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.