Metric and Vector Space

Paper Code: 
24CMAT511
Credits: 
4
Contact Hours: 
60.00
Max. Marks: 
100.00
Objective: 

This course will enable the students to -

  1. Explain the basic ideas of analysis for Metric spaces and vector spaces etc.
  2. Emphasize on Cauchy’s sequences, continuous mappings.
  3. Differentiate between connected, compact sets and related theorems.

 

Course Outcomes: 

 Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

 

 

 

 

 

 

 

24CMAT

511

 

 

 

 

Metric and Vector Space

(Theory)

 

 

 

 

 

 

CO89: Create metric spaces, types and various properties.

CO90: Depict continuous mappings, sequences and their properties.

CO91: Classify the types of various spaces like separable spaces, compact, connected product spaces etc.

CO92: apply the knowledge of vector spaces and subspaces.

CO93: Explore the knowledge of dependence and independence of vectors and bases in various applications like signal system processing, digital signal propagation etc. 

CO94: Contribute effectively in course-specific interaction.

Approach in teaching:

Interactive Lectures, Discussion, Power Point Presentations, Informative videos

 

Learning activities for the students:

Self learning assignments, Effective questions, presentations, and assigned tasks

 

 

 

Quiz,

Individual or group projects,

Open Book Test, Semester End Examination

 

 

 

 

Unit I: 
Metric Space
12.00

Definition with examples, Bounded set, Open set, Closed set, Neighborhood, Boundary points and limit points, Exterior point, Closure of a set.

 

 

Unit II: 
Mappings and Sequences
12.00

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

                                                 

Unit III: 
Types of Metric spaces
12.00

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

Unit IV: 
Vector space
12.00

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

Unit V: 
Linearly dependent and independent vectors
12.00

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.

e- RESOURCES

JOURNALS

 

 

 

 

 

 

Academic Year: 
2024-2025
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