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# 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:

1. K. Jain and K. Ahmad, Metric Spaces, Narosa Publishing House, New Delhi.
2. Methods of Real analysis-R.G. Goldberg.   Chapters: 4(4.2,and 4.3), 5,and  6 (up to 6.8)

Reference Books:

1. S.C. Malik and S. Arora – Mathematical Analysis (New Age International)
2. Principles of Mathematical Analysis – Walter Rudin, III Edition, McGRAW-Hill Book Co.

## Session 2

Examples on Metric Spaces

Study Material: http://more examples

## Session 3

Project-1: Write a report on applications of metric spaces.

## Session 4

Theorems on Metric Spaces.

Study Material: Theorems on metric space

## Session 5

Project-2: Show that the class denote a set of all bounded sequence of real numbers is a metric space.

Open spheres.

Closed spheres

## Session 8

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

## Session 9

Neighbourhood of a point

## Session 10

Interior points and theorems

## 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

## Session 13

Project-5(Flip class): Prove theorems on Neighbourhoods, Interior points

## Session 14

closed set

study material: closed set and theorems

## Session 15

limit points and isolated point

## Session 16

Project-6: Write a report on difference between open set closed set and limit point , interior point with appropriate examples.

closure of a set

## Session 18

Theorems on closurere of a set

dense set

## 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

## Session 23

sequences in metric space

## Session 24

convergent sequences

## 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.

cauchy sequences

## Session 28

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

## Session 29

subspaces and theorems

## Session 30

Complete metric space

## Session 31

Project-12(Flip class): Prove theorems on Complete metric space, subspace.

## Session 32

Continuous functions

## Session 33

Project- 13: Prove that addition, subtraction, multiplication and division (with condition) of continuous functions are continuous.

## Session 34

Theorems on Continuous functions

## Session 35

Uniform continuity and Theorems

Connectedness

## Session 37

Project-14(Flip class): Prove theorems on Uniform continuity and Connectedness.

## Session 38

Theorems on Connectedness

## 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

## Session 41

Project-16 (Flip class): Prove theorems on connected subsets of separated sets.

## Session 42

Disconnected sets

## Session 43

Theorems on Disconnected sets

## Session 44

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

## Session 45

Banach fixed point theorem.

## Session 46

contraction mapping

## Session 47

contraction mapping and theorems

## Session 48

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

## Case Studies

### Dr. Bhairaba Kumar Majhi

##### Asst.Professor
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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.