Analysis

Teacher

Dr. Sasi Bhusan Padhi

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Course Name :  Analysis

Code(Credit) : CUNT2480(3-0-1)

Course Objectives

  • To get sufficient knowledge about different analytical structures, concepts and Theorems.
  • In much of analysis, the emphasis is not on finding explicit solutions to specific problems, but rather on determining which problems can be solved and what general properties solutions may share.

Learning Outcomes

Upon successful completion of this course, students will be able to:

  • Understand Linear maps, Normed Linear spaces, Inner product spaces.
  • Know about Closed Graph and Open Mapping Theorems, Bounded Inverse Theorem.

Course Syllabus

Module-I: Linear space, Hamel basis, Span of a linear space, Quotient space, Product space.

 

Project-1: A study on Linearspaces.

 

Module -II: Linear map, Range space, Zero space, Hyperspace, Linear maps on finite dimensional linear spaces.

 

Project-2: A study on Linear maps.

 

Module - III: Compactness, Some fundamental theorems regarding compactness.

 

Project-3: A study on Compactness.

 

Module - IV: Normed space, Euclidian norm, Sequence spaces, Lp space, Function spaces, Inner product spaces.

 

Project-4: A study on Normed spaces.

 

Module – V : Quotient norm, Riesz Lemma, Some theorems regarding normed spaces.

 

Module – VI : Continuity of linear maps, Complete space, Some fundamental theorems.

 

Project-5: A study on Continuous linear maps.

 

Module – VII : Bounded Linear maps, Some fundamental theorems,

 

Project-6: A study on Bounded Linear maps.

 

 

 

BOOK PRESCRIBED  :

 

Functional Analysis—B. V. Limayee (New Age— International Limited,

Publishers, Second Edition)

Chapters: 2, 2.1, 2.2, 2.3, 2.4, 2.5, 3.5, 3.6, 3.7, 5, 5.1, 5.2, 5.3, 5.4, 6, 6.1, 6.2, 6.3, 6.6, 6.7, 6.8.

 

Session Plan

Session 1

 

Linear space

Session 2

 

Hamel basis

Session 3

 

Span of a linear space

Session 4

 

Quotient space

Session 5

 

Product space

Session 6&7

 

Project-1: A study on Linear spaces

Session 8

 

Linear map

Session 9

 

Range space

Session 10

 

Zero space

Session 11

 

Hyperspace

Session 12

 

Linear maps on finite dimensional linear spaces

Session 13&14

 

Project-2: A study on Linear maps

Session 15

 

Compactness

Session 16

 

Theorem regarding compactness

Session 17

 

Theorem regarding compactness

Session 18&19

 

Project-3: A study on Compactness

Session 20

 

Normed space

Session 21

 

Euclidian norm

Session 22

 

Sequence spaces

Session 23

 

Lp space

Session 24

 

Function spaces

Session 25

 

Inner product spaces

Session 26&27

 

Project-4: A study on Normed spaces

Session 28

 

Quotient norm

Session 29

 

Riesz Lemma

Session 30

 

Theorems regarding normed spaces

Session 31

 

Theorems regarding normed spaces

Session 32

 

Continuity of linear maps

Session 33

 

Complete space

Session 34

 

Some fundamental theorems

Session 35&36

 

Project-5: A study on Continuous linear maps

Session 37

 

Bounded Linear maps

Session 38

 

Theorem

Session 39

 

Theorem

Session 40

 

Theorem

Session 41&42

 

Project-6: A study on Bounded Linear maps

Our Main Teachers

Dr. Sasi Bhusan Padhi

Asst. Prof. Department of Mathematics
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Dr. Sasi Bhusan Padhi works as an Assistant Professor, Department of Mathematics, Centurion University of Technology and Management. He has received M.Sc degree in Mathematics from Berhampur University in the year 2000 with Gold Medal and Ph. D from Centurion University of Technology and Management. He has more than 20 years of teaching experience in […]