Analysis-II

Paper Code: 
CMAT 613
Credits: 
4
Contact Hours: 
60.00
Max. Marks: 
100.00
Objective: 

This course will enable the students to -

  1. Introduce the basic ideas of analysis for Fourier Series, convergence of Sequences, Metric spaces etc.
  2. Emphasis has been laid on Cauchy’s sequences, continuous mappings, connected, compact sets and related theorems.

Course Outcomes (Cos):

 

 Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

 

 

 

 

 

 

 

 

CMAT 612

 

 

 

 

Analysis-II

(Theory)

 

 

 

 

 

 

The students will be able to –

 

CO121: Describe the types of Fourier series, related properties and theorems.

CO122: Differentiate between convergence, uniformly convergence and absolute convergence.

CO123: Identify sequences and series.

CO124: Describe metric spaces, their types and various properties.

CO125: Use the knowledge of Fourier series in various applications like signal system processing, digital signal propagation etc. 

CO126: Describe special spaces like compact, connected, product etc and their applications in real life problems.

Approach in teaching:

 

Interactive Lectures, Discussion, Power Point Presentations, Informative videos

 

Learning activities for the students:

Self learning assignments, Effective questions, presentations, Giving tasks

 

Quiz, Poster Presentations,

Power Point Presentations, Individual and group projects,

Open Book Test, Semester End Examination

 

 

 

 

 

Unit I: 
I
9.00
Fourier series: Periodic and piecewise continuous function, Dirichlet’s conditions, Fourier series representation of function on intervals [-π , π], [0 , π] and on arbitrary intervals, Fourier series of odd and even functions.
 
Unit II: 
II
9.00
Sequence and series of function: Point wise and uniform convergence, Cauchy criterion and Weirstrass M- Test (including proof), Abel’s and Dirichlet’s Test (Without proof),Uniform convergence and continuity, Term by term differentiation and integration.
 
Unit III: 
III
9.00
Metric Space: Definition with examples, Bounded set, Open set, Closed set, Neighborhood, Boundary points and limit points, Exterior point, Closure of a set, Metric subspace.
 
Unit IV: 
IV
9.00
Continuous mappings, Sequence in a metric space, Cauchy sequence, Subsequence, Completeness of metric space.
 
Unit V: 
V
9.00
Separable spaces, Compact spaces and Compact sets, Connected spaces and Connected sets, Bolzano’s theorem, Product spaces.
 
Essential Readings: 
  • T.M. Apostol, Mathematical Analysis, Narosa Publishing House, New Delhi, 2002. 
  • G. F. Simmons, Introduction to Topology and Modern Analysis, Tata McGraw-Hill Education Pvt. Ltd., 2016.
  • Savita Arora and S. C. Malik, Mathematical Analysis, New Age International, 2017.
  • Robert T Seeley, An Introduction to Fourier Series and Integrals, Dover Publications Inc. 2006.
 
References: 
  • Michael O'Searcoid, Metric Spaces, Springer, 2007.
  • Irving Kaplansky, Set Theory and Metric Space, AMS Chelsea Publishing, 2003.
  • Heinonen, Juha, Lectures on Analysis on Metric Spaces, Springer, 2001.
  • P.K. Jain and K. Ahmad, Metric Spaces, Narosa Publishing House, New Delhi, 2004.
  • Shanti Narayan, A course of Mathematical Analysis, S. Chand and Co New Delhi, 2005. 
  • K.C. Sarangi, Real Analysis and Metric spaces, Ramesh Book Depot Jaipur, 2006.
 
Academic Year: 
2022-2023
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