Metric and Vector Space

Paper Code: 
CMAT 511
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 Metric spaces and vector 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 511

 

 

 

 

 

 

 

Metric and Vector Space

(Theory)

 

 

 

 

 

 

 

The students will be able to –

 

CO65: Describe metric spaces, types and various properties.

CO66: Identify continuous mappings.

CO67: Describe the types separable spaces, compact spaces etc.

CO68: Use the knowledge of vector spaces in various applications like signal system processing, digital signal propagation etc. 

CO69: Describe special spaces like 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
Metric Space: Definition with examples, Bounded set, Open set, Closed set, Neighborhood, Boundary points and limit points, Exterior point, Closure of a set.
 
Unit II: 
II
9.00

Continuous mappings, Sequence in a metric space, Cauchy sequence, Subsequence, Completeness of metric space.

Unit III: 
III
9.00

Separable spaces, Compact spaces and Compact sets, Connected spaces and Connected sets, Bolzano’s theorem, Product spaces.

Unit IV: 
IV
9.00

Vector space: Definition with Examples, Sub-space, Linear combination of vectors, Linear Span.

Unit V: 
V
9.00

Linearly dependent and independent vectors and their simple properties, Bases and dimensions.

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.
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: 
2023-2024
  • Printer-friendly version
  • PDF version