the APS theorem is the calculus of 6-pseudodifferential operators. This allows the mapping, especially Fredholm, and spectral properties of to be readily understood. As motivation for the analytic part of the discussion the one-dimensional case will first be considered, although the result is not proved in detail. This case is ‘easy’ for many reasons, not least because the dimension is odd, which means there is no interior contribution to the index, and the boundary dimension is zero, so the boundary operator is a matrix, i.e. has finite rank. However, from an analytic point of view the one-dimensional cases serves as quite a good guide to the general case. 1.1. Operators and coordinates.