Analysis II

This
is part two of a two-volume book on real analysis and is intended for senior
undergraduate students of mathematics who have already been exposed to
calculus. The emphasis is on rigour and foundations of analysis. Beginning with
the construction of the number systems and set theory, the book discusses the
basics of analysis (limits, series, continuity, differentiation, Riemann
integration), through to power series, several variable calculus and Fourier
analysis, and then finally the Lebesgue integral. These are almost entirely set
in the concrete setting of the real line and Euclidean spaces, although there
is some material on abstract metric and topological spaces. The book also has appendices
on mathematical logic and the decimal system. The entire text (omitting some
less central topics) can be taught in two quarters of 25-30 lectures each. The
course material is deeply intertwined with the exercises, as it is intended
that the student actively learn the material (and practice thinking and writing
rigorously) by proving several of the key results in the theory.