Metric and Vector Space

Paper Code: 
25DMAT501(A)
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. Emphasis has been laid on Cauchy’s sequences, continuous mappings, connected, compact sets and related theorems.

 

Course Outcomes: 

 Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

25DMAT

501(A)

 

 

 

 

 

 

 

Metric and Vector Space

(Theory)

 

 

 

 

 

 

CO45: Create metric spaces, Types and various properties.

CO46: Depict continuous mappings, Sequences and their properties.

CO47: Classify the types of various spaces like separable spaces, Compact, Connected, Product spaces etc.

CO48: Apply the knowledge of vector spaces and subspaces.

CO49: Explore the knowledge of dependence and independence of vectors and bases in various applications like signal system processing, Digital signal propagation etc.

CO50: Contribute effectively in course-specific interaction.

Approach in teaching:

Interactive Lectures, Discussion, PowerPoint Presentations, Informative videos

 

Learning activities for the students:

Self learning assignments, Effective questions, Assigned tasks

 

 

Quiz, Individual or group project,

Open Book Test, Continuous Assessment, 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
Definition 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: 
2025-2026
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