Analysis-II

Paper Code: 
MAT602
Credits: 
3
Contact Hours: 
45.00
Max. Marks: 
100.00
Objective: 

This course will enable the students to -

  1. Introduce the basic ideas of analysis for the Fourier Series, the 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

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

 

 

 

 

 

MAT 602

 

 

 

 

 

 

Analysis-II

(Theory)

 

 

 

 

 

 

The students will be able to –

 

CO87: Students will describe the types of Fourier series, related properties, and theorems.

CO88: Students will differentiate between convergence, uniformly convergence, and absolute convergence.

CO89: Students will identify sequences and series.

CO90: Students will describe metric spaces, their types, and their various properties.

CO91: Students will use the knowledge of the Fourier series in various applications like signal system processing, digital signal propagation, etc. 

CO92: Students will describe special spaces like compact, connected, product, etc, and their applications in real-life problems.

The approach in teaching:

Interactive Lectures, discussions, PowerPoint Presentations, Informative videos

 

Learning activities for the students:

Self-learning assignments, Effective questions, presentations, Giving tasks

Quiz, Poster Presentations,

PowerPoint 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: 
  • Shanti Narayan, A course of Mathematical Analysis, S. Chand and Co New Delhi, 2005. 
  • 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.
  • K.C. Sarangi, Real Analysis and Metric spaces, Ramesh Book Depot Jaipur, 2006.
 
Academic Year: 
2022-2023
  • Printer-friendly version
  • PDF version