Lang S. Real and Functional Analysis

3 ed. 1993 Springer-Verlag New York, Inc. 596 p.This book is meant as a text for a first year graduate course in analysis.
Any standard course in undergraduate analysis will constitute sufficient
preparation for its understanding, for instance, my Undergraduate
Analysis. I assume that the reader is acquainted with notions of uniform
convergence and the like.
In this third edition, I have reorganized the book by covering
integration before functional analysis. Such a rearrangement fits the way
courses are taught in all the places I know of. I have added a number of
examples and exercises, as well as some material about integration on the
real line (e.g. on Dirac sequence approximation and on Fourier analysis),
and some material on functional analysis (e.g. the theory of the Gelfand
transform in Chapter XVI). These upgrade previous exercises to sections
in the text.