MT2223  Real analysis-I

This page contains information about the course MT2223 Real analysis-I that  I taught  in the semester January-May 2022 in IISER Pune for 4th semester (2nd year) BS-MS students. 

First day of instruction : 17th January 2022. 

There will be 2 week teaching break from 7th March 2022 to 18th March 2022. One week (7-11 March) for mid sem exams (MT2223 is on 11th of March). Second week (14-18 March) are holidays :)

One way to make use of these holidays is to have a quick look at lecture notes/YouTube videos and  make sure you understand what we have seen in the last 16 sessions (before the teaching break starts) .  

 Last day of instruction : 6th May 2022

Lecture videos:

The playlist for youtube videos is https://youtube.com/playlist?list=PLZbgNdSTyWDaB2w9R5L5LjkNXIIhSrKQR 

The introduction video is uploaded in YouTube at  https://youtu.be/aMD4-sF04aI

Session 1 (17th January 2022): We gave the tentative descritption of the "real numbers". We also saw the notion of an order on the set of real numbers. The recorded  video is available at https://youtu.be/fbkH0S4eYyg 

Session 2 (20th January 2022): We have the notion of compatibility between the order structure and the field structure of R. We mentioned about the completeness axiom. The recorded video is available at https://youtu.be/oyg6rWUFxyE 

Session 3 (21st January 2022) : We will see some conseqeunces of completeness axiom. We will also see the notions of infimum and supremum of bounded subsets of R. The recorded video is available at https://youtu.be/9w4B-vEL2YU 

Session 4 (24th January 2022) : We see some properties of infimum, supremum maps. Further, we see that the set of rational numbers covers most of the set of real numbers (some people call it denseness of Q in R). The recorded video is available at https://youtu.be/CasMpxl4v1A 

Session 5 (27th January 2022) : In this session, we have seen the notion of sequence of real numbers and gave two special classes of sequences of real numbers (bounded seqeunces and monotone sequences). The recorded video is available at https://youtu.be/nF_z8oO5Gfw 

Session 6 (28th January 2022) : In this session, we have seen two more special classes of sequences of real numbers (convergent seqeunces, and Cauchy sequences). The recorded video is available at https://youtu.be/72X__ISlmWo  

Session 7 (31st  January  2022): In this session, we have seen some examples of sequences, and realised which of them are bounded, which of them are convergent. The recorded video is available at https://youtu.be/-hJzirdWJsg 

Session 8 (3rd February 2022): In this session, we have seen some results about sequences of real numbers (including the so called monotone+bounded implies convergence theorem). The recorded video is available at https://youtu.be/0OyGxJx5Vso 

Session 9 (7th February 2022) : In this session, we have seen some results about Cauchy sequences of real numbers (including the so called Cauchy completeness theorem). The recorded video is available at https://youtu.be/rbbHFlxFiKM 

Session 10 (10th February 2022): In this session, we mentioned about subsequences of real-number sequences and some results (including Bolzano-Weirstrass theorem). The recorded video is available at https://youtu.be/20aSE7vxXUk 

Session 11 (14th Februrary 2022): In this sessinon, we mentioned a definition of continuous function, and some examples of continuous functions (including polynomial functions). The recorded video is available at https://youtu.be/WP36EfbXCfs 

Session 12 (18th February 2022) : In this session, we saw some properties of continuous functions (including the so called Intermediate value theorem). The recorded video is available at https://youtu.be/beLmNUbyPIQ .

Session 13 (21st February 2022) : In this session, we have seen the notion of bounded functions, monotone functions and the possible relation between bounded functions, montone functions and continuous functions. The recorded video is available at https://youtu.be/0Cm3G5zhJgE .

Session 14 (24th February 2022): In this session, we have seen the notion of ''limit of a function'' and the notion of ''uniform continuity''. The recorded video is available at https://youtu.be/3kZ0o2PKuiY . 

Session 15 (28th February 2022): In this session, we have seen some computations related to sequences. The recorded video is available at https://youtu.be/BoZvYLJCy-M. 

Session 16 (3rd March 2022): In this session, we have seen some computations related to continuous functions, and uniform continuous functions. The recorded video is available at https://youtu.be/UY-g71ImRzs 

Session 17 (21st March 2022): In this session, we introduce the notion of differentiable functions and do some examples. The recorded video is available at https://youtu.be/v7vVHJ2JjNI 

Session 18 (25th March 2022) : In this session, we have seen some results about differentiable functions; in particular Rolle's theorem and the mean value theorem. The recorded video is available at https://youtu.be/WoGfpDxpNqs 

Session 19 (4th April 2022): In this session, we have seen some results about differentiable functions; in particular Taylor's theorem and L'Hospital rule . The recorded video is available at https://youtu.be/UWbZVzR5NtY 

Session 20 (7th April 2022):  In this session, we have seen some relation between convex/concave functions and differentiable functions. The recorded video is available at https://youtu.be/4mP970ebUqQ 

Session 21 (11th April 2022) : In this session, we have seen the notion of summable sequences ("the series"), some examples and a result. The recorded video is available at https://youtu.be/WXvFp3nRMG0 

Session 22 ( 18th April 2022) : In this session, we have seen certain special kind of series; namely the series of non-negative terms. We have also seen some tests for convergence of series (comparision test). The recorded video is available at https://youtu.be/K5vu0Z5KABg 

Session 23 (21st April 2022): In this session, we have seen two tests for convergence of series; ratio test and root test. The recorded video is available at  https://youtu.be/V0-RqjqWP9k 

Session 24 (22nd April 2022) : In this session, we have seen some examples of convergent series, a convergent test, and the notion of power series. The recorded video is available at https://youtu.be/hfogzY9qu_Q 

Session 25 ( 25th April 2022): In this session, we have seen two (equivalent) definitions of an integrable function. We have also seen Riemann condition for integrability of a function. The recorded video is available at https://youtu.be/oWT82Ku3jLk 

Session 26 (28th April 2022) : In this session, we have seen some (algebraic) properties of integrable functions. The recorded video is available at https://youtu.be/hmr51XC5Ol0 https://youtu.be/iyWRElRpFwg 

Session 27 (29th April 2022) : In this session, we have seen some examples of integrable functions, and the fundamental theorem of calculus. The recorded video is avilable at https://youtu.be/Az8wA84wY6Y 

Session 28 (2nd May 2022) : In this session, we have introduced the notion of metric spaces. We have seen some examples. The recorded video is available at https://youtu.be/OElP19mOTsw 

Session 29 (5th May 2022)  : In this session, we have seen the idea of continuity of functions between metric spaces. The recorded video is avialable at https://youtu.be/a2PTiBvy3E8 

Session 30 (6th May 2022) : In this session, we have seen the notions of compact metric spaces, connected metric spaces and complete metric spaces. The recorded video is available at https://youtu.be/iyWRElRpFwg 

Assignment sheets:

1. Assignment 1 

2. Assignment 2 

3. Assignment 3  

4. Assignment 4  

5. Assignment 5  

Mid semester, end semester, repeat exam papers

Mid semester examination paper 

End semester examination paper 

The rules are simple:

1. The assignment is not about yes or no answers. 

2. If you think something is true, then prove it.

3. If you think something is not true, then give a counterexample. 

4. If you think with some extra conditions an implication is true, you should mention those extra conditions (and prove it).

References for the course:

1. A Course in Calculus and Real Analysis by Sudhir R. Ghorpade Balmohan V. Limaye

2. A Basic Course in Real Analysis by S. Kumaresan and Ajith Kumar

3. Real Mathematical Analysis by Charles C. Pugh

4. Mathematical Analysis I by Vladimir A. Zorich

5. Metric Spaces by Satish Shirali and Harkrishan L. Vasudeva. This book is freely available on the link mentioned. 

Some more textbook references will be added as the course progresses.

Some other useful links:

1. Resonance – Journal of Science Education All articles are freely accessible. Please contact IISER Pune library if you have difficulty in downloading any article.

2. Expository articles written by S. Kumaresan. All are freely accessible. 

3. Mathematics Stack Exchange