There is a new and attractive book on Index theory and applications to Physics 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.