Advanced Real Analysis-I

Paper Code: 
24MAT324(C)
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
Objective: 

This course will enable the students to –

  1. Explore their knowledge in the area of real analysis. 
  2. Get sufficient knowledge of the subject which can be used by students for further applications in their respective domains of interest.
  3. Understand the Introduction of Ordinal number, perfect sets, Borel measureable functions.
  4. Get ideas about approximate continuous function, Henstock integration.

 

Course Outcomes: 

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

 

 

 

 

 

 

 

24MAT

324(C)

 

 

 

Advanced Real Analysis-I

   (Theory)

 

 

 

 

CO113: Outline the concept of ordinal numbers.

CO114: Determine properties of perfect set and prove related theorems.

CO115: Explore properties of Borel measurable functions and Darbous function of Baire class one.

CO116: Analyze characteristics of approximate continuous function.

CO117: Explain the concept of Henstock integration on the real line.

CO118: Contribute effectively in course-specific interaction.

 

Approach in teaching:

Interactive Lectures, Discussion, Informative videos

 

Learning activities for the students:

Self learning assignments, Effective questions,  Topic  presentation, Assigned tasks

 

 

Quiz, Class Test, Individual projects,

Open Book Test, Continuous Assessment, Semester End Examination

 

 

 

Unit I: 
Ordinary number:
15.00

Order types, Well-ordered sets, Transfinite induction, Ordinal numbers, Comparability of ordinal numbers, Arithmetic of ordinal numbers, First uncountable ordinal Ω.

 

Unit II: 
Descriptive properties of sets:
15.00

Perfect sets, Decomposition of a closed set in terms of perfect sets of first category, 2nd category and residual sets, Characterization of a residual set in a complete metric space, Borel sets of class α, ordinal α < Ω,  Density point of a set in R, Lebesgue density theorem.

 

Unit III: 
Baire and Borel functions:
15.00

Borel measurable functions of class α (α < Ω) and its basic properties, Comparison of Baire and Borel functions, Darboux functions of Baire class one.

 

Unit IV: 
Properties of Baire one functions:
15.00

Nature of the sets of points of discontinuity of Baire one functions, approximate continuity and its fundamental properties, Characterization of approximate continuous functions.

 

Unit V: 
Henstock integration on the real line:
15.00

Concepts of δ-fine partition of the closed interval [a,b] where δ is a positive function on [a,b], Cousin’s lemma, definition of Henstock integral of a functions over the interval [a,b] and its basic properties, Saks-Henstock lemmas and its applications, Continuity of the indefinite integral, Fundamental theorem, Convergence theorems, Absolute Henstock integrability, Characterization of Lebesgue integral by absolute Henstock integral.

 

Essential Readings: 
  • A.M. Bruckner, J.B. Bruckner and B.S. Thomson, Real Analysis, Prentice-Hall, New York 1997.
  • H.S. Gaskill and P.P. Narayanswami, Elements of Real Analysis, PHI, 1988.
  • W.P. Parzynski and P.W. Zipse, Introduction to Mathematical Analysis, MC Graw-Hill Company, 1982.
  • I.P. Natanson, Theory of Functions and Real Variable, Vol. I& II, Frederic Ungar Publishing, 1955.
  • C. Goffman, Real Functions, Rinehart Company, N.Y. 1953.

 

SUGGESTED READING

  • P.Y. Lee, Lanzhou Lectures on Henstock Integration, World Scintific Press, 1990.
  • J.F. Randolph, Basic Real and Abstract Analysis, Academic Press, N.Y. 2014.
  • S.M. Srivastava, A Course on Borel Sets, Springer, N.Y. 1998.
  • R.G. Rartle, Introduction to Real Analysis, John Willey and Sons, 2000.
  • A.J. Kosmala, Introductory Mathematical Analysis, WCB Company, 1995.

 

e- RESOURCES

JOURNALS

https://link.springer.com/book/10.1007/0-8176-4442-3

Academic Year: 
2024-2025
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