Instructor: Demetrio Labate

• OFFICE:   PGH 694 
• EMAIL ADDRESS:   dlabate@math.uh.edu 
• HOMEPAGE:   http://www.math.uh.edu/~dlabate 

When and Where

• MEETING TIME:   MW 1-2:30,
• MEETING PLACE:   PGH 350 
• OFFICE HOURS:   Mon: 12-1; Wed: 2:30-3:30 (or by appointment) 

Course Description:

This course is the first part of a two-semester sequence about elementary functional analysis. The core of the course in an introduction to the theory of Hilbert and Banach spaces and the main properties of Linear Operators acting on these spaces. A number of applications from numerical analysis, harmonic analysis and PDEs and will also be presented in class. The second semester will be a more technical development of the theory of linear operators on Hilbert spaces; unbounded operators; topics from Fourier Analysis. The selection of topics for the second semester will be based, in part, on the interest and feedback from interested students. 

Textbook:

Introductory Functional Analysis with Applications, by Kreyszig, Wiley, 1989.

Prerequisites:

Real Analysis (MA 4331 or, better, MA 6320-6321) and Linear Algebra (MA 4377). The course and the textbook do not require a specific knowledge of measure theory, so that students don't need be too concerned if they lack that background. However, a solid background on elementary linear algebra (e.g., matrices, linear independence), analysis (e.g., convergence) and topology (e.g., open/closed sets) are necessary to successfully atten this class.

Course outline:

• Metric Spaces
• Normed and Banach Spaces
• Inner product spaces and Hilbert Spaces.
• Fundamental theorems for normed and Banach spaces
• Banach Fixed Point Theorem and its applications
• Applications from approximation theory and harmonic analysis

HOMEWORK:

• Homework 1 - Due: 9/1 
• Homework 2: Sec.1.3, Ex 10,12,14; Sec.1.4, Ex 8; Sec. 1.5, Ex 10,13,14 - Due 9/13 
• Review Problems (for Test 1 on 9/22 - Not to be collected): Sec 1.2, Ex 7,8,9,10; Sec 1.3, Ex 12; Sec 1.4, Ex 1,2,3; Sec 1.5, Ex 6,7,8; Sec 1.6, Ex 13,14; Sec. 2.2 Ex 11,12, Sec. 2.3, Ex 10 
• Homework 3: Sec. 5.1: 10,12; Sec.2.4:1;Sec.2.5:10; also: prove that C^1[0,1] is a complete metric space - Due: 10/11 - Solution
• Homework 4: Sec.2.6, Ex 14; Sec.2.7, Ex 5,6; Sec. 2.8, Ex 6; Sec. 2.10, Ex. 8 - Due 10/25 
• Homework 5: Sec.3.1, Ex 6; Sec.3.2, Ex 10; Sec. 3.3, Ex 8; Sec. 3.5, Ex. 8; Sec 3.6, Ex 4,10 - Due 11/8 
• Review problems for Test 2: 2.2: 11-14; 2.3:12-13; 2.6: 7,11-13; 2.7:7-10; 2.8: 9-10; 2.10: 6-7; 3.2: 7-10; 3.3: 2,6,8; 3.4:5-7; 3.5: 5-6; 3.6: 4-5; 3.8: 2,5; 3.9: 3-5; 3.10: 4-5, 8,9,11,12.  
• Review Problems for Final Exam: Sec 4.2: 2,5; Sec 4.3: 4,6,8,11; Sec 4.5: 1; Sec 4.6: 3; Sec 4.7: 8-12  

Tests and Exam Dates:

The dates for the midterm exams are Wed Sept 22 and Mon Nov 15. The final exam will be posted on Thu Dec 9 and collected on Mon Dec 13, 9:30AM.

Grading:

Grades will be based on homework assignments counting 40% towards the final grade, on two midterm exams counting 30% towards the final grade and one final counting 30% towards the final grade. 

The grade will be determined according to a set point scale: 90%-100%: A, 80%-89%: B, 70%-79%: C, 60-69% D; F is less than 60% (+ and - will also be used).