ADVANCED REAL ANALYSIS-II (Optional Paper)
- Understand the advanced concept of real analysis and give sufficient knowledge of the subject which can be used by students for further applications in their respective domains of interest.
- An introduction to derivative and integrability of absolutely continuous functions, general measure and integration, fourier series of functions of class L, distribution theory.
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Learning Outcomes |
Learning and teaching strategies |
Assessment |
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After the completion of the course the students will be able to: CLO138- Analyze properties derivative and integrability of absolutely continuous functions . CLO139- Describe properties of measure and integration. CLO140-Understand the concept of fourier series of functions of class L and prove related theorem. CLO141- Discuss operation , properties and convergence on Disributions CLO142- Apply differentiation on distribution and define direct product of distributions. |
Approach in teaching: Interactive Lectures, Discussion, Tutorials, Reading assignments, Demonstration, Team teaching Learning activities for the students: Self learning assignments, Effective questions, Simulation, Seminar presentation, Giving tasks, Field practical
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Presentations by Individual Student Class Tests at Periodic Intervals. Written assignment(s) Semester End Examination
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Derivative: Banach-Zareckitheorem, Derivative and integrability of absolutely continuous functions, Lebesgue point of a function, Determining a function by its derivative.
General Measure and Integration : Additive set functions, Measure and signed measure, Limit theorems, Jordan and Hahn decomposition theorems, Complete measures, Integrals of non- negative functions, Integrable functions, Absolute continuous and singular measures, Radon-Nikodym theorem, Radon- Nikodyn derivative in a measure space.
Fourier Series:Fourier series of functions of class L, Fejer-Lebesgue theorem, Integration of Fourier series, Cantor- Lebesgue theorem on trigonometric series, Riemann’s theorem on trigonometric series, Uniqueness of trigonometric series.
Distribution Theory: Test functions, Compact support functions, Distributions, Operation on distributions, Local properties of distributions, Convergence of distributions.
Differentiation of distributions and some examples, Derivative of locally integrable functions, Distribution of compact support, Direct product of distributions and its properties, Convolution and properties of convolutions.
- A.M. Bruckner, J.B.Bruckner, B.S. Thomson, Real Analysis, Prentice-Hall, N.Y. 1997.
- H.S. Gakill, P.P. Narayanswami, Elements of Real Analysis, Prentice-HallIndia, 1988.
- W.P. Parzynski, P.W. Zipse, Introduction to Mathematical Analysis, Mc Graw-Hill Company, 1982.
- C. Goffman, Real Functions, Rinehart Company, N.Y. 1953.
- J.F. Randolph, Basic Real and Abstract Analysis, Academic Press, N.Y. 1968.
- Robert G. Bartle, Donald R. Sherbert, Introduction to Real Analysis, John Wiley & Sons Canada, 2011.
- A.J. Kosmala, Introductory Mathematical Analysis, WCB Company, 1995.
