Book: Advanced Real Analysis
This book and its companion volume Basic Real Analysis systematically develop
concepts and tools in real analysis that are vital to every mathematician, whether
pure or applied, aspiring or established. The two books together contain what the
young mathematician needs to know about real analysis in order to communicate
well with colleagues in all branches of mathematics.
The books are written as textbooks, and their primary audience is students
who are learning the material for the first time and who are planning a career in
which they will use advanced mathematics professionally. Much of the material
in the books corresponds to normal course work. Nevertheless, it is often the
case that core mathematics curricula, time-limited as they are, do not include all
the topics that one might like. Thus the book includes important topics that are
sometimes skipped in required courses but that the professional mathematician
will ultimately want to learn by self-study.
The content of the required courses at each university reflects expectations of
what students need before beginning specialized study andwork on a thesis. These
expectations vary from country to country and from university to university. Even
so, there seems to be a rough consensus about what mathematics a plenary lecturer
at a broad international or national meeting may take as known by the audience.
The tables of contents of the two books represent my own understanding of what
that degree of knowledge is for real analysis today.