Math 510 - Introduction to Analysis I

This is the first graduate course on Analysis. Together with Math 511, it prepares graduate students for the Real Analysis Qualifying Exam.
Time: Tue & Thur 11 a.m. - 12:15 p.m. Room: SMLC 352
 
Instructor Info: Anna Skripka, skripka [at] math [dot] unm [dot] edu
Office hours: Tue and Thur 3:30 - 4:45 p.m., and by appointment, SMLC 212
 
Topics: Real number fields, sets and mappings. Basic point set topology, sequences, series, convergence issues. Continuous functions, differentiation, Riemann integral. General topology and applications: Weierstrass and Stone-Weierstrass approximation theorems. We will cover most of the material from Chapters 1-14 of Carothers' book.
 
Textbook: N. L. Carothers, Real Analysis, Cambridge University Press, 1st edition, 2000, ISBN-10: 0521497566.
Supplementary textbook: W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Science/Engineering/Math, 3d edition, 1976, ISBN-10: 007054235X.
Review of Advanced Calculus: T. Tao, Analysis I, Hindustan Book Agency, 2nd edition, 2009, ISBN-10: 8185931941   related website
 
Prerequisites: Advanced calculus (Math 401) and linear algebra (Math 321) are required; complex analysis (Math 313) is strongly recommended.
Grades: 25% homework, 25% each of two midterms, 25% final exam.
 
American Disabilities Act: In accordance with University Policy 2310 and the American Disabilities Act (ADA), academic accommodations may be made for any student who notifies the instructor of the need for an accommodation. It is imperative that you take the initiative to bring such needs to the instructor's attention, as the instructor is not legally permitted to inquire. Students who may require assistance in emergency evacuations should contact the instructor as to the most appropriate procedures to follow. Contact Accessibility Services at 505-661-4692 for additional information.
 
Video lectures (the content is not identical with the content of our course)
 
TENTATIVE Schedule (will be updated at the end of each week)
 
Week
Important Dates
Chapters
Topics
Homework
1
Aug 19 (Pretest), 21
Chapter 1
Review of Advanced Calculus.
HW 1
2
Aug 26, 28 (HW1)
Chapter 2
Countable and uncountable sets. Cantor set. Monotone functions.
HW 2
3
Sep 2, 4 (HW 2);   Drop with no grade: Sep 5
Chapter 3
Metric and normed spaces. ℓp spaces.
HW 3
4
Sep 9, 11 (HW 3)
Chapter 4
Open and closed sets. Relative metric.
HW 4
5
Sep 16, 18 (HW 4)
Chapters 5, 6
Continuous functions. Connected sets.
Prepare for Exam 1
6
Sep 23, 25 (Exam 1)
Chapter 6
Connectedness.
7
Sep 30, Oct 2
Chapter 7
Complete metric spaces. Fixed points.
HW 5
8
Oct 7 (HW 5)
Chapter 8
Compactness. Continuous functions on compact spaces.
HW 6
9
Oct 14, 16 (HW 6)
Chapters 8, 10
Uniform continuity. Pointwise and uniform convergence. Interchanging limits.
HW 7
10
Oct 21, 23 (HW 7)
Chapter 10
Interchanging limits. Series of functions. Space B(X). The Weierstrass M-test. Power series. Nowhere differentiable function.
Prepare for Exam 2
11
Oct 28 (Exam 2), 30
Chapter 11
The Weierstrass approximation theorem.
HW 8
12
Nov 4, 6 (HW 8);   Withdrawal deadline: Nov 7
Chapter 11
Approximations in C(R). Equicontinuity. Compact subsets of C(X).
HW 9
13
Nov 11, 13 (HW 9)
Chapter 13
Functions of bounded variation.
HW 10
14
Nov 18, 20 (HW 10)
Chapter 14
Riemann-Stieltjes integral with an increasing integrator.
HW 11
15
Nov 25
Chapter 14
Properties of Riemann-Stieltjes integrable functions.
16
Dec 2, 4
Chapter 14
Riemann-Stieltjes integral with a general integrator. Properties of the Riemann integral. Review.
Prepare for Final Exam
17
Final Exam Dec 9 (Tuesday), 12:30 - 2:30 p.m.
Office hour: Dec 8, 2:30 - 4 pm