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- 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

- Understand the Euclidean distance function on R
and appreciate its properties, and state and use the Triangle and Reverse Triangle Inequalities for the Euclidean distance function on R^{n}^{n} - Explain the definition of continuity for functions from R
to R^{n}and determine whether a given function from R^{m}to R^{n}is continuous^{m} - 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.

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.

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-3: Show that the class of all sequences is a metric space.

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

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.

sequences in metric space

PPT link: https://www.slideshare.net/VishangSingh/analysis-sequences-and-bounded-sequences

convergent sequences

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-11(Flip class): Prove theorems on convergent sequences and Cauchy sequences.

**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

**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.

Project- 17(Flip class): Geometrical interpretation of connected and disconnected sets.

Project-18: Write some applications of Banach fixed point theorem.

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.

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