II MSc

Updated on Date: 26/12/2017

FUNCTIONAL ANALYSIS

IIT Kharagpur Course , Prof. P.D. Srivastava

Dear Students,

Find the useful video lecture for ” Functional Analysis” given by IIT Professor. Match your syllabus with this video Lecture and choose a Topic for your seminar. Your Seminar starts soon…

Lecture Details :

Functional Analysis by Prof. P.D. Srivastava, Department of Mathematics, IIT Kharagpur. For more details on NPTEL visit http://nptel.iitm.ac.in

Course Description :

Contents:

1. Metric Spaces : Metric spaces with examples,Holder inequality & Minkowski inequality,Various concepts in a metric space,Separable metric space with examples,Convergence, Cauchy sequence , Completeness,Examples of Complete & Incomplete metric spaces,Completion of Metric spaces +Tutorial,Vector spaces with examples

2. Normed / Banach Spaces : Normed Spaces with examples,Banach Spaces & Schauder Basis,Finite Dimensional Normed Spaces & Subspaces,Compactness of Metric/Normed spaces,Linear Operators-definition & examples,Bounded linear operators in a Normed Space,Bounded linear Functionals in a Normed space,Concept of Algebraic Dual & Reflexive space,Dual Basis & Algebraic Reflexive Space,Dual spaces with examples,Tutorial

3. Inner-Product Space & Hilbert Space : Inner Product & Hilbert space,Further properties of Inner product spaces,Projection Theorem & Orthonormal Sets & Sequences,Representation of functionals on a Hilbert Spaces,Hilbert Adjoint Operator,Self Adjoint, Unitary & normal Operators,Tutorial,Annihilator in an IPS,Total Orthonormal Sets & Sequences

4. Fundamental Theorems for Normed & Banach Spaces : Partially Ordered Set & Zorns Lemma,Hahn Banach Theorem for Real Vector Spaces,Hahn Banach Theorem for Complex V.S. & Normed Spaces,Baires Category & Uniform Boundedness Theorems,Open Mapping Theorem,Closed Graph Theorem,Adjoint Operator,Strong & Weak Convergence,Convergence of Sequence of Operators & Functionals,Tutorial,Banach Fixed Point Theorem

VIDEO LECTURE WEB SITE:

http://freevideolectures.com/Course/3145/Functional-Analysis/38

Click here to watch the video on this below topic

• .Mod-01 Lec-01 Metric Spaces with Examples
• 2.Mod-01 Lec-02 Holder Inequality and Minkowski Inequality
• 3.Mod-01 Lec-03 Various Concepts in a Metric Space
• 4.Mod-01 Lec-04 Separable Metrics Spaces with Examples
• 5.Mod-01 Lec-05 Convergence, Cauchy Sequence, Completeness
• 6.Mod-01 Lec-06 Examples of Complete and Incomplete Metric Spaces
• 7.Mod-01 Lec-07 Completion of Metric Spaces + Tutorial
• 8.Mod-01 Lec-08 Vector Spaces with Examples
• 9.Mod-01 Lec-09 Normed Spaces with Examples
• 10.Mod-01 Lec-10 Banach Spaces and Schauder Basic
• 11.Mod-01 Lec-11 Finite Dimensional Normed Spaces and Subspaces
• 12.Mod-01 Lec-12 Finite Dimensional Normed Spaces and Subspaces
• 13.Mod-01 Lec-13 Linear Operators-definition and Examples
• 14.Mod-01 Lec-14 Bounded Linear Operators in a Normed Space
• 15.Mod-01 Lec-15 Bounded Linear Functionals in a Normed Space
• 16.Mod-01 Lec-16 Concept of Algebraic Dual and Reflexive Space
• 17.Mod-01 Lec-17 Dual Basis & Algebraic Reflexive Space
• 18.Mod-01 Lec-18 Dual Spaces with Examples
• 19.Mod-01 Lec-19 Tutorial – I
• 20.Mod-01 Lec-20 Tutorial – II
• 21.Mod-01 Lec-21 Inner Product & Hilbert Space
• 22.Mod-01 Lec-22 Further Properties of Inner Product Spaces
• 23.Mod-01 Lec-23 Projection Theorem, Orthonormal Sets and Sequences
• 24.Mod-01 Lec-24 Representation of Functionals on a Hilbert Spaces
• 25.Mod-01 Lec-25 Hilbert Adjoint Operator
• 26.Mod-01 Lec-26 Self Adjoint, Unitary & Normal Operators
• 27.Mod-01 Lec-27 Tutorial – III
• 28.Mod-01 Lec-28 Annihilator in an IPS
• 29.Mod-01 Lec-29 Total Orthonormal Sets And Sequences
• 30.Mod-01 Lec-30 Partially Ordered Set and Zorns Lemma
• 31.Mod-01 Lec-31 Hahn Banach Theorem for Real Vector Spaces
• 32.Mod-01 Lec-32 Hahn Banach Theorem for Complex V.S. & Normed Spaces
• 33.Mod-01 Lec-33 Baires Category & Uniform Boundedness Theorems
• 34.Mod-01 Lec-34 Open Mapping Theorem
• 35.Mod-01 Lec-35 Closed Graph Theorem
• 36.Mod-01 Lec-36 Adjoint Operator
• 37.Mod-01 Lec-37 Strong and Weak Convergence
• 38.Mod-01 Lec-38 Convergence of Sequence of Operators and Functionals
• 39.Mod-01 Lec-39 LP – Space
• 40.Mod-01 Lec-40 LP – Space (Contd.)

Updated on Date:18/12/2017

Dear Students

•  Welcome to next semester which begin Tomorrow (18 Dec 2017). I hope You all have done the exam pretty  good.
• In this Semester I will be handling FUNCTIONAL ANALYSIS for II MSc Students.

• Clik here for Syllabus: 8 – M.Sc Mathematics Syllabi (2014-17)

A Brief History of Functional Analysis

1. Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure.
2. Functional analysis was born in the early years of the twentieth century as part of a larger trend toward abstraction—what some authors have called the “arithmetization” of analysis.
3. This same trend toward“axiomatics” contributed to the foundations of abstract linear algebra, modern geometry, and topology.
4. Functional analysis is now a very broad field, encompassing much of modern analysis. In fact, it would be difficult to give a simple definition of what functional analysis means today.
5. The usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function and the name was first used in Hadamard‘s 1910 book on that subject.
6. The theory of nonlinear functionals was continued by students of Hadamard, in particular Fréchet and Lévy. Hadamard also founded the modern school of linear functional analysis further developed by Riesz and the group of Polish mathematicians around Stefan Banach
7. Rather than discuss its current meaning, we will concentrate on its foundations and settle for an all too brief description of modern trends.

8. One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis

II YEAR – IV SEMESTER

COURSE CODE: 4MMA4C1

CORE COURSE XII – FUNCTIONAL ANALYSIS

 

Unit I : Normed spaces, continuity of linear Maps.

Unit II : Hahn – Banach theorems, Banach limits, Banach spaces.

Unit III: Uniform boundedness Principle – Closed graph and open mapping theorems

 Unit IV: Duals and Transposes, Duals of L^p ([a, b]) and C ([a, b]) (excluding moment sequences)

 Unit V: Inner product spaces, orthonormal sets, projection and Reisz Representation theorems.

Text Book

 1.Functional Analysis by B.V Limaye, Second Edition, New Age International Pvt. Ltd., Publishers.

Chapter II       :           (Section 5, 6, 7, 8)

Chapter III      :           Section 9 (Subsections 9.1, 9.2, & 9.3 only)

                                    & Sections 10

Chapter IV      :           (Sections 13, 14)

                            (excluding Moment Sequences Subsections 14.6 & 14.7)

Chapter VI      :           (Sections 21, 22, and 24.1, 24.2, 24.3 & 24.4)

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FUNCTIONAL ANALYSIS NOTES: (PDF Format)

1. Coursetitle
2. contents
3. Preface
4. chapter1
5. chapter2
6. chapter3
7. chapter4
8. Reference

FUNCTIONAL ANALYSIS books:

1. Kreyszig – Introductory Functional Analysis with Applications