Syllabus
Course Name: Functional Analysis, Sobolev Spaces and Partial Differential Equations(I)
Instructor: Tianyu Zhang
Prerequest:
I. Mathematic Analysis(or above)
II. Linear Algebra(or equivalently, Advanced Linear Algebra)
Course Material:
book: Functional Analysis, Sobolev Spaces and Partial Differential Equations, Haim Brezis
web page: https://tymathdb.com/home/tutorial/functional-analysisi-course-video/
contact: bidenbaka@gmail.com
Schedule:
time: August 24th, 2022 – September 28th, 2022, Wednesday, Thursday, Friday, Saturday, Sunday
Arrangements:
Lecture 1: (3.3)Weak Topology, Convex Set, and Linear Operators, 8.2
Lecture 2: (3.4)The Weak* Topology, 8.25
Lecture 3: (3.5)Reflexive Spaces, 8.26
Lecture 4: (3.6)Separable Spaces, 8.27
Lecture 5: (3.7)Uniformly Convex Spaces, 8.28
Recitation I: Chapter III, 8.31
Review Session I: Chapter III, 8.31
Lecture 6: (4.2)Definition and Elementary Properties of L^p Spaces. 9.1
Lecture 7: (4.3)Reflexivity, Separability, Dual of L^p, 9.2
Lecture 8: (4.4)Convolution and Regularization, 9.3
Lecture 9: (4.5)Criterion for Strong Compactness in L^p, 9.4
Recitation II: Chapter IV, 9.7
Review Session II: Chapter IV, 9.7
Lecture 10: (5.1)Definitions and Elementary Properties. Projection onto a Convex Set, 9.8
Lecture 11: (5.2)The Dual Space of a Hilbert Space, 9.9
Lecture 12: (5.3)The Theorems of Stampacchia and Lax-Milgram, 9.10
Lecture 13: (5.4)Hilbert Sums, Orthonormal Bases, 9.11
Recitation II: Chapter V, 9.14
Review Session III: Chapter V, 9.14
Lecture 14: (6.1)Definitions, Elementary Properties, Adjoint, 9.15
Lecture 15: (6.2)The Riesz-Fredholm Theory, 9.16
Lecture 16: (6.3)The Spectrum of a Compact Operator, 9.17
Lecture 17: (6.4)Spectral Decomposition of Self-Adjoint CompactOperators, 9.18
Recitation III: Chapter VI, 9.21
Review Session IV: Chapter VI, 9.21
Lecture 18: (7.2)Solution of the Evolution Problem, 9.22
Lecture 19: (7.3)Regularity, 9.23
Lecture 20: (7.4)The Self-Adjoint Case, 9.24
Recitation IV: Chapter VII, 9.25
Review Session V: Chapter VII, 9.25
Review Session VI: 9.28
Acknowledgement:
problems or errors may occur, please point it out, thanks.