To the Student
OUR SUBJECT IS CALLED "ANALYSIS"Analysis is the mathematical subject that underlies and extends the theory of
calculus. It is a deep and extensive subject that has been under development for
centuries. It has itself evolved into a number of distinct fields of study, two of
which may be called real analysis and complex analysis, according to whether
the underlying number system is taken to be the real number system or the
complex number system. In this course we focus on real analysis, although most
of what we discuss finds use in all areas of analysis.
Because analysis has its origins in calculus, it will look somewhat familiar
to you. However, you will be exploring the subject at a level much deeper than
that of your introductory calculus courses. This course will demand careful,
critical thought. Indeed, it is designed to help you gain what mathematicians
call "mathematical maturity."
FOUNDATIONS DO MATTERYou are already familiar with some of the powerful results of analysis; you
have seen them and used them in your calculus courses. However, you most
likely did not prove all of these results rigorously from a small set of beginning
assumptions. Consequently, your understanding of why they are true may be
somewhat clouded in mystery. How can you be sure that all this theory is really
"true?" In what sense can these results be proved?
We cannot answer these questions by looking "forward." We must look
backward, and trace the subject back to its logical (but not necessarily histor-
ical2) origins. After we have laid a secure foundation for the subject, and have
reconstructed its core framework by rigorous logical deduction, we will have a
fresh understanding of the analysis we once naively thought we knew. We will
also push forward toward new and deeper results.
Thus, the course brings a shift of emphasis: from the development of math-
ematical techniques and applications, to a critique of the subject itself. We shall
- For insights on the historical development of analysis, consult references [34], [37], [56],
[57], [58], [59], [60], [61], [66], [75], [88], and [127].
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