Functional Analysis

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Contents:
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

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

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

Fundamental Theorems for Normed & Banach Spaces : Partially Ordered Set & Zorn’s Lemma,Hahn Banach Theorem for Real Vector Spaces,Hahn Banach Theorem for Complex V.S. & Normed Spaces,Baire’s 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

Questions & Worked out answers : Problems on Metric Spaces,Problems on Normed & Banach Spaces,Problems on IPS & Hilbert Spaces – Assignment Sheet & Cumulative Question Papers

Course Curriculum

Mod-01 Lec-01 Metric Spaces with Examples Details 53:51
Mod-01 Lec-02 Holder Inequality and Minkowski Inequality Details 55:51
Mod-01 Lec-03 Various Concepts in a Metric Space Details 52:50
Mod-01 Lec-04 Separable Metrics Spaces with Examples Details 50:44
Mod-01 Lec-05 Convergence, Cauchy Sequence, Completeness Details 53:57
Mod-01 Lec-06 Examples of Complete and Incomplete Metric Spaces Details 51:19
Mod-01 Lec-07 Completion of Metric Spaces + Tutorial Details 54:21
Mod-01 Lec-08 Vector Spaces with Examples Details 53:41
Mod-01 Lec-09 Normed Spaces with Examples Details 55:9
Mod-01 Lec-10 Banach Spaces and Schauder Basic Details 56:24
Mod-01 Lec-11 Finite Dimensional Normed Spaces and Subspaces Details 54:14
Mod-01 Lec-12 Finite Dimensional Normed Spaces and Subspaces Details 55:35
Mod-01 Lec-13 Linear Operators-definition and Examples Details 56:22
Mod-01 Lec-14 Bounded Linear Operators in a Normed Space Details 55:38
Mod-01 Lec-15 Bounded Linear Functionals in a Normed Space Details 54:36
Mod-01 Lec-16 Concept of Algebraic Dual and Reflexive Space Details 56:57
Mod-01 Lec-17 Dual Basis & Algebraic Reflexive Space Details 54:31
Mod-01 Lec-18 Dual Spaces with Examples Details 56:4
Mod-01 Lec-19 Tutorial – I Details 55:48
Mod-01 Lec-20 Tutorial – II Details 58:14
Mod-01 Lec-21 Inner Product & Hilbert Space Details 56:21
Mod-01 Lec-22 Further Properties of Inner Product Spaces Details 55:51
Mod-01 Lec-23 Projection Theorem, Orthonormal Sets and Sequences Details 54:27
Mod-01 Lec-24 Representation of Functionals on a Hilbert Spaces Details 51:13
Mod-01 Lec-25 Hilbert Adjoint Operator Details 57:17
Mod-01 Lec-26 Self Adjoint, Unitary & Normal Operators Details 55:24
Mod-01 Lec-27 Tutorial – III Details 56:11
Mod-01 Lec-28 Annihilator in an IPS Details 55:47
Mod-01 Lec-29 Total Orthonormal Sets And Sequences Details 55:17
Mod-01 Lec-30 Partially Ordered Set and Zorns Lemma Details 54:7
Mod-01 Lec-31 Hahn Banach Theorem for Real Vector Spaces Details 55:5
Mod-01 Lec-32 Hahn Banach Theorem for Complex V.S. & Normed Spaces Details 55:7
Mod-01 Lec-33 Baires Category & Uniform Boundedness Theorems Details 58:27
Mod-01 Lec-34 Open Mapping Theorem Details 57:15
Mod-01 Lec-35 Closed Graph Theorem Details 52:49
Mod-01 Lec-36 Adjoint Operator Details 0:57
Mod-01 Lec-37 Strong and Weak Convergence Details 54:46
Mod-01 Lec-38 Convergence of Sequence of Operators and Functionals Details 54:37
Mod-01 Lec-39 LP – Space Details 53:38
Mod-01 Lec-40 LP – Space (Contd.) Details 52:58

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