Explore a comprehensive course on Functional Analysis covering advanced mathematical concepts and theorems. Delve into topics such as Hilbert adjoint operators, self-adjoint operators, unitary and normal operators, and inner product spaces. Learn about the Hahn-Banach theorem for real and complex vector spaces, Baire's category theorem, and the uniform boundedness theorem. Study metric spaces, normed spaces, and Banach spaces, along with their properties and examples. Examine concepts like strong and weak convergence, LP-spaces, and bounded linear operators. Investigate the projection theorem, orthonormal sets and sequences, and dual spaces. Engage with tutorials and examples throughout the course to reinforce understanding of these complex mathematical topics.
Overview
Syllabus
Hilbert Adjoint Operator.
Self adjoint, Unitary and Normal operators.
Tutorial - III.
Annihilator in an IPS.
Total Orthonormal Sets and Sequences.
Partially Ordered Set and Zorns Lemma.
Hahn Banach Theorem for Real Vector Spaces.
Hahn Banach Theorem for Complex V.S. & Normed Spaces.
Baires Category & Uniform Boundedness Theorems.
Open Mapping Theorem.
Closed Graph Theorem.
Adjoint operator.
Strong and Weak Convergence.
S30 2074.
LP - Space.
LP - space (contd.).
Completion of Metric Spaces + Tutorial.
Examples of Complete and Incomplete Metric Spaces.
Holder inequality and Minkowski Inequality.
Convergence, Cauchy Sequence, Completeness.
Metric Spaces with Examples.
Separable Metrics Spaces with Examples.
Banach Spaces and Schauder Basic.
Normed Spaces with Examples.
Vector Spaces with Examples.
Various Concepts in a Metric Space.
Bounded Linear Operators in a Normed Space.
Linear Operators - Definitions and Examples.
Compactness of Metric/Normed Spaces.
Finite Dimensional Normed Spaces and Subspaces.
Concept of Algebraic Dual and Reflexive Space.
Bounded Linear Functionals in a Normed Space.
Tutorial - II.
Representation of Functionals on a Hilbert Spaces.
Tutorial - I.
Projection theorem, Orthonormal Sets and Sequences.
Dual Spaces with Examples.
Dual Basis & Algebraic Reflexive Space.
Further properties of inner Product Spaces.
Inner product & Hilbert space.
Taught by
Ch 30 NIOS: Gyanamrit
