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