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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)

Lecture 02 - Holder Inequality and Minkowski Inequality

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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.)