Advanced Analysis
advance-analysis
Syllabus
Faculty: Dr. Bhairaba Kumar Majhi
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:
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
Examples on Metric Spaces
Study Material: http://more examples
Youtube link: https://youtu.be/etP21xln8iQ
Project-1: Write a report on applications of metric spaces.
Theorems on Metric Spaces.
Study Material: Theorems on metric space
youtube link: https://youtu.be/Af03P1xVNSs?list=PLbMVogVj5nJSxFihV-ec4A3z_FOGPRCo-
Project-2: Show that the class denote a set of all bounded sequence of real numbers is a metric space.
Open spheres.
youtube link: https://youtu.be/QC2n_Y2XJDM
Closed spheres
youtube link: https://youtu.be/QC2n_Y2XJDM
Project-3: Show that the class of all sequences is a metric space.
Neighbourhood of a point
youtube link: https://youtu.be/pvGWK65vw7A
Interior points and theorems
youtube link:https://youtu.be/VsL8Tq8nDow
Project-4: Write a report on relation between boundary points, limit points. And their applications
Open set
Study material: open set
youtube link: https://youtu.be/EE5FTMBkQjE
Project-5(Flip class): Prove theorems on Neighbourhoods, Interior points
closed set
study material: closed set and theorems
youtube link: https://youtu.be/PcnSgIoASSk
limit points and isolated point
youtube link:https://youtu.be/Z0e4gZZCxBM
Project-6: Write a report on difference between open set closed set and limit point , interior point with appropriate examples.
closure of a set
youtube link: https://youtu.be/kx7-juddElg
Theorems on closurere of a set
youtube link: https://youtu.be/kx7-juddElg
dense set
youtube link: https://youtu.be/tI7tC7lkkic
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.
Separable metric spaces
youtube link: https://youtu.be/lv-j2KK-prc
sequences in metric space
PPT link: https://www.slideshare.net/VishangSingh/analysis-sequences-and-bounded-sequences
convergent sequences
PPT link :https://www.slideshare.net/math266/20-sequences-x-138311036?qid=7d545255-01be-4c5f-a7f3-057f45830dc7&v=&b=&from_search=2
youtube link: https://youtu.be/DZT8o7Y08Mw
youtube link: https://youtu.be/FgRslqEYTJI
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.
cauchy sequences
youtube link: https://youtu.be/jD5s2jvnH1k
Project-11(Flip class): Prove theorems on convergent sequences and Cauchy sequences.
subspaces and theorems
youtube link: https://youtu.be/eY-WLtxKpBw
Complete metric space
yo utube link:https://youtu.be/X_weB_pYMV4
Project-12(Flip class): Prove theorems on Complete metric space, subspace.
Continuous functions
PPT link: https://www.slideshare.net/sumanmathews/continuous-functions
youtubelink : https://youtu.be/hXkQqCBLRp8
Project- 13: Prove that addition, subtraction, multiplication and division (with condition) of continuous functions are continuous.
Theorems on Continuous functions
PPT link: https://www.slideshare.net/VishveshJasani/continuity-of-a-function
Uniform continuity and Theorems
youtube lonk: https://youtu.be/IAN7TUbNOCs
Connectedness
youtube link: https://youtu.be/_06QIROBDko
Project-14(Flip class): Prove theorems on Uniform continuity and Connectedness.
Theorems on Connectedness
youtube link: https://youtu.be/mdCU7TlZupI
Project- 15: Write the difference between continuity and uniform continuity and explain through a proper example
Theorems on Connected and separated sets
youtube link: https://youtu.be/77KtqjqknYk
Project-16 (Flip class): Prove theorems on connected subsets of separated sets.
Disconnected sets
youtube link: https://youtu.be/mdCU7TlZupI
Theorems on Disconnected sets
youtube link : https://youtu.be/VDqhDsT40wU
Project- 17(Flip class): Geometrical interpretation of connected and disconnected sets.
Banach fixed point theorem.
you tube link: https://youtu.be/9jL8iHw0ans
contraction mapping
youtube link: https://youtu.be/vcQSOm2-hEI
contraction mapping and theorems
youtube link: https://youtu.be/cwmnjP1rxHM
Project-18: Write some applications of Banach fixed point theorem.
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
Learning Outcomes
- 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
- Explain the definition of continuity for functions from Rn to Rm and determine whether a given function from Rn to Rm is continuous
- Explain the geometric meaning of each of the metric space properties and be able to verify whether a given distance function is a metric
- 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
- State the definition of continuity of a function between two metric spaces.
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)
- 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 6
Open spheres.
youtube link: https://youtu.be/QC2n_Y2XJDM
Session 7
Closed spheres
youtube link: https://youtu.be/QC2n_Y2XJDM
Session 8
Project-3: Show that the class of all sequences is a metric space.
Session 9
Neighbourhood of a point
youtube link: https://youtu.be/pvGWK65vw7A
Session 10
Interior points and theorems
youtube link:https://youtu.be/VsL8Tq8nDow
Session 11
Project-4: Write a report on relation between boundary points, limit points. And their applications
Session 12
Open set
Study material: open set
youtube link: https://youtu.be/EE5FTMBkQjE
Session 13
Project-5(Flip class): Prove theorems on Neighbourhoods, Interior points
Session 14
closed set
study material: closed set and theorems
youtube link: https://youtu.be/PcnSgIoASSk
Session 15
limit points and isolated point
youtube link:https://youtu.be/Z0e4gZZCxBM
Session 16
Project-6: Write a report on difference between open set closed set and limit point , interior point with appropriate examples.
Session 17
closure of a set
youtube link: https://youtu.be/kx7-juddElg
Session 18
Theorems on closurere of a set
youtube link: https://youtu.be/kx7-juddElg
Session 19
dense set
youtube link: https://youtu.be/tI7tC7lkkic
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 22
Separable metric spaces
youtube link: https://youtu.be/lv-j2KK-prc
Session 23
sequences in metric space
PPT link: https://www.slideshare.net/VishangSingh/analysis-sequences-and-bounded-sequences
Session 24
convergent sequences
PPT link :https://www.slideshare.net/math266/20-sequences-x-138311036?qid=7d545255-01be-4c5f-a7f3-057f45830dc7&v=&b=&from_search=2
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 27
cauchy sequences
youtube link: https://youtu.be/jD5s2jvnH1k
Session 28
Project-11(Flip class): Prove theorems on convergent sequences and Cauchy sequences.
Session 29
subspaces and theorems
youtube link: https://youtu.be/eY-WLtxKpBw
Session 30
Complete metric space
yo utube link:https://youtu.be/X_weB_pYMV4
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 35
Uniform continuity and Theorems
youtube lonk: https://youtu.be/IAN7TUbNOCs
Session 36
Connectedness
youtube link: https://youtu.be/_06QIROBDko
Session 37
Project-14(Flip class): Prove theorems on Uniform continuity and Connectedness.
Session 38
Theorems on Connectedness
youtube link: https://youtu.be/mdCU7TlZupI
Session 39
Project- 15: Write the difference between continuity and uniform continuity and explain through a proper example
Session 40
Theorems on Connected and separated sets
youtube link: https://youtu.be/77KtqjqknYk
Session 41
Project-16 (Flip class): Prove theorems on connected subsets of separated sets.
Session 42
Disconnected sets
youtube link: https://youtu.be/mdCU7TlZupI
Session 43
Theorems on Disconnected sets
youtube link : https://youtu.be/VDqhDsT40wU
Session 44
Project- 17(Flip class): Geometrical interpretation of connected and disconnected sets.
Session 45
Banach fixed point theorem.
you tube link: https://youtu.be/9jL8iHw0ans
Session 46
contraction mapping
youtube link: https://youtu.be/vcQSOm2-hEI
Session 47
contraction mapping and theorems
youtube link: https://youtu.be/cwmnjP1rxHM
Session 48
Project-18: Write some applications of Banach fixed point theorem.
Case Studies
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Course Materials
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