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Functional Analysis-II [1]

Paper Code: 
24MAT421
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
Objective: 

This course will enable the students to -

  1. Create the concept of Normed linear spaces, Banach spaces, Hilbert Spaces
  2. Apply different types of operators on these spaces.

 

Course Outcomes: 

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

 

 

 

 

 

 

 

24MAT

421

 

Functional Analysis-II

(Theory)

CO142: Explain the fundamental concepts of Hilbert space and projections.

CO143: Apply the properties of Banach space and derivatives.

CO144: Analyse mean value theorem and their applications.

CO145: Investigate the continuously differentiable maps and theorems.

CO146: Calculate the integral analytically using step function.

CO147: 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: 
Hilbert space:
15.00

Adjoint of an operator on a Hilbert space: Self-adjoint, positive, normal and unitary operators and their properties, Projection on a Hilbert space.

 

Unit II: 
Derivatives of continuous map:
15.00

Derivatives of a continuous map from an open subset of Banach space to a Banach space, Rules of derivation, Derivative of a composite, Directional derivative.

 

Unit III: 
Mean value theorem:
15.00

Mean value theorem and its applications, Partial derivatives and Jacobian Matrix.

 

Unit IV: 
Continuously differentiable maps:
15.00

Continuously differentiable maps, Higher derivatives, Taylor’s formula, Inverse function theorem, Implicit function theorem.

 

Unit V: 
Integration:
15.00

Step function, Regulated function, Primitives and integrals, Differentiation under the integral sign, Riemann integral of function of real variable with values in normed linear space.

 

Essential Readings: 
  • G.F. Simmons, Topology and Modern Analysis, McGraw Hill, 1963.
  • George Bachman and Lawrence Narici, Functional Analysis, Academic Press, 1964.
  • Dileep S. Chauhan, Functional Analysis and calculus in Banach space, JPH, 2016.
  • B.V. Limaye, Functional Analysis, New age international, 2017.
  • B.V. Limaye, Linear Functional Analysis for Scientists and Engineers, Springer, 2016.

SUGGESTED READING

  • Erwin Kreyszig, Introductory Functional Analysis with Application, Willey, 2007.
  • A.E. Taylor, Introduction to Functional Analysis, John Wiley and sons, 1958.
  • 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.
  • A.L. Brown and A. Page, Elements of Functional Analysis, Van Nostrad Reinlold, 1970.
  • Walter Rudin, Functional Analysis, McGraw- Hill, 1973.
  • Barbara D. Maccluer, Elementary Functional Analysis, Springer, 2009.

e- RESOURCES

 

JOURNALS

 

Academic Year: 
2024-2025 [9]

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Source URL: https://maths.iisuniv.ac.in/courses/subjects/functional-analysis-ii-3

Links:
[1] https://maths.iisuniv.ac.in/courses/subjects/functional-analysis-ii-3
[2] https://nptel.ac.in/courses/111105037
[3] https://nptel.ac.in/courses/111106147
[4] https://www.digimat.in/nptel/courses/video/111105037/L10.html
[5] https://www.mimuw.edu.pl/~aswiercz/AnalizaF/lecture.pdf
[6] https://www.hindawi.com/journals/jfs/2020/6666229/
[7] https://www.degruyter.com/journal/key/agms/html?lang=en
[8] https://www.springer.com/journal/41
[9] https://maths.iisuniv.ac.in/academic-year/2024-2025