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Functional Analysis

Functional Analysis. Instructor: Prof. P.D. Srivastava, Department of Mathematics, IIT Kharagpur. This course provides an introduction to functional analysis. The aim of the course is to familiarize the students with basic concepts, principles and methods of functional analysis and its applications. Topics include: Metric spaces with example, Complete metric spaces, Separable metric space, Compact sets, Normed and Banach spaces, Convergence, Bounded linear functionals and operators, Dual spaces, Reflexive 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, Open mapping theorem and Closed graph theorem. (from nptel.ac.in)

Metric Spaces with Examples


Lecture 01 - Metric Spaces with Examples
Lecture 02 - Holder Inequality and Minkowski Inequality
Lecture 03 - Various Concepts in a Metric Space
Lecture 04 - Separable Metric Spaces with Examples
Lecture 05 - Convergence, Cauchy Sequence, Completeness
Lecture 06 - Examples of Complete and Incomplete Metric Spaces
Lecture 07 - Completion of Metric Spaces and Tutorial
Lecture 08 - Vector Spaces with Examples
Lecture 09 - Normed Spaces with Examples
Lecture 10 - Banach Spaces and Schauder Basis
Lecture 11 - Finite Dimensional Normed Spaces and Subspaces
Lecture 12 - Compactness of Metric/Normed Spaces
Lecture 13 - Linear Operators: Definition and Examples
Lecture 14 - Bounded Linear Operators in a Normed Space
Lecture 15 - Bounded Linear Functionals in a Normed Space
Lecture 16 - Concept of Algebraic Dual and Reflexive Space
Lecture 17 - Dual Basis and Algebraic Reflexive Space
Lecture 18 - Dual Spaces with Examples
Lecture 19 - Tutorial I
Lecture 20 - Tutorial II
Lecture 21 - Inner Product and Hilbert Space
Lecture 22 - Further Properties of Inner Product Spaces
Lecture 23 - Projection Theorem, Orthonormal Sets and Sequences
Lecture 24 - Representation of Functionals on a Hilbert Space
Lecture 25 - Hilbert Adjoint Operator
Lecture 26 - Self Adjoint, Unitary and Normal Operators
Lecture 27 - Tutorial III
Lecture 28 - Annihilator in an Inner Product Space
Lecture 29 - Total Orthonormal Sets and Sequences
Lecture 30 - Partially Ordered Set and Zorn's Lemma
Lecture 31 - Hahn Banach Theorem for Real Vector Spaces
Lecture 32 - Hahn Banach Theorem for Complex Vector Spaces and Normed Spaces
Lecture 33 - Baire's Category and Uniform Boundedness Theorems
Lecture 34 - Open Mapping Theorem
Lecture 35 - Closed Graph Theorem
Lecture 36 - Adjoint Operator
Lecture 37 - Strong and Weak Convergence
Lecture 38 - Convergence of Sequence of Operators and Functionals
Lecture 39 - Lp-Space
Lecture 40 - Lp-Space (cont.)

References
Functional Analysis
Instructor: Prof. P.D. Srivastava, Department of Mathematics, IIT Kharagpur. This course provides an introduction to functional analysis.