Advanced Real Analysis-I
This course will enable the students to –
- Explore their knowledge in the area of real analysis.
- Get sufficient knowledge of the subject which can be used by students for further applications in their respective domains of interest.
- Understand the Introduction of Ordinal number, perfect sets, Borel measureable functions.
- Get ideas about approximate continuous function, Henstock integration.
Course Outcomes (COs):
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Course |
Learning outcomes (at course level) |
Learning and teaching strategies |
Assessment Strategies |
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Course Code |
Course Title |
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MAT 324C |
Advanced Real Analysis-I (Theory)
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The students will be able to –
CO111: Introduce the concept of ordinal numbers. CO112: Describe properties of perfect set and prove related theorems. CO113: Discuss properties of Borel measurable functions and Darbous function of Baire class one CO114: Analyze characteristics of approximate continuous function. CO115: Understand the concept of Henstock integration on the real line. |
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 |
Ordinary number: Order types, Well-ordered sets, Transfinite induction, Ordinal numbers, Comparability of ordinal numbers, Arithmetic of ordinal numbers, First uncountable ordinal Ω.
Descriptive properties of sets: 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 compete metric space, Borel sets of class α, ordinal α < Ω, Density point of a set in R, Lebesgue density theorem.
Functions of some special classes: Borel measurable functions of class α (α < Ω) and its basic properties, Comparison of Baire and Borel functions, Darboux functions of Baire class one.
Continuity: Nature of the sets of points of discontinuity of Baire one functions, Approximate continuity and its fundamental properties, Characterization of approximate continuous functions.
Henstock integration on the real line: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.
- 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
- 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.
