Lecture 1 - An Overview and Metric Spaces with Examples
Lecture 2 - Basic Inequalities and their Applications
Lecture 3 - Complete Metric Spaces with Examples
Lecture 4 - Vector Spaces and Subspaces with Examples
Lecture 5 - Linear Transformation with Examples
Lecture 6 - Normed Linear Space and Pseudo-Normed Linear Space with Examples
Lecture 7 - Convergence, Cauchy Sequence in Normed Linear Spaces and Banach Spaces with Examples
Lecture 8 - Further examples of Banach Spaces
Lecture 9 - The Linear Space C[a, b] as Normed Linear Space but not Banachspace
Lecture 10 - The Linear Spaces ?p and ?? as Banach Spaces
Lecture 11 - The Quotient Space as Banach Space
Lecture 12 - The Lp spaces as Banach Spaces
Lecture 13 - Bounded Linear Transformations and Equivalent Norms with Examples
Lecture 14 - Norm of Bounded (Continuous) LinearTransformations and Some Related Theorems
Lecture 15 - Continuity of Functions in Normed Linear Spaces and some related Theorems
Lecture 16 - The set of all Bounded Linear Transformations of Normed Linear Space N into N? as a
Lecture 17 - Further Theorems on Bounded Linear Transformations
Lecture 18 - Linear Functionals, Dual Vector Space and Dual Basis
Lecture 19 - Continuous (Bounded) Linear Functionals on Normed linear spaces
Lecture 20 - Consequences of Equivalent Norms with Examples and on a Finite-Dimensional space
Lecture 21 - Further Theorems on Finite-Dimensional Normed Linear Spaces
Lecture 22 - Dual Space of ?p^n, ?1^n and ??^n
Lecture 23 - Dual Space of ?p is ?q
Lecture 24 - Dual Space of ?1 is ??
Lecture 25 - Dual Space of ????0 is ?1
Lecture 26 - Dual Space of c is ?1
Lecture 27 - Inner Product Space with Examples
Lecture 28 - Further Example of Inner Product Space and Schwartz’s Inequality
Lecture 29 - Continuity of Inner Product, Parallelogram Law and Polarization Identity
Lecture 30 - Hilbert Space with Examples
Lecture 31 - Application of Polarization Identity
Lecture 32 - Applications of the Parallelogram Law
Lecture 33 - Weak Convergence and Strong Convergence
Lecture 34 - Orthogonality of Vectors
Lecture 35 - Orthogonal Complement of an Orthogonal Complement
Lecture 36 - The Orthogonal Decomposition (Projection) Theorem
Lecture 37 - Orthonormal Sets in Hilbert Spaces and Bessel’s Inequality
Lecture 38 - Complete Orthonormal Sets
Lecture 39 - Further Theorems on Complete Orthonormal Sets and Riesz-Fischer Theorem
Lecture 40 - Gram-Schmidt Orthogonalization Process
Lecture 41 - The Conjugate Space H* and The Riesz Representation Theorem
Lecture 42 - Further Theorems on Conjugate Space H*
Lecture 43 - The Conjugate of an Operator
Lecture 44 - The Adjoint of an Operator on a Hilbert Space
Lecture 45 - Properties of Adjoint Operators on Hilbert Spaces
Lecture 46 - Self Adjoint Operators
Lecture 47 - Partially Ordered Set and Positive Operators
Lecture 48 - Normal Operators
Lecture 49 - Further Theorems of Normal Operators
Lecture 50 - Unitary Operators
Lecture 51 - Partial Order Relations, Total Order, Maximaland Minimal Elements, Zorn’s Lemma
Lecture 52 - Hahn-Banach Theorem - Part I
Lecture 53 - Hahn-Banach Theorem - Part II and Proofof the Main Theorem
Lecture 54 - Sublinear Functional and Generalized Hahn-Banach Theorem
Lecture 55 - Uniform Boundedness Principle
Lecture 56 - Application of the Uniform Boundedness Principle and Uniform Boundedness Principle
Lecture 57 - Applications of Uniform Boundedness Principle for Continuous Linear Functionals
Lecture 58 - Open Mapping Theorem
Lecture 59 - Applications of Open Mapping Theorem
Lecture 60 - The Closed Graph Theorem
