Course Description
Metric and normed spaces, convergence in normed spaces, completeness in metric spaces, Banach spaces and dual spaces, linear functionals and linear operators and their properties, compact operators, inner product spaces and Hilbert spaces, orthonormal bases, orthogonal complements and direct sums, Riesz representation theorem, adjoint operators, fundamental theorems in functional analysis(Baire theorem, Banach-Steinhauss theorem, open mapping theorem, closed graph theorem and Hahn-Banach theorem), strong and weak convergence.
Course Objectives & Outcomes
Objectives:
- Introduce the concepts of metric, normed, and inner product spaces and discuss the relations between them.
- Equip student with features of Hilbert and Banach spaces.
- Provide students with concepts of linear operators and functional.
- Explain the fundamental theorems of functional analysis and their applications.
- Develop students skills to deal with applications of convergence, and adjoint operarors
Outcomes: Upon successful completion of this course, the student will be able to:
- Identify metric, normed and inner product spaces, and discuss the mathematical connections between them.
- Check the completeness in metric spaces and find the completion of non-complete spaces,
- Explain properties of Banach and Hilbert spaces, and discuss the applications related to them.
- Introduce the concepts and properties of linear operators and functional
- Define Hilbert spaces and discuss the notions of orthonormal bases, direct sums and projection operators related to Hilbert spaces.
- Discuss fundamental theorems in functional analysis (Baire theorem, Banach-Steinhauss theorem, open mapping theorem, closed graph theorem and Hahn-Banach theorem),
- Solve applied problems using theorems of functional analysis.
- Recognize adjoint operators and show familiarity with basic notions of adjoint operators.
References
1. Kreyszig, K., (1978) Introductory functional analysis with applications, John Wiley & sons, ISBN-13: 9780471507314, ISBN-10: 0471507318.
2. Rudin, W., (1991) Functional Analysis, McGraw Hill, ISBN-13: 9780070542365, ISBN-10: 0070542368.
3. Yosida,K., (1980) Functional Analysis, Springer-Verlag,ISBN-13: 9780387102108, ISBN-10: 0387102108.
Course ID: MATH 552
Credit hours | Theory | Practical | Laboratory | Lecture | Studio | Contact hours | Pre-requisite | 3 | 3 | 3 | MATH 503 - MATH 507 |
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