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).
