# Analysis

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# 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.

Linear space

Hamel basis

## Session 3

Span of a linear space

Quotient space

Product space

## Session 6&7

Project-1: A study on Linear spaces

Linear map

Range space

Zero space

Hyperspace

## Session 12

Linear maps on finite dimensional linear spaces

## Session 13&14

Project-2: A study on Linear maps

Compactness

## Session 16

Theorem regarding compactness

## Session 17

Theorem regarding compactness

## Session 18&19

Project-3: A study on Compactness

Normed space

Euclidian norm

Sequence spaces

Lp space

Function spaces

## Session 25

Inner product spaces

## Session 26&27

Project-4: A study on Normed spaces

Quotient norm

Riesz Lemma

## Session 30

Theorems regarding normed spaces

## Session 31

Theorems regarding normed spaces

## Session 32

Continuity of linear maps

Complete space

## Session 34

Some fundamental theorems

## Session 35&36

Project-5: A study on Continuous linear maps

## Session 37

Bounded Linear maps

Theorem

Theorem

Theorem

## Session 41&42

Project-6: A study on Bounded Linear maps

### Our Main Teachers

##### Asst. Prof. Department of Mathematics
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MR. 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 2001 with Gold Medal. He has 17 years of teaching experience in different Graduate and Post Graduate colleges. Now he is doing research in […]