# Mathematics - Functional Analysis

Indian Institute of Technology Kharagpur

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.

Course 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.

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