Functional Analysis-I

Paper Code: 
24MAT321
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
Objective: 

This course will enable the students to -

  1. Cover theoretical needs of Partial Differential Equations and Mathematical Analysis. 
  2. Solve the problems arising in Partial Differential Equations, Measure Theory and other branches of Mathematics.
  3. Compare various spaces such as Normed Linear Spaces, Banach spaces.
  4. Apply the operators on these spaces.

 

Course Outcomes: 

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

 

 

 

 

 

 

 

24MAT

321

 

 

Functional Analysis-I

(Theory)

 

 

 

CO71: Explain the fundamental concepts of normed linear spaces and quotient spaces.

CO72: Test elementary properties of normed linear spaces and compactness.

CO73: Apply the knowledge of various theorems in solving the numerical.

CO74: Explain the concept of Hilbert space.

CO75: Analyze various inequalities and Construct orthonormal sets.

CO76: Contribute effectively in course-specific interaction.

 

Approach in teaching:

Interactive Lectures, Discussion, Informative videos

 

Learning activities for the students:

Self learning assignments, Effective questions,  Topic  presentation, Assigned tasks

 

 

Quiz, Class Test, Individual projects,

Open Book Test, Continuous Assessment, Semester End Examination

 

 

 

Unit I: 
Normed linear and Banach spaces:
15.00

Normed linear spaces, Quotient space of normed linear spaces and its completeness, Banach spaces and examples, Bounded linear transformations.

 

Unit II: 
Normed linear spaces of linear transformations:
15.00

Normed linear space of bounded linear transformations, Equivalent norms, Basic properties of finite dimensional normed linear spaces and compactness, Riesz lemma.

 

Unit III: 
Theorems in Banach spaces:
15.00

Open mapping theorem, Closed graph theorem, Uniform boundness theorem, Continuous linear functional, Hahn-Banach theorem and its consequences.

 

Unit IV: 
Hilbert space:
15.00

Hilbert space and its properties, Orthogonality and functionals in Hilbert spaces, Phythagorean theorem, Projection theorem, Orthonormal sets.

 

Unit V: 
More properties on Hilbert spaces:
15.00

Bessel’s inequality, Complete orthonormal sets, Parseval’s identity, Structure of a Hilbert space, Riesz representation theorem.

 

Essential Readings: 
  • B.V. Limaye, Functional Analysis, New Age International, New Delhi, 2017.
  • D. S. Chauhan, Functional Analysis and calculus in Banach space, Jaipur Publishing House, 2013.
  • G.F. Simmons, Topology and Modern Analysis, Mc-Graw Hill, 2017.
  • G. Bachman, Lawrence Narici, Functional Analysis, Academic Press, 2003.

SUGGESTED READING

  • A.E. Taylor, Introduction to Functional analysis, John Wiley and Sons, 1980.
  • A.L. Brown and A. Page, Elements of Functional Analysis, Van Nostrand Reinhold, 1970.
  • Erwin Kreyszig, Introductory Functional Analysis with Application, Willey, 2007.
  • F. Riesz and B. Sz. Nagay, Functional Analysis, Dover Publications, 2003.
  • Graham Allan and H. Garth Dales, Introduction to Banach Spaces and Algebras, Oxford            University Press, 2010.
  • Reinhold Meise, Dietmar Vogt and M. S. Ramanujan, Introduction to Functional analysis, Oxford University Press, 1997.

e- RESOURCES

JOURNALS

 

Academic Year: 
2024-2025
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