ADVANCED REAL ANALYSIS-II (Optional Paper)

Paper Code: 
MAT 424C
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
Objective: 
This course will enable the students to -
  1. 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. 
  2. An introduction to derivative and integrability of absolutely continuous functions, general measure and integration, fourier series of functions of class L, distribution theory.

 

Course Outcomes (COs):

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Paper Code

Paper Title

 

 

MAT 424C

 

 

 

 

 

 

Advanced Real Analysis-II

 (Theory)

 

 

 

The students will be able to –

 

CO147: Analyze    properties derivative   and   integrability of absolutely continuous   functions.

CO148: Describe properties of measure and integration.

CO149: Understand the concept of Fourier series of functions of class L and prove related theorem.

CO150: Discuss operation , properties and convergence of distribution’s

CO151: Apply differentiation on distribution and define direct product of distributions.

 

Approach in teaching:

Interactive Lectures, Discussion, Power Point Presentations, Informative videos

Learning activities for the students:

Self learning assignments, Effective questions, presentations, Field trips

Quiz, Poster Presentations,

Power Point Presentations, Individual and group projects,

Open Book Test, Semester End Examination

 

 

 

 

Unit I: 
I
15.00
Derivative: Banach-Zareckitheorem.Derivative and integrability of absolutely continuous functions.Lebesgue point of a function.Determining a function by its derivative.                                                    
Unit II: 
II
15.00
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. 
                                                    
Unit III: 
III
15.00
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.
                                                                                                                                   
Unit IV: 
IV
15.00
Distribution Theory: Test functions. Compact support functions.Distributions.Operation on distributions.Local properties of distributions.Convergence of distributions. 
Unit V: 
V
15.00
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.
 
 
Essential Readings: 
 
  • A.M. Bruckner, J.B. Bruckner and B.S. Thomson, Real Analysis, Prentice-Hall, N.Y. 1997.
  • H.S. Gakill and P.P. Narayanswami, Elements of Real Analysis, Prentice-Hall India, 1988.
  • W.P. Parzynski and 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 and Donald R. Sherbert, Introduction to Real Analysis, John Wiley & Sons Canada, 2011.
  • A.J. Kosmala, Introductory Mathematical Analysis, WCB Company, 1995.

 

Academic Year: 
2020-2021
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