Evaluation: There will be five assignments
(handed out Wednesdays, due the following Wednesday) and
a final exam.
Please discuss lectures and homework problems among yourselves
and with me, and consult other sources. But write up
your assignments in your own words, and be ready to defend them!
Your work will be judged on the clarity of your presentation
as well as correctness and completeness.
Recommmended books:
- Eliott H. Lieb and Michael Loss,
Analysis. AMS Graduate Texts in Mathematics, 14
(available on my dropbox)
Not really a textbook but rather a path to research
in Analysis, PDE, and the Calculus of Variations.
The stated goal of this book is "to guide beginning
students through" [the topics of Analysis] "with a minimum of fuss and
to lead them to the point where they can read current literature
with some understanding. At the same time, everything
is done in a rigorous and, hopefully, pedagogical way."
- G. Folland, Real Analysis: Modern Techniques and their Applications.
Wiley
A thorough introduction to measure theoy
and integration with excellent excercises and
applications. Later chapters cover advanced topics
from Functional Analysis and Probability.
Not easy to read by yourself but well worth the effort !
- A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis. Dover,
1975
A classic. Clear, concise (and cheap).
- H.L. Royden: Real Analysis, Macmillan, 1988 (available
in the UCC Math/Stats library)
Contains a detailed construction of
the Lebesgue measure and integral
on R^n, at the advanced undergraduate level.
-
E. Stein and R. Shakarchi, Measure Theory,
Integration, and Hilbert Spaces (Vol. 3 of their undegraduate
series in Analysis).
A beautiful introduction
for advanced undergraduates and beginning graduate
students. Excellent for self-study.
- C.C. Pugh, Real Mathematical Analysis, Springer Verlag 2002.
A (very advanced) undergraduate-level text, with many examples
and challenging exercises. Highlights are the Arzela-Ascoli
Theorem and the Weierstrass Approximation Theorem
(about the topology of uniform convergence), the
Contraction Mapping Theorem (with applications
to ODE and the Implicit Function Theorem), and
the Lebesgue Theory on R^n (including the Banach-Tarski
Paradox, Vitali coverings, and Lebesgue Differentiation).
- H. Brezis, Functional Analysis, Sobolev Spaces, and
Partial Differential Equations.
A classic, finally translated into English just a couple
of years ago. An introduction
to Functional Analysis for beginning and active
researchers in PDE, Nonlinear Analysis,
and the Calculus of Variations. The English edition
also contains an extensive collection
of problems, with partial solutions.
Other sources: