Functional Analysis

Department: 
MATH
Course Number: 
7338
Hours - Lecture: 
3
Hours - Lab: 
0
Hours - Recitation: 
0
Hours - Total Credit: 
3
Typical Scheduling: 
Every even fall semester

Topics include the Hahn-Banach theorems, the Baire Category theorem and its consequences, duality in Banach spaces, locally convex spaces, additional topics.

Prerequisites: 

MATH 6337 or MATH 6338

Course Text: 

At the level of Conway: A Course in Functional Analysis, 2nd edition, Springer-Verlag

Topic Outline: 
  • Normed, Banach, and Hilbert spaces
  • Operators on Banach and Hilbert spaces, linear functionals, the Riesz Representation Theorem
  • Hahn-Banach theorem, Baire Category theorem, open mappings, theorem, closed graph theorem, and the uniform boundedness principle
  • Locally convex spaces, weak topologies, duality, Alaoglu's theorem, Krein-Milman theorem
  • Other applicatins and topics at the discretion of the instructor, such as
    • Distributions
    • The Fourier transform
    • Sturm-Liouville systems
    • Sobolev spaces
    • Banach algebras
    • Compact operators on a Banach space, Riesz functional calculus
    • Fixed point theorems