**About the course**

This is a basic course in Real Analysis which is a back bone of any course on pure & applied Mathematics and Statistics. This is a very useful course for any branch of science and engineering. The present course has been designed to introduce the subject to undergraduate/postgraduate students in science and engineering. The course contains a good introduction to each topic and an advance treatment of theory at a fairly understandable level to the students at this stage. Each concept has been explained through examples and application oriented problems.

**Industries that will recognize this course**

All universities , Engineering colleges , IITs ,IISER etc.

**Course
instructor**

** **

**P.D.Srivastava** is a Professor (in
HAG scale) in the department of Mathematics at IIT KGP. Professor Srivastava
has 36 years of teaching and research experience. He has taught many PG and UG
courses such as Mathematics I and II for
B.Tech. students, Real analysis, complex analysis, functional analysis, measure
theory, sequence space etc. are also taught
by him for undergraduate and post graduate students of integrated
courses in Mathematics. Professor Srivastava has supervised so far 13 students
for their PhD degrees and approx. 50
students for M.Sc. projects. He has more than 65 research publications in
national and international journal of high repute. Professor Srivastava’s main
research interest is functional
analysis, in particular, Operator theory and sequence spaces. He has also worked in fuzzy sequence spaces
and Cryptography.

**Course layout**

** **

**Week 1: **countable & uncountable sets (3 lectures)

Concepts of Metric Space (1 lectures)

Open ball, closed ball, limit point of a set (1 lectures)

**Week 2: **Some
theorems on Open & closed set
(1 lectures)

Ordered set, least upper bound, greatest lower bound (2 lectures)

Compact set & some properties of Compact set (2 lectures)

**Week 3: **Heine Borel Theorem (1 lecture)

Weierstrass Theorem, connected set (1 lecture)

Cantor Set & its properties (1 lecture)

Dense set & derived set (1 lecture)

Limit of sequences of real numbers & Monotone sequence (1 lecture)

**Week 4: **Some
important limits of sequences (1 lecture)

Ratio tests, Cauchy theorems on limits of sequence of real numbers (1 lectures)

Fundamental theorems on limit (1 lecture)

Some results on limit & Bolzano-Weierstrass Theorem (1 lecture)

Criteria for convergent sequence (1 lecture)

**Week 5: **Criteria
for Divergent sequence (1 lecture)

Cauchy sequence (1 lecture)

Cauchy convergence criteria for sequences (1 lecture)

Infinite series of Real numbers (1 lecture)

Convergence Criteria for series of positive real no. (1 lecture)

**Week 6: **Comparison test for series (1 lecture)

Absolutely and Conditional convergent series and Tests (2 lectures)

Ratio & Integral Tests for convergence of series (1 lecture)

Raabe’s test for convergence of series (1 lecture)

**Week 7: **Limit of functions & cluster point
(2 lectures

Divergence criteria for limit (1 lecture)

Various properties of limit of functions (1 lecture)

Left & Right hand limits for functions (1 lecture)

**Week 8: **Limit of functions at infinity (1
lecture)

Continuity functions (Cauchy‘s definition) (1 lecture)

Continuity functions (Heine‘s definition) (1 lecture)

Properties of continuous functions (2 lectures)

**Week 9: **Boundedness Theorem and Max-Min
theorem (1 lecture)

Location of root and Bolzano’s theorem (1 lecture)

Uniform continuity & related theorems (1 lecture)

Absolute continuity& related theorems (1 lecture)

Types of discontinuities & Continuity in a Metric Space (1 lectures)

**Week 10:** Types of discontinuities &
Continuity in a Metric Space (1 lectures)

Relation between continuity & compact sets (1 lecture)

Differentiability of real valued functions (1 lecture)

Local Max. – Min. Cauchy’s and Lagrange’s Mean value theorem (1 lecture)

Rolle’s Mean value theorems & Applications (1 lecture)

**Week 11: **Applications of Derivatives (1
lecture)

Application of MVT & Darboux’s theorem (1 lecture)

L’Hospital Rule (1 lecture)

Taylor’s Theorem (1 lecture)

Riemann/Riemann Steiltjes Integral (1lecture)

**Week 12: **Riemann/Riemann Steiltjes Integral (1lecture)

Existence of Riemann Stieltjes Integral (1 lecture)

Riemann Stieltjes Integrable functions (1 lecture)

Properties of Riemann Stieltjes Integral (1 lecture)

Various results of Riemann Stieltjes Integral using step function (1 lecture)

Some more Results on Riemann Stieltjes Integral (1 lecture)

** **

**Certification exam:**

• The exam is optional for a fee. Exams will be on 23 April 2017.

• Time: Shift 1: 9am-12 noons; Shift 2: 2pm-5pm

• Any one shift can be chosen to write the exam for a course.

• Registration url: Announcements will be made when the registration form is open for registrations.

• The online registration form has
to be filled and the certification exam fee needs to be paid. More details will
be made available when the exam registration form is published.

**Certificates:**

• Final score will be calculated as : 25% assignment score + 75% final exam score.

• 25% assignment score is calculated as 25% of average of 12 weeks course: Best 8 out of 12 assignments.

• E-Certificate will be given to those who register and write the exam and score greater than or equal to 40% final score. Certificate will have your name, photograph and the score in the final exam with the breakup. It will have the logos of NPTEL and IIT KHARAGPUR. It will be e-verifiable at nptel.ac.in/noc