Introduction to Functional Analysis

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Lecturer:

Tutor:

Supervisor:

Department of Technical Mathematics

Synopsis:

The unifying role of functional analysis in formulation and solution of mathematical problems. Normed linear space and linear space with a scalar product, operators, the Banach fixed point theorem, generalized Fourier series.

Requirements:

Syllabus of lectures:

1. The role of functional analysis, its position in the system of mathematical disciplines. Metric space, examples. Open and closed sets, boundary, closure.

2. Convergence of a sequence in the metric space, complete metric space, the theorem on completion.

3. Contractive mapping, Banach?s fixed point theorem.

4. Applications of Banach?s fixed point theorem: iterative methods of solution of a system of linear algebraic equations, approximate solution of a differential equation, approximate solution of an integral equation.

4. Linear space, basis, dimension, subspace. Normed linear space, Banach space.

5. Examples of normed linear spaces: Rn with various norms, spaces of continuous and continuously differentiable functions, spaces of integrable functions.

6. Linear space with a scalar product, Schwarz inequality. Hilbert space. Orthogonal elements, orthogonal complement.

7. Orthogonal and orthonormal system in the Hilber space. Question of the best approximation, generalized Fourier series.

8. Complete orthogonal system, Bessel?s inequality, Parseval?s equality.

9. Linear operators in normed linear spaces. Domain, range, null space of a linear operator.

10. Bounded operator, continuous operator, connectedness of the continuity with the boundedness in the case of a linear operator. Examples. Spectrum of a linear operator.

11. Linear functional, dual space. Reflexive Banach space. Weak convergence.

12. Compact and precompact set in the Banach space. Compact operator. Compact imbedding.

13. Applications of methods of functional to problems of mathematical physics.

14. Applications of methods of functional to problems of mathematical physics

Syllabus of tutorials:

1. The role of functional analysis, its position in the system of mathematical disciplines. Metric space, examples. Open and closed sets, boundary, closure.

2. Convergence of a sequence in the metric space, complete metric space, the theorem on completion.

3. Contractive mapping, Banach?s fixed point theorem.

4. Applications of Banach?s fixed point theorem: iterative methods of solution of a system of linear algebraic equations, approximate solution of a differential equation, approximate solution of an integral equation.

4. Linear space, basis, dimension, subspace. Normed linear space, Banach space.

5. Examples of normed linear spaces: Rn with various norms, spaces of continuous and continuously differentiable functions, spaces of integrable functions.

6. Linear space with a scalar product, Schwarz inequality. Hilbert space. Orthogonal elements, orthogonal complement.

7. Orthogonal and orthonormal system in the Hilber space. Question of the best approximation, generalized Fourier series.

8. Complete orthogonal system, Bessel?s inequality, Parseval?s equality.

9. Linear operators in normed linear spaces. Domain, range, null space of a linear operator.

10. Bounded operator, continuous operator, connectedness of the continuity with the boundedness in the case of a linear operator. Examples. Spectrum of a linear operator.

11. Linear functional, dual space. Reflexive Banach space. Weak convergence.

12. Compact and precompact set in the Banach space. Compact operator. Compact imbedding.

13. Applications of methods of functional to problems of mathematical physics.

14. Applications of methods of functional to problems of mathematical physics

Study Objective:

Study materials:

A. Taylor: Introduction to Functional Analysis, Krieger pub., 1986

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans: