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.