FUNCTIONAL ANALYSIS-II (Compulsory Paper)

Paper Code: 
MAT 421
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. Inter-relate the problems arising in Partial Differential Equations, Measure Theory and other branches of Mathematics.
  3. Know about various spaces such as Banach spaces, Hilbert Spaces.
  4. Use the operators on these spaces.

Course Outcomes (COs):

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Paper Code

Paper Title

 

MAT 421

 

 

Functional Analysis-II

(Theory)

The students will be able to –

 

CO116: Explain the fundamental concepts of functional analysis in applied contexts.

CO117: Use elementary properties of Banach space and Hilbert space.

CO118: Identify normal, self adjoint or unitary operators.

CO119: Communicate the spectrum of bounded linear operators.

CO120: Construct orthonormal sets.

 

Approach in teaching:

Interactive Lectures, Discussion, Power Point Presentations, Informative videos

Learning activities for the students:

Self learning assignments, Effective questions, presentations, Field trips

Quiz, Poster Presentations,

Power Point Presentations, Individual and group projects,

Open Book Test, Semester End Examination

 

Unit I: 
I
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: 
II
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: 
III
15.00
Mean value theorem and its applications, Partial derivatives, Jacobian Matrix.                  
Unit IV: 
IV
15.00
Continuously differentiable maps, Higher derivatives, Taylor’s formula, Inverse function theorem, Implicit function theorem.                                                 
Unit V: 
V
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.
  • 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.

 

Academic Year: 
2020-2021
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