MA6201E

Real analysis

This page contains information about the course MA6201E Real analysis that I will be teaching in the semester July - November 2024 in NIT Calicut for the 1st year M. Sc (Mathematics) students.

MA6201E should technically be a second course in real analysis. We assume that you have done a first course in real analysis and you know the following topics:

sequences/series of real numbers, their convergence
"real valued"continuous, differentiable, integrable functions

You may visit https://sites.google.com/view/praphulla-koushik/mt2223-real-analysis-i to get some idea of what is assumed in this course.

Nevertheless, we will recall all those ideas with a n-times speed of how you might have learnt it in the first course (for n anywhere between 1 and 10). This should take around 2 weeks.

The new things will start from the notion of sequence/series of real valued functions (topic 24 mentioned below)

First day of Instruction :

Tentative list of topics to be covered (one per session from topic 24):

Real number system and its structure
infimum, supremum,
LUB Axiom
Countable and uncountable sets
Sequences and series of real numbers
subsequences
monotone sequences
limit inferior, limit superior,
convergence of sequences and series, Cauchy criterion.
Functions of a single real variable
limits of functions
continuity of functions
uniform continuity
Differentiation,
properties of derivatives,
chain rule,
Rolle’s theorem
mean value theorems
L’Hopital’s rule
Riemann Integration
Darboux Integrability
Properties of the Integral
Fundamental theorem of calculus.
Sequences and series of functions,
pointwise and uniform convergence
Consequences of Uniform convergence
Series of functions
Power series
equicontinuity
pointwise and uniform boundedness
Arzela-Ascoli’s theorem
Metric spaces
Definition and examples
open balls and open sets
Convergent sequences in metric spaces
limit and cluster points,
Cauchy sequences
Bounded sets
Dense sets
Compact spaces and their properties
Continuous functions on Compact spaces
Characterization of Compact Metric spaces
Connected spaces
Complete metric spaces
Examples
Baire Category Theorem
Banach Contraction Principle.

Main references:

Principles of Mathematical analysis by Rudin (Chapter 7 and Chapter 8 in 3rd Edition)
Real analysis by Royden and Fitzpatrick (Chapter 9 and Chapter 10 in 4th Edition)

Suggested further reading (mathematics oriented)

Suggested futher reading (physics oriented)