There is a new and attractive book on [Index theory and applications to Physics](http://inspirehep.net/record/1205569/files/BlckBss_2012_04_25.pdf) by Booss and Bleecker which covers all the necessary analysis background. To quote from its preface : >In order to enjoy reading or even work through Parts I-III, we expect the readerto be familiar with the concept of a smooth function and a complex separableHilbert space. Nothing more.) Note that this is different from the 1984 book 'Topology and Analysis : ASIT and Gauge theoretic Physics' by the same authors which has been mentioned above.