Advanced Analysis
Teacher
Dr. Bhairaba Kumar Majhi
Category
Core Courses
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Course Name : Advanced Analysis
Code(Credit) : CUTM1519(3-0-3)
Course Objectives
- To point out that iterative processes and convergence of sequences occur in many areas of mathematics, and to develop a general context in metric spaces.
- To provide a basic course in analysis.
- To reinforce ideas of proof
Course Outcomes
| COs | Course outcomes | Mapping Cos with POs (High-3, Medium-2, Low-1) |
| CO1 | Understand the Euclidean distance function on Rn and appreciate its properties, and state and use the Triangle and Reverse Triangle Inequalities for the Euclidean distance function on Rn
|
PO1(3), PO2(2) |
| CO2 | Explain the definition of continuity for functions from Rn to Rm and determine whether a given function from Rn to Rm is continuous
|
PO2(2) |
| CO3 | Explain the geometric meaning of each of the metric space properties and be able to verify whether a given distance function is a metric
|
PO9(2) |
| CO4 | Distinguish between open and closed balls in a metric space and be able to determine them for given metric spaces
Define convergence for sequences in a metric space and determine whether a given sequence in a metric space converges
|
PO4(3) |
Course Syllabus
Module-I (T-5hr P-0hr P-6hr)
Metric Spaces: Definition and examples, open and closed spheres,
Project-1: Write a report on applications of metric spaces.
Project-2: Show that the class denote a set of all bounded sequence of real numbers is a metric space.
Project-3: Show that the class of all sequences is a metric space.
Module-II (T-3hr P-0hr P-4hr)
Neighbourhoods, Interior points. Open set
Project-4: Write a report on relation between boundary points, limit points. And their applications.
Project-5(Flip class): Prove theorems on Neighbourhoods, Interior points
Module-III (T-5hr P-0hr P-6hr)
Closed set, boundary points, limit points and isolated points, closure of a set, dense sets.
Project-6: Write a report on difference between open set closed set and limit point , interior point with appropriate examples.
Project-7: Write the difference between boundary points, limit points and isolated points explain through examples.
Project-8 (Flip class): Prove theorems on closed set and dense sets.
Module-IV (T-4hr P-0hr P-6hr)
Separable metric spaces, sequences in metric space, convergent sequences, Cauchy sequences
Project-9: Write a report on applications of convergent sequences and Cauchy sequences.
Project-10: Every convergent sequence is bounded. Is the reverse true? Justify your answer with suitable example.
Project-11(Flip class): Prove theorems on convergent sequences and Cauchy sequences.
Module-V (T-4hr P-0hr P-4hr)
Complete metric space, subspace, Cantor’s theorem. Continuous functions,
Project- 12(Flip class): Prove theorems on Complete metric space, subspace.
Project-13(Flip class): : Prove that addition, subtraction, multiplication and division (with condition) of continuous functions are continuous.
Module-VI (T-4hr P-0hr P-6hr)
Uniform continuity, Connectedness, connected subsets of separated sets.
Project-14(Flip class): Prove theorems on Uniform continuity and Connectedness.
Project- 15: Write the difference between continuity and uniform continuity and explain through a proper example..
Project-16 (Flip class): Prove theorems on connected subsets of separated sets.
Module-VII (T-5hr P-0hr P-4hr)
Disconnected sets, contraction mappings, Banach fixed point theorem.
Project- 17(Flip class): Geometrical interpretation of connected and disconnected sets.
Project-18: Write some applications of Banach fixed point theorem.
Text Books:
- K. Jain and K. Ahmad, Metric Spaces, Narosa Publishing House, New Delhi.
- Methods of Real analysis-R.G. Goldberg. Chapters: 4(4.2,and 4.3), 5,and 6 (up to 6.8)
Reference Books:
- S.C. Malik and S. Arora – Mathematical Analysis (New Age International)
- Principles of Mathematical Analysis – Walter Rudin, III Edition, McGRAW-Hill Book Co.
Session Plan
Session 1
Introduction
PPT link: https://www.slideshare.net/NaliniSPatil/metric-space-151515844
Study Material: Introduction to Metric spaces with examples
youtube link: https://youtu.be/6CLl5xx5X-Y
Session 2
Examples on Metric Spaces
Study Material: http://more examples
Youtube link: https://youtu.be/etP21xln8iQ
Session 3
Project-1: Write a report on applications of metric spaces.
Session 4
Theorems on Metric Spaces.
Study Material: Theorems on metric space
youtube link: https://youtu.be/Af03P1xVNSs?list=PLbMVogVj5nJSxFihV-ec4A3z_FOGPRCo-
Session 5
Project-2: Show that the class denote a set of all bounded sequence of real numbers is a metric space.
Session 8
Project-3: Show that the class of all sequences is a metric space.
Session 11
Project-4: Write a report on relation between boundary points, limit points. And their applications
Session 13
Project-5(Flip class): Prove theorems on Neighbourhoods, Interior points
Session 14
Session 16
Project-6: Write a report on difference between open set closed set and limit point , interior point with appropriate examples.
Session 20
Project-7: Write the difference between boundary points, limit points and isolated points explain through examples
Session 21
Project-8 (Flip class): Prove theorems on closed set and dense sets.
Session 23
sequences in metric space
PPT link: https://www.slideshare.net/VishangSingh/analysis-sequences-and-bounded-sequences
Session 24
convergent sequences
youtube link: https://youtu.be/DZT8o7Y08Mw
youtube link: https://youtu.be/FgRslqEYTJI
Session 25
Project-9: Write a report on applications of convergent sequences and Cauchy sequences
Session 26
Project-10: Every convergent sequence is bounded. Is the reverse true? Justify your answer with suitable example.
Session 28
Project-11(Flip class): Prove theorems on convergent sequences and Cauchy sequences.
Session 31
Project-12(Flip class): Prove theorems on Complete metric space, subspace.
Session 32
Continuous functions
PPT link: https://www.slideshare.net/sumanmathews/continuous-functions
youtubelink : https://youtu.be/hXkQqCBLRp8
Session 33
Project- 13: Prove that addition, subtraction, multiplication and division (with condition) of continuous functions are continuous.
Session 34
Theorems on Continuous functions
PPT link: https://www.slideshare.net/VishveshJasani/continuity-of-a-function
Session 37
Project-14(Flip class): Prove theorems on Uniform continuity and Connectedness.
Session 39
Project- 15: Write the difference between continuity and uniform continuity and explain through a proper example
Session 41
Project-16 (Flip class): Prove theorems on connected subsets of separated sets.
Session 44
Project- 17(Flip class): Geometrical interpretation of connected and disconnected sets.
Session 48
Project-18: Write some applications of Banach fixed point theorem.
Case Studies
Case Studies
Our Main Teachers
My Self Dr.Bhairaba Kumar Majhi,M,Sc,B.Ed,M.Tech(CSE),Ph.D(Mathematics ) ,Asst.Professor And HOD Department of Mathematics School Of Applied Sciences,Presently working at CUTM,Bolangir Campus.I have 16 Years of Experience in Teaching at Different Levels Of UG,PG & Engineering Mathematics.