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Course11110561063 lectures

Functional Analysis and its Applications

Prof. Akhilesh Prasad · IIT-ISM Dhanbad

Start with lecture 1 At NPTEL

Lectures

Week : 01

  1. Lecture 1: An Overview and Metric Spaces with Examples Video
  2. Lecture 2: Basic Inequalities and their Applications Video
  3. Lecture 3: Complete Metric Spaces with Examples Video
  4. Lecture 4: Vector Spaces and Subspaces with Examples Video
  5. Lecture 5: Linear Transformation with Examples Video

Week : 02

  1. Lecture 6: Normed Linear Space and Pseudo-Normed Linear Space with Examples Video
  2. Lecture 7: Convergence, Cauchy Sequence in Normed Linear Spaces and Banach Spaces with Examples Video
  3. Lecture 8: Further examples of Banach Spaces Video
  4. Lecture 9: The Linear Space C[a, b] as Normed Linear Space but not Banachspace Video
  5. Lecture 10: The Linear Spaces ?p and ?? as Banach Spaces Video

Week : 03

  1. Lecture 11: The Quotient Space as Banach Space Video
  2. Lecture 12: The Lp spaces as Banach Spaces Video
  3. Lecture 13: Bounded Linear Transformations and Equivalent Norms with Examples Video
  4. Lecture 14: Norm of Bounded (Continuous) LinearTransformations and Some Related Theorems Video
  5. Lecture 15: Continuity of Functions in Normed Linear Spaces and some related Theorems Video

Week : 04

  1. Lecture 16: The set of all Bounded Linear Transformations of Normed Linear Space N into N? as a... Video
  2. Lecture 17: Further Theorems on Bounded Linear Transformations Video
  3. Lecture 18: Linear Functionals, Dual Vector Space and Dual Basis Video
  4. Lecture 19: Continuous (Bounded) Linear Functionals on Normed linear spaces Video
  5. Lecture 20: Consequences of Equivalent Norms with Examples and on a Finite-Dimensional space,.... Video

Week : 05

  1. Lecture 21: Further Theorems on Finite-Dimensional Normed Linear Spaces Video
  2. Lecture 22: Dual Space of ?p^n, ?1^n and ??^n Video
  3. Lecture 23: Dual Space of ?p is ?q Video
  4. Lecture 24: Dual Space of ?1 is ?? Video
  5. Lecture 25: Dual Space of ????0 is ?1 Video

Week : 06

  1. Lecture 26: Dual Space of c is ?1 Video
  2. Lecture 27: Inner Product Space with Examples Video
  3. Lecture 28: Further Example of Inner Product Space and Schwartz’s Inequality Video
  4. Lecture 29: Continuity of Inner Product, Parallelogram Law and Polarization Identity Video
  5. Lecture 30: Hilbert Space with Examples Video

Week : 07

  1. Lecture 31: Application of Polarization Identity Video
  2. Lecture 32: Applications of the Parallelogram Law Video
  3. Lecture 33: Weak Convergence and Strong Convergence Video
  4. Lecture 34: Orthogonality of Vectors Video
  5. Lecture 35: Orthogonal Complement of an Orthogonal Complement Video

Week : 08

  1. Lecture 36: The Orthogonal Decomposition (Projection) Theorem Video
  2. Lecture 37: Orthonormal Sets in Hilbert Spaces and Bessel’s Inequality Video
  3. Lecture 38: Complete Orthonormal Sets Video
  4. Lecture 39: Further Theorems on Complete Orthonormal Sets and Riesz-Fischer Theorem Video
  5. Lecture 40: Gram-Schmidt Orthogonalization Process Video

Week : 09

  1. Lecture 41: The Conjugate Space H* and The Riesz Representation Theorem Video
  2. Lecture 42: Further Theorems on Conjugate Space H* Video
  3. Lecture 43: The Conjugate of an Operator Video
  4. Lecture 44: The Adjoint of an Operator on a Hilbert Space Video
  5. Lecture 45: Properties of Adjoint Operators on Hilbert Spaces Video

Week : 10

  1. Lecture 46: Self Adjoint Operators Video
  2. Lecture 47: Partially Ordered Set and Positive Operators Video
  3. Lecture 48: Normal Operators Video
  4. Lecture 49: Further Theorems of Normal Operators Video
  5. Lecture 50: Unitary Operators Video

Week : 11

  1. Lecture 51: Partial Order Relations, Total Order, Maximaland Minimal Elements, Zorn’s Lemma Video
  2. Lecture 52: Hahn-Banach Theorem (Part-I) Video
  3. Lecture 53: Hahn-Banach Theorem (Part-II) and Proofof the Main Theorem Video
  4. Lecture 54: Sublinear Functional and GeneralizedHahn-Banach Theorem Video
  5. Lecture 55: Uniform Boundedness Principle Video

Week : 12

  1. Lecture 56: Application of the Uniform BoundednessPrinciple and Uniform Boundedness Principle..... Video
  2. Lecture 57: Applications of Uniform Boundedness Principle for Continuous Linear Functionals Video
  3. Lecture 58: Open Mapping Theorem Video
  4. Lecture 59: Applications of Open Mapping Theorem Video
  5. Lecture 60: The Closed Graph Theorem Video

Live Session

  1. Live Session 25-02-2025 Video
  2. Live Session 18-03-2025 Video
  3. Live Session 11-04-2025 Video

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IIT-ISM Dhanbad
Department
Mathematics
Teacher
Prof. Akhilesh Prasad
Level
Undergraduate
Language
English
Subjects
Mathematics
Source
NPTEL
Licence
CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
Attribution
NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge. Coordinated by IIT Kharagpur.
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