# Analysis

Teacher

Category

### Course Attendees

Still no participant

Still no reviews

# 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