Skip to the content

Source document. This page reproduces the syllabus this course was written to, as published — its semesters, credits and paper numbers are that document’s, not this site’s. The course itself is studied on its own, in any order.

This page reproduces the prescribed outline for the paper, so that the teaching pages can be checked against it line by line. It is the syllabus, not a summary of it. Nothing about marks, duration or examination pattern appears on this site.

Course Objectives

  1. Able to understand the Theoretical Probability and Distributions and Statistical Inference which are based on Advanced concepts of Real Analysis, Complex Analysis.
  2. Able to understand the concepts of Limits, continuity, completeness of set, supremum, infimum, sequence and series, Convergences, R-S Integrations, Multiple integrations, etc.

Course Outcomes

  1. Able to understand the basic concepts of real analysis.
  2. Strong foundations in understanding Probability & Distribution concepts.
  3. Able to understand convergence of sequence and series of real valued function.
  4. Understand existence of integral and their evaluation.
  5. Understand change of multiple integral and Lebesgue integral.

Unit I

AS PRESCRIBED

Metric spaces — Compact sets — Perfect sets — Connected sets. Limits of functions — Continuous functions — Continuity and compactness, Continuity and connectedness, Discontinuities — Monotonic functions, Differentiation.

→ Unit 1 notes

Unit II

AS PRESCRIBED

Riemann–Stieltjes (R-S) Integral and its linear properties. Integration by parts, Euler's summation, Riemann's condition.

→ Unit 2 notes

Unit III

AS PRESCRIBED

Integrators of Bounded variations. Statements of necessary and sufficient conditions of Riemann–Stieltjes integral. Differentiation under the integral sign. Interchanging the order of integration.

→ Unit 3 notes

Unit IV

AS PRESCRIBED

Sequences and Series of Functions: Uniform convergence — Uniform convergence and continuity — Uniform convergence and integration — Uniform convergence and differentiation — The Stone–Weierstrass theorem.

→ Unit 4 notes

References

  1. Walter Rudin (2013): Principles of Mathematical Analysis, McGraw-Hill International, 3rd Edition. (Unit I: pp 30–46 and pp 83–102) (Unit II: pp 120–133 and 135–142) (Unit III: pp 143–154, 159–161, 165–171 and 220–222).
  2. H. L. Royden (1988): Real Analysis, PHI, 3rd edition.
  3. Apostol, T. M. (2002): Mathematical Analysis, Narosa, Indian Ed.
  4. Malik, S. C. (1984): Mathematical Analysis, Wiley–Eastern.
  5. Mathematical Analysis Vol I (2013), by D. J. H. Garling.
← Mathematical Analysis Unit 1 →