MA224: Real Analysis, during January-April 2026

Preamble: This course presents a geometric and application-driven pathway through modern analysis. We begin with metrics and norms, which formalize distance and size far beyond Euclidean space, and use them to study convergence, completeness (spaces with no "missing points"), and compactness (no escape to infinity). These ideas provide the language behind stability of approximations, existence of solutions, and continuity principles that appear throughout analysis, optimization, and applied models.

We then turn to functions of several variables, viewing differentiability as the geometry of the best linear approximation to a nonlinear map. The chain rule and Taylor's theorem quantify how local behavior propagates, while the inverse and implicit function theorems explain when nonlinear equations can be solved locally, when constraints define smooth solution sets, and when coordinate changes are valid - tools central to nonlinear systems and constrained optimization.

Finally, we develop Lebesgue measure and integration as a strengthened notion of "measuring physical quantities" such as length/area/volume. While Riemann integration works well for regions with reasonably regular boundaries (e.g., sectors bounded by almost continuous curves), it fails for many rough, highly discontinuous, or unbounded objects. Lebesgue's breakthrough is to decompose sets into countably many elementary pieces (built from nearly open/closed sets) and "reassemble" them without changing total size; sets for which this is possible are measurable sets, while objects too wild for any consistent assignment are non-measurable. When boundaries cannot be captured by finitely many almost continuous parametrizations, one must also broaden the class of functions: "almost continuous" is replaced by measurable functions, which are nearly continuous in a precise sense. This leads to the Lebesgue integral and its powerful limit theorems - Fatou's lemma, monotone convergence, and dominated convergence - culminating in L^p spaces, the natural setting for probability, Fourier analysis, PDE, and many modern applications.

 

Syllabus: Metrics and norms - metric spaces, normed vector spaces, convergence in metric spaces, completeness, compactness; Functions of several variables - differentiability, chain rule, Taylor's theorem, inverse function theorem, implicit function theorem; Lebesgue measure and integral - sigma-algebra of sets, measure space, Lebesgue measure, measurable functions, Lebesgue integral, Fatou's lemma, dominated convergence theorem, monotone convergence theorem, L^p spaces.


Texts/References:

1. J. E. Marsden and M. J. Hoffman, Elementary Classical Analysis, 2nd Edition, W. H. Freeman, 1993.
2. M. Capinski and E. Kopp, Measure, Integral and Probability, 2nd Ed., Springer, 2007.
3. N. L. Carothers, Real Analysis, Cambridge University Press, 2000.
4. G. de Barra, Measure Theory and Integration, New Age International, 1981.
5. R. C. Buck, Advanced Calculus, Waveland Press Incorporated, 2003.
6. M. Capinski and E. Kopp, Measure, Integral and Probability, 2nd Ed., Springer, 2007.
7. W. Rudin, Principles of Mathematical Analysis, 3rd Edition, McGraw Hill, 1976.


Course policy (click here)

Classroom and slot: 5104, Slot - A (Tue 09:00-09:55, Wed 10:00-10:55, Thu 11:00 -11:55)

This course closely resembles MA 224 (Real Analysis). For reference, you may consult the (older) course materials available at:
https://www.iitg.ac.in/rksri/MA224_2018.htm

 
Lecture Notes: MA224 (Real Analysis) Lecturenotes

 

 Assignments: Assignment 1, Assignment 2, Assignment 3


Exams:

Class Discipline:

Students must ensure that mobile phones are switched off or set to silent mode and kept securely inside their bags, which must remain under the desk throughout the class. Only a notebook and a pen are permitted on the desk - no other items are allowed. Any student found with a mobile phone or any electronic device on the desk, regardless of intent, will be required to leave the classroom immediately and the incident might be reported to the Academic Section.