Functional Analysis-I

Paper Code: 
MAT 321
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 Normed Linear Spaces, Banach 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 321

 

 

 

 

 

Functional Analysis-I

(Theory)

 

 

 

 

 

The students will be able to –

 

CO69: Explain the fundamental concepts of functional analysis in applied contexts.

CO70: Use elementary properties of Banach space and Hilbert space.

CO71: Identify normal, self adjoint or unitary operators.

CO72: Communicate the spectrum of bounded linear operator.

CO73: Construct orthonormal sets.

CO74:Analyse various inequalities and their applications

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
Normed linear spaces, Quotient space of normed linear spaces and its completeness, Banach spaces and examples, Bounded linear transformations.                       
Unit II: 
II
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: 
III
15.00

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

Unit IV: 
IV
15.00

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

Unit V: 
V
15.00
Bessel’s inequality, Complete orthonormal sets, Parseval’s identity, Structure of a Hilbert space, Riesz representation theorem.
 
Essential Readings: 
  • G.F. Simmons, Topology and Modern Analysis, Mc-Graw Hill, 1963.
  • G. Bachman, Lawrence Narici, Functional Analysis, Academic Press, 1966.
  • Dileep S. Chauhan, Functional Analysis and calculus in Banach space, Jaipur Publishing House, 2013.
  • B.V. Limaye, Functional Analysis, New Age International, New Delhi, 2017.
  • 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 Nostrand Reinhold, 1970.
  • F. Riesz and B. Sz. Nagay, Functional Analysis, Dover Publications, 1965.
 
Academic Year: 
2021-2022
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