Functional Analysis-II

Paper Code: 
MAT 421
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
Objective: 

This course will enable the students to -

  • Understand the concept of Normed linear spaces, Banach spaces, Hilbert Spaces, and operators on these spaces.

Course Outcomes (COs):

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

 

 

 

 

 

MAT 421

Functional Analysis-II

(Theory)

The students will be able to –

 

CO140: Explain the fundamental concepts of functional analysis in applied contexts.

CO141: Use the properties of continuity, series expansions etc.

CO142: Identify normal, self adjoint or unitary operators.

CO143: Analyse various inequalities and their applications

CO144:  Identify various operators like differential, integral etc.ct orthonormal sets.

CO145: Calculate the integral analytically using R-integrals

 

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 and 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.

 

References: 
  • 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: 
2023-2024
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