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Lecture 12 of 63Week : 03
Lecture 12: The Lp spaces as Banach Spaces
Prof. Akhilesh Prasad · IIT-ISM Dhanbad
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All the lectures
- 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 GeneralizedHahn-Banach Theorem
- Lecture 55: Uniform Boundedness Principle
- Lecture 56: Application of the Uniform BoundednessPrinciple 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
- Live Session 25-02-2025
- Live Session 18-03-2025
- Live Session 11-04-2025