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- 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.

**U**pon 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.

**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.

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

Project-5: A study on Continuous linear maps.

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

Project-6: A study on Bounded Linear maps.

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**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

Span of a linear space

Quotient space

Product space

Project-1: A study on Linear spaces

Linear map

Range space

Zero space

Hyperspace

Linear maps on finite dimensional linear spaces

Project-2: A study on Linear maps

Compactness

Theorem regarding compactness

Theorem regarding compactness

Project-3: A study on Compactness

Normed space

Euclidian norm

Sequence spaces

Lp space

Function spaces

Inner product spaces

Project-4: A study on Normed spaces

Quotient norm

Riesz Lemma

Theorems regarding normed spaces

Theorems regarding normed spaces

Continuity of linear maps

Complete space

Some fundamental theorems

Project-5: A study on Continuous linear maps

Bounded Linear maps

Theorem

Theorem

Theorem

Project-6: A study on Bounded Linear maps

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 […]