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Course Co-ordinated by IIT Kharagpur
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Prof. P.D. Srivastava
IIT Kharagpur

 

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It is a first level course on Functional Analysis. The motto is to familiarize the students with basic concepts, principles and methods of Functional analysis and its applications.

CONTENTS : Metric spaces with example, Complete metric spaces , Separable Metric Space , Compact sets, Normed & Banach spaces , Convergence , Bounded linear functionals and operators , Dual spaces , Refexive Spaces,Adjoint Operator, Inner Product Space and Hilbert Spaces with example, Projection theorem , Orthonormal sets and sequences ,Total Orthonormal Sets , Riesz Representation theorem, Self adjoint, Unitary and Normal operators, Hilbert -Adjoint Operator,The Hahn Banach Extension theorem, Uniform boundedness theorem (The Banach Steinhaus theorem ), Open mapping theorem and Closed graph theorem .

Module No.

Topic/s

Lectures

1

Metric Spaces

  1. Metric spaces with examples

  2. Holder inequality & Minkowski inequality

  3. Various concepts in a metric space

  4. Separable metric space with examples

  5. Convergence, Cauchy sequence , Completeness

  6. Examples of Complete & Incomplete metric spaces

  7. Completion of Metric spaces +Tutorial

  8. Vector spaces with examples

2

2

Normed / Banach Spaces

  1. Normed Spaces with examples

  2. Banach Spaces & Schauder Basis

  3. Finite Dimensional Normed Spaces & Subspaces

  4. Compactness of Metric/Normed spaces

  5. Linear Operators-definition & examples

  6. Bounded linear operators in a Normed Space

  7. Bounded linear Functionals in a Normed space

  8. Concept of Algebraic Dual & Reflexive space

  9. Dual Basis & Algebraic Reflexive Space

  10. Dual spaces with examples

  11. Tutorial

  12. Tutorial

10

3

Inner-Product Space & Hilbert Space

  1. Inner Product & Hilbert space

  2. Further properties of Inner product spaces

  3. Projection Theorem & Orthonormal Sets & Sequences

  4. Representation of functionals on a Hilbert Spaces

  5. Hilbert Adjoint Operator

  6. Self Adjoint, Unitary & normal Operators

  7. Tutorial

  8. Annihilator in an IPS

  9. Total Orthonormal Sets & Sequences

15

4

Fundamental Theorems for Normed & Banach Spaces

  1. Partially Ordered Set & Zorn’s Lemma

  2. Hahn Banach Theorem for Real Vector Spaces

  3. Hahn Banach Theorem for Complex V.S. & Normed Spaces

  4. Baire’s Category & Uniform Boundedness Theorems

  5. Open Mapping Theorem

  6. Closed Graph Theorem

  7. Adjoint Operator

  8. Strong & Weak Convergence

  9. Convergence of Sequence of Operators & Functionals

  10. Tutorial

  11. Banach Fixed Point Theorem

5

5

Questions & Worked out answers

  1. Problems on Metric Spaces

  2. Problems on Normed & Banach Spaces

  3. Problems on IPS & Hilbert Spaces

** Assignment Sheet & Cumulative Question Papers

8

Calculus & Linear Algebra


  1. Erwin Kreyszig : - Introdutory Functional Analysis with Applications , John Wiley& Sons, New York

  2. W.Rudin :- Functional Analysis , Tata McGraw-Hill Pub.Co.

  3. I.J.Maddox :- Elements of Functional Analysis, Cambridge university Press

  4. B.Limaye :- Functional Analysis , New age international Ltd,pub.


  1. Yosida, K.- Functional analysis, Springer

  2. Wilansky, A.-Functional Analysis,Blaisdell Pub. Co.,London



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