MATHEMATICAL SCIENCES DEPARTMENT

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Spring Semester 2002-2003 (022)

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__MATH 538 : Applied
Functional Analysis__

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*Catalog Data *: __MATH 538: Applied Functional
Analysis__

Credit 3. A quick review of basic properties of Topoligical, Metric, Banach and Hilbert Spaces. Introduction to Hausdorff metric and iterated function systems. Fixed point theorems and their applications. Introduction to calculus in infinite dimensional spaces - Frechet and Gateau derivatives, Bochner Integral. Introduction to weak and w*-topologies. Algorithmic optimization including complementarity problems and variational inequalities.

*Prerequisite: Math 411.*

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*Textbook/ *

*References::** *1)
A. H. Siddiqi __Applied Functional Analysis__.

2) E.
Zeidler *APPLIED FUNCTIONAL ANALYSIS*. *Applications to Mathematical
Physics*

3) E. Zeidler *APPLIED
FUNCTIONAL ANALYSIS*. *Main
Principles and Their Applications*

**Instructor** Dr. M. El-Gebeily, Associate
Professor, Mathematical Sciences.

**Goals** The
course provides an essential overview of the main principles of functional
analysis and their application to a wide variety of fields in physics,
engineering, economics, … etc.

*Prerequisites by topic:*

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1) Basic properties of convergence of sequences in topological and metric spaces

2) Basic theorems of analysis as covered in a course of advanced calculus

3) Elements of linear algebra

4) Basic definitions and manipulations in the calculus of several variables.

1) Metric Spaces and Banach Fixed Point Theorem. (1 Class)

2) Banach Spaces. (3 Classes)

3) Hilber Spaces. (3 Classes)

4) Fundamental Theorems. (4 Classes)

*5) *Differential
and Integral Calculus in Banach Spaces (3 Classes)

6) Optimization Problems (4 Classes)

7) Operator Equations and Variational Methods (3 Classes)

8) Variational formulation and approximation. (3 Classes)

9) Variational Inequalities and Applications (4 Classes)

10) Tests. (2 Classes)

*Homework:*

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Homework problems will be assigned, collected and graded. Your homework grade will be based on the quality of the submitted work as well as the timely submission of the assignments.

*Projects:*

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A term project will be assigned that includes the application of the above topics. Reports are required.

*Grading:*

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*The distribution of grade is as follows:*

*Classwork and projects 40%*

*Major 1 15%*

*Major 2 15%*

*Final Exam (Compehensive) 30%*

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*Total
100%*