# InfoCoBuild

## 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.