## 414/514 Advanced Analysis (Analysis I) – Syllabus

Instructor: Andrés E. Caicedo
Fall 2014

Time: MWF 10:30-11:45 am.
Place: Mathematics building, Room 139.

Contact Information

• Office: 239-A Mathematics building.
• Phone number: (208)-426-1116. (Not very efficient.)
• Office Hours: W 12:00-1:15 pm. (Or by appointment.)
• Email: caicedo@math.boisestate.edu

Text
We will use three textbooks and complement with papers and handouts for topics not covered there.

• MR1886084 (2003e:00005).
Pugh, Charles Chapman
Real mathematical analysis.
Undergraduate Texts in Mathematics. Springer-Verlag, New York, 2002. xii+437 pp.
ISBN: 0-387-95297-7.
• MR0655599 (83j:26001).
van Rooij, A. C. M.; Schikhof, W. H.
A second course on real functions.
Cambridge University Press, Cambridge-New York, 1982. xiii+200 pp.
ISBN: 0-521-23944-3; 0-521-28361-2.
• MR1996162. See also MR0169961 (30 #204).
Gelbaum, Bernard R.; Olmsted, John M. H.
Counterexamples in analysis.
Corrected reprint of the second (1965) edition. Dover Publications, Inc., Mineola, NY, 2003. xxiv+195 pp.
ISBN: 0-486-42875-3.

The book by van Rooij and Schikhof will be our primary reference, supplemented naturally by the Counterexamples book.  The book assumes some knowledge beyond what is covered in our undergraduate course Math 314: Foundations of Analysis, and does not cover the theory in dimension $n>1$; for these topics, we will follow Pugh’s text.

Contents
Math 414/514 covers Analysis on Euclidean spaces (${\mathbb R}^n$) with emphasis on the theory in dimension one. The approach is theoretical, as opposed to the more computational approach of calculus, and a certain degree of mathematical maturity is required. The course is cross-listed, and accordingly the level will be aimed at beginning graduate students.

From the Course Description on the Department’s site:

Introduction to fundamental elements of analysis on Euclidean spaces including the basic differential and integral calculus. Topics include: infinite series, sequences and series of function, uniform convergence, theory of integration, implicit function theorem and applications.

Here is a short list of topics we expect to cover. This list may change based on students’ interest:

1. Set theoretic preliminaries.
• Cantor’s approach to infinite cardinalities. Countable vs. uncountable sets. Sets of size continuum. The Bernstein-Cantor-Schröder theorem.
• The axiom of choice. Zorn’s lemma. Countable and dependent choice.
• Transfinite recursion. The first uncountable ordinal $\omega_1$.
2. Axiomatization and construction of the set of reals.
• The least upper bound property; uniqueness of $\mathbb R$ up to isomorphism.
• Dedekind cuts, and complete orders.
• Metric spaces, and Cauchy completions. Banach contraction mapping theorem.
3. Topology on $\mathbb R$.
• Open and closed sets. Compact sets and Cantor sets. Baire space.
• Borel sets. Analytic sets.
• Notions of smallness.
• Meagerness and the Baire category theorem. The Baire-Cantor stationary principle.
• Sets of Jordan content zero and of measure zero.
• Introduction to the theory of strong measure zero sets.
4. Continuity.
• Sets of discontinuity of functions.
• Monotonicity. Functions of bounded variation.
5. Differentiability.
• The problem of characterizing derivatives. Baire class one functions. The intermediate value property. Sets of continuity of derivatives.
• The mean value theorem. L’Hôpital’s rule.
• The dynamics of Newton’s method.
• The Baire hierarchy of functions.
• Continuous nowhere differentiable functions.
6. Power series.
• Real analytic functions. Taylor series.
• $C^\infty$ functions. Zahorsky’s characterization of the sets of points where a $C^\infty$ function fails to be analytic.
7. Integration.
• Riemann integration. Lebesgue’s characterization of Riemann integrability.
• Weierstraß approximation theorem.
• Lebesgue integration. The fundamental theorem of calculus.
• The Henstock-Kurzweil integral. Denjoy’s approach to reconstructing primitives.
8. Introduction to multivariable calculus.
• (Frechet) derivatives.
• The inverse and implicit function theorems.

Based on homework. No late homework is allowed. Collaboration is encouraged, although students must turn in their own version of the solutions, and give credit to books/websites/… that were consulted and people with whom the problems where discussed.

There will be no exams. However, an important component of being proficient in mathematics is a certain amount of mental agility in recalling notions and basic arguments. I plan to assess these by requesting oral presentations of solutions to some of the homework problems throughout the term. If I find the students lacking here, it will be necessary to have an exam or two. The final exam is currently scheduled for Wednesday, December 17, 2014, 12:00-2:00 pm.

Additional information will be posted in this blog, and students are encouraged to use the comments feature. Please use full names, which will simplify my life filtering spam out.

On occasion, I post links to supplementary material on Google+ and Twitter.