MATH 4332/6313  Spring 2018
Introduction to Real Analysis, II
Course Info. View syllabus.
Week 1. Metric spaces. Open and closed
sets. Closure and interior.
Homework Set 1 (Solutions), due
February 1.
Week 2. Compactness. Completeness.
Homework Set 2 (Solutions), due
February 8.
Week 3. Compactness vs. sequential compactness.
Homework Set 3 (Solutions), due
February 15.
Week 4. Continuity and its
characterization. Continuity and compactness.
Homework Set 4 (Solutions), due
February 22.
Week 5. Metric completion. Baire's theorem.
Exam 1, March 1, in class. When you feel prepared,
try a practice
run (Solutions).
Special office hours: Wed, Feb 28, 9:3010:50am.
Week 6. The Contraction Mapping Theorem.
Homework Set 5 (Solutions), due
March 8.
Week 7. Newtons's method and
finding solutions to ordinary differential equations.
Homework Set 6 (Solutions), due
March 22.
Week 8. Polynomial
approximation.
Taylor polynomials and Taylor series.
Homework Set 7 (Solutions), due
March 29.
Week 9.
Weierstrass theorem.
Exam 2, April 5, in class. When you feel prepared,
try a practice
run (Solutions).
Week 10. Best
approximation with
polynomials. Equioscillation
condition. Chebyshev polynomials.
Homework Set 8, due
April 19.
Week
11.
Approximation with
trigonometric
polynomials. Fourier
series.
Homework Set 9, due
April 26.
MATH 4331/6312  Fall 2017
Introduction to Real Analysis
Course Info. View syllabus.
Office hours: PGH 604, Tu 11:30am12:30pm, We 12pm.
Week 1. The topology of
R^{n}. Cauchy sequences and completeness. Open and closed
sets.
Homework Set 1 (Solutions), due
date deferred to September 7.
Week 2. Closure of a set. Compactness.
Homework Set 2 (Solutions), due
September 14.
Week 3. Heine Borel property and other
properties of compact sets. Limits and continuity of functions.
Homework Set 3 (Solutions), due
September 21.
Week 4. Discontinuous
functions. Uniform continuity.
Connected sets.
Homework Set 4 (Solutions), due
September 28.
Week 5. Connectedness and the
Intermediate Value Theorem in higher dimensions.
Summary of the material in a handout.
Exam 1, October 5, in class. When you feel prepared,
try a practice
run (Solutions).
Week 6. Differentiablity and the
Mean Value Theorem.
Homework Set 5 (Solutions), due October
19.
Week 7. The Riemann integral and its
properties.
Homework Set 6 (Solutions), due October
26.
Week 8. The Fundamental Theorem of Calculus,
see handout. Normed vector spaces.
Homework Set 7 (Solutions), due November
2.
Week 9. Inner product spaces. CauchySchwarz
inequality. Relation between inner product and norm.
Exam 2, November 9, in class. Covers material
from the Intermediate Value Theorem (and its
higherdimensional generalization) to normed vector spaces.
When you feel prepared,
try a practice
run.
Week 10. Hölder and Minkowski
inequalities for functions and sequences.
Homework Set 8 (Solutions), due Tuesday, November
21.
Week 11. Limits of sequences of
functions. Uniform convergence. Completeness of C(K).
Homework Set 9 (Solutions), due November
30.
Week 12. Equicontinuity. Total
boundedness. Characterization of compact subsets in C(K).
Final exam. Dec. 12, 11am2pm, in
classroom. Material from Homework Sets 19. When you feel prepared,
try a practice
run (Solutions).
Review session on Tuesday, Dec 5, 10amnoon, PGH 646.
