MATH 6360  Fall 2018
Applied Analysis
Course Info. View syllabus.
Week 1. Review of metric spaces, completeness,
characterization of compactness, extreme value
theorem. Contraction mappings and fixed points.
Homework Set
1 (Solutions), due Friday, August 31.
Week 2. Applications of contractions
mappings:
integral equations, solutions to initial value
problems. Local existence and uniqueness of solutions, stability.
Homework Set
2 (Solutions), due Friday, September 7.
Week 3. The contraction mapping
theorem in multivariate calculus.
Homework Set
3 (Solutions), due Friday, September 14.
Week 4. The inverse
and implicit function theorems.
Homework Set
4 (Solutions), due Friday, September 21.
Week 5. Towards L^{p} spaces. Exam 1, Sep. 28. To see whether you are prepared,
take a practice run (Solutions).
Week 6. Extending the Riemann integral to L^{p} spaces. Banach spaces.
Homework Set
5 (Solutions), due
Friday, October 5.
Week 6. Dual spaces. Uniform boundedness
Homework Set
6 (Solutions),
due Friday, October 19.
Week 6. Consequences of uniform boundedness for
Fourier series and polynomial interpolation.
Homework Set
7 (Solutions),
due Friday, October 26.
Weeks 7 and 8. Uniform convexity, best
approximation propert and duality for L^{p}spaces. Bounded inverse, closed graph theorem.
Homework Set
8 (Solutions),
due Friday, November 9.
Week 9. Hilbert spaces. Exam 2, Nov 16. To see whether you are prepared,
take a practice
run (Solutions).
Week 10. Homework Set
9 (Solutions),
deferred to Monday, December 3.
Week 11. Orthonormal bases and their
characterization. Final exam, Dec 10. Review session:
Wednesday, Dec 5, 9am, 646PGH.
To see whether you are prepared,
take a practice run (Solutions).

MATH 6361  Spring 2019
Applied Analysis II
Course Info. View syllabus.
Week 1. Review of Hilbert
spaces. Characterization of best approximation by
orthogonal projection. Orthonormal bases.
Homework Set
1 (Solutions), due
Friday, January 25.
Week 2. Fourier
series. Convergence in L^{2} and
pointwise convergence. Weak convergence.
Homework Set
2 (Solutions), due
Friday, February 1.
Week 3. Relationships between weak and
norm convergence. Weak compactness in Hilbert spaces.
Homework Set
3 (Solutions), due
Friday, February 8.
Week 4. Linear and convex programming in Hilbert spaces.
The LaxMilgram theorem.
Homework Set
4 (Solutions), due
Friday, February 22.
Week 5. The HilbertSchmidt norm and
HilbertSchmidt operators. Exam 1, March 1. To see whether you are prepared,
take a practice
run (Solutions).
Week 6. Compact selfadjoint
operators. Towards the spectral theorem.
Homework Set
5 (Solutions), due
Friday, March 8.
Week 7. The spectral theorem for
compact, selfadjoint operators. Exam 1 retake,
March 22. Material from Homework Sets 15.
Homework Set
6 (Solutions), due
Friday, March 29.
Week 8. Diagonalizing normal
operators. Fredholm theory.
Homework Set
7 (Solutions), deferred to
Monday, April 8.
Week 9. Solutions to
Schrödinger's eigenvalue problem and
compact integral operators.
Reading assignment. Review the
Contraction Mapping Principle with the help
of these notes and
memorize the theorem until Monday.
Week 10. Calculus in
normed vector spaces.
Homework Set
8, due
Friday, April 19.
Reading assignment. Read
the Inverse Function Theorem notes and
memorize the theorem until
Wednesday.
Week 11.
The Inverse Function Theorem and
applications. Introduction to the Calculus of
Variations.
Homework Set
9, Part I, due
Friday, April 26.
Exam 2, April
29. Material from Homework Sets 69.
