## Winter 2010 Math 540

# What, When, and Where:

### Course Description:

Linear Analysis: Theory and applications of Banach and Hilbert spaces. Topics include complete normed linear spaces, duality, fundamental theorems, operator theory, representation theorems, spectral theory.

### Text:

Linear Analysis: An Introductory Course (Paperback) Bela Bollobas, Cambridge University Press; 2ed 256 pages.

(Available on Amazon)

### Lectures:

TuTh 8:00-9:15a 323 TMCB

### Office Hours:

TuTh 5:00-6:00p 306 TMCB

### Grading Scheme:

60% Homework

20% Midterm Exam

20% Final Exam

# Course Schedule: (subject to change)

Jan 05 Inequalities I

Jan 07 Inequalities II

Jan 12 Normed Spaces and Bounded Linear Operators I

Jan 14 Normed Spaces and Bounded Linear Operators II

Jan 19 Linear Functionals and the Hahn Banach Theorem I

Jan 21 Linear Functionals and the Hahn Banach Theorem II

Jan 26 Linear Functionals and the Hahn Banach Theorem III

Jan 28 Linear Functionals and the Hahn Banach Theorem IV

Feb 02 Baire Category, Open Mapping and Closed Graph Theorems I

Feb 04 Baire Category, Open Mapping and Closed Graph Theorems II

Feb 09 Baire Category, Open Mapping and Closed Graph Theorems III

Feb 11 Continuous Functions on Compact Spaces and the Stone-Weierstrass Theorem I

Feb 16 NO CLASS

Feb 23 Continuous Functions on Compact Spaces and the Stone-Weierstrass Theorem II

Feb 25 EXAM PREP

Mar 02 Contraction Mapping Principle

Mar 04 Weak Topologies and Duality

Mar 09 Orthonormal Systems I

Mar 11 Orthonormal Systems II

Mar 16 Adjoint Operators

Mar 18 Algebra of Bounded Linear Operators I

Mar 23 Algebra of Bounded Linear Operators II

Mar 25 Compact Operators on a Banach Space I

Mar 30 Compact Operators on a Banach Space II

Apr 01 More Fixed Point Theorems I

Apr 06 More Fixed Point Theorems II

Apr 08 Invariant Subspaces

# Assignments:

Assignment #1: B-Chapter 1, Exercises 2, 3, 5, 20-22, 26-29 (Due Jan 14)

Assignment #2: B-Chapter 2, Exercises 7-10, 20, 22, 26-29 (Due Jan 28)

Assignment #3: B-Chapter 3, Exercises 1-2, 4-8, 14-16 (Due Feb 11)

Assignment #4: B-Chapter 5, Exercises 3, 6, 11-14, 18-21 (Due Feb 25)

Assignment #5: B-Chapter 6, Exercises 8, 12, 14, 15 & Chapter 7, Exercises 2-7 (Due Mar 09)

Assignment #6: DM-Chapter 3, Exercises 7, 9-13, 15, 20, 22, 24 (Due Mar 30)

Assignment #7: DM-Chapter 3, Exercises 34, 39, 40, 43-45, 47, 51, 52, 59 (Due April 06)

Assignment #8: DM-Chapter 4, Exercises 15, 24, 25, 43, 48-51, 64, 71 (Due April 13)