Metric Spaces

By Prof. Qamrul Hasan Ansari, Prof. Monirul Islam | Aligarh Muslim University

Learners enrolled: 300 | Exam registration: 11

ABOUT THE COURSE:
The primary objective of this proposed online course is to give the idea of the distance between two elements in an arbitrary set and to extend the concepts, namely, open sets, closed sets, convergence of sequences, compact sets, continuity of functions etc., from real line to a metric space. The course focuses on basic notions of metric spaces and their properties. A large number of both Multiple-Choice Questions and Subjective Questions will be provided for the practice of the students. Several real-world problems will be discussed in the frame work of metric spaces.

INTENDED AUDIENCE: B.Sc. / B.C.A. / M.Sc. / M.C.A. / B.Tech

PREREQUISITES: Basic Real Analysis

Summary

Course Status :	Completed
Course Type :	Core
Language for course content :	English
Duration :	12 weeks
Category :	Mathematics
Credit Points :	4
Level :	Undergraduate/Postgraduate
Start Date :	22 Jul 2024
End Date :	11 Oct 2024
Enrollment Ends :	05 Aug 2024
Exam Registration Ends :	16 Aug 2024
Exam Date :	25 Oct 2024 IST
ALERT ON EXAM DATE :	Exam date is subject to change.

Note: This exam date is subject to change based on seat availability. You can check final exam date on your hall ticket.

Page Visits

Course layout

Week 1: Basic Concepts: Definition and examples of metric spaces, bounded and unbounded metric spaces, Distance between sets, Diameter of a set.

Week 2: Open and Closed Sets: Open and closed balls, lnterior points and interior of a set, Open set, Neighbourhood of a point, Limit point of a set, Closure of a set, Closed set

Week 3: Boundary of a Set and SubSpaces: Boundary points and boundary of a set, Exterior points and exterior of a set, Subspace of a metric space.

Week 4: Convergent Sequences: Sequences and subsequence in a metric space, Convergent and Cauchy sequences

Week 5: Complete Metric Spaces: Complete metric spaces, Relation between completeness and closedness, Cantor lntersection Theorem, Completion Theorem

Week 6: Dense Sets and Related Results: Dense sets, Separable spaces, Nowhere dense sets, Categories and Baire Category Theorem.

Week 7: Compact Metric Spaces: Cover of a metric space, Compact metric spaces, Compact sets and their criterion, Properties of compact sets

Week 8: Finite lntersection Property: Relation between compactness, completeness and closedness, Finite lntersection property

Week 9: Some lmportant Theorems: Bolzano-Weierstrass property, Sequential compactness, Totally bounded spaces

Week 10: Continuous Functions: Continuous functions between two metric spaces, Characterizations of Continuous functions

Week 11: Continuous functions, Uniform continuity: Continuous functions on compact spaces, Uniform continuous functions, Homeomorphism and lsometry,

Week 12: Equicontinuity and Fixed Point Theory: Equicontinuity and Ascoli-Arzela Theorem, Fixed points, Banach contraction theorem.

Books and references

1. Q. H. Ansari: Metric Spaces Including Fixed Point Theory and Selvalued Maps, Narosa Publishing House, New Delhi. 2010.
2. E. T. Copson: Metric spaces, Cambridge University Press, 1968.
3. M. O. Searcoid: Metric spaces, Springer, 2007.
4. S. Kumaresan: Topology of Metric Spaces, Narosa Publishing House, 2nd Ed,2011.

Instructor bio

Prof. Qamrul Hasan Ansari

Aligarh Muslim University

Prof. Qamrul Hasan Ansari is a professor in the Department of Mathematics, Aligarh MuslimUniversity, Aligarh, lndia. At present, he is Dean of Faculty of Science, and Chairperson of Department of Mathematics, Aligarh Muslim University, Aligarh, lndia. He is the recipient of AMU Outstanding Research Award of the Years 2021 and ranked among lop2o/o scientists in the World by Stanford University in the years 2020,2021,2022. He is listed in the top cited 800 mathematicians of the world since 2007. ln the year 2017, he is placed at 200th position. He has published more than 270 research papers in internationally repute journal. He is the associate editor of Journal of Optimization Theory and Applications, Fixed Point Theory and Applications, Carpathian Journal of Mathematics, Fixed Point Theory (Romania), etc. He has acted as a guest editor of several internationally repute journals, namely, Positivity, Journal of Global Optimization, Applicable Analysis, Journal of Fixed Point Theory, etc. He is regular visitor of several universities around the world. He is the author of a book on Metric Spaces published by Narosa and co-author of two books namely, Generalized Convexity and Nonsmooth Variational lnequalities, published by Taylor & Francis, CRC Press, and Fixed Point Theory and Variational Principles in Metric Spaces, published by Cambridge University Press. He has edited more than 8 books published by Springer, Taylor & Francis, Elsevier, etc. He has delivered plenary / invited talks in more than 70 international conference / workshops across the World. He has supervised eight PhD and two M. Phil. Students. He is one of the most cited persons in his department with more than 5000 citations and with h-index 40.

Prof. Monirul Islam

Prof. Monirul lslam is an Assistant Professor in the Department of Mathematics at Aligarh Muslim University. He obtained his Ph.D. degree in Mathematics in 2019. He is working in variational analysis and optimization on Riemannian/ Hadamard manifolds. He has published several research papers in the Journal of lnternational Repute, e.g., Applicable Analysis, Journal of Nonlinear and Convex Analysis, Carpathian Journal of Mathematics, Journal of lnequalities and Applications, Mathematics, etc. Dr. lslam has eight months of research collaboration with Prof. J.C. Yao (Department of Applied Mathematics, National Sun Yalsen University, and Research Center for Nonlinear Analysis and Optimization and General Education Center, Kaohsiung Medical University, Kaohsiung City, Taiwan). Prof. lslam is a reviewer of many international Journals, namely, the Journal of Nonlinear and Convex Analysis, Carpathian Journal of Mathematics, Nonlinear Functional Analysis and Applications, MathSciNet- Mathematical Reviews (American Mathematicalsociety), etc. He has delivered an invited talk (Research Center for Nonlinear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung, Taiwan) and presented research papers at lnternational conferences/workshops in lndia and abroad. He is a lifetime member of the lndian Mathematical Society. Besides all the work, he is actively involved in administrative work, e.9., as an observer (AMU-B. Tech. entrance exam, Patna Centre, two times) and veriflTing officer (AMU-B.Sc. admission, two times).

Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.

The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).

Date and Time of Exams: 25 October 2024 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.

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. If there are any changes, it will be mentioned then.

Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

CRITERIA TO GET A CERTIFICATE

Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.

Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup. It will have the logos of AMU and INI.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- INI Team

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