This course on Functional Analysis aims to teach students the following learning outcomes and goals: understanding Hilbert Adjoint Operator, Self adjoint, Unitary, and Normal operators, exploring concepts like Annihilator in an IPS, Total Orthonormal Sets and Sequences, and learning about various theorems such as Hahn Banach Theorem, Baires Category, Uniform Boundedness Theorem, Open Mapping Theorem, and Closed Graph Theorem.
Students will also acquire skills in analyzing LP-Spaces, Completion of Metric Spaces, Holder inequality, Minkowski Inequality, Convergence, Cauchy Sequence, Completeness, and understanding various concepts in Metric Spaces, Separable Metric Spaces, Banach Spaces, Schauder Basic, Vector Spaces, Bounded Linear Operators, Compactness of Metric/Normed Spaces, Finite Dimensional Normed Spaces, Algebraic Dual and Reflexive Space, and Bounded Linear Functionals.
The teaching method includes tutorials, examples, and theoretical explanations. This course is intended for students and professionals interested in advanced mathematics, particularly in the field of Functional Analysis.
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
