I.The Basic Information of the Course
Course Number:202012420007
The EnglishName of theCourse：Modern Analysis
The ChineseName of theCourse：现代分析学
In-classHours andAllocationTotal class hours:48, classroom teaching:40classhours,classroom discussion:8classhours.
Credit(s):3
Semester:1
AppliedDiscipline(ProfessionalDegreeCategory)：Mathematics
CourseObjectOriented：AcademicDoctorand Academic Master
EvaluationMode:Closed-bookExamination.
TeachingMethod:BlendedTeaching
CourseOpening Department:School of Mathematical Science
II.Prerequisite Course
Mathematical analysis, Advanced Mathematics and Function of Real Variable
III. The Objectives and Requirements of the Course
This course is a professional elective course of mathematics and applied mathematics. The purpose of the course is to cultivate the ability of induction and abstract thinking of postgraduates from special to general, master the basic concepts and important theorems of modern analytical science and the basic ideas, theories and methods of proving these theorems, To cultivate students' understanding of infinite dimensional space, have strong abstract thinking ability and logical reasoning ability, so as to form a strict and accurate mathematical literacy, and lay a foundation for further learning of non-linear functional analysis, operator semigroup theory and other follow-up courses.
By learning this course, the students should reach the following basic requirements:
(1) Read through the basic knowledge in the book, and pay attention to the words;
(2) Pay attention to the ability of abstract thinking and logical reasoning, which requires some theoretical proof;
(3) After class, they can consult the relevant literature and understand the relevant conclusions.
IV. The Content of the Course
Chapter 1Metricspace and topological space(8classhours)
Understand the basic concepts of distance space, including the determinant condition inmetricspace and thestructureofcompletemetricspace. Master the principle and application of compressionmapping, understand the basic concept of topological space, compactness in topological space and compactness inmetricspace. Can apply these theories to solve some problems in real life.
Chapter2Normed linear space（10 classhours）
Understand the basic concept of normed space, grasp the structure and some analytical properties of space, understand the properties of general normed space, understand the difference between the structure of finite dimension normed space and infinite dimension normed space, accurately analyze the internal properties, and solve some practical application problems.
Ideological and political courses: introduced by two examples:
(1)Open a map of the world in one place. There is a point on it. Its position on the map overlaps its actual position.
(2)Imagine a spherical hedgehog with fur on its surface. Now, if you comb all the hairs, that is, map the points on the sphere to another point, then there is one hair that cannot be turned down.
The above example shows that there is always a permanent part in the phenomenon of dynamic change. That is, no point. When explaining the principle of Banach compression image, students will not only continuously experience the perfection of the construction proof process, but also think about the eternal truth. For example,
(1)If life is a functional relation (y = f (x)), then most people will play the role of x and y. they are fickle and have countless changeable characters. How many people will do the eternal x, that is, x= f (x);
(2)When we talk about fixed points, we can use the verse of Zhi-Zhang He, a poet of Tang Dynasty: "I left home young and not till old do i come back, my accent is unchanged,my hair no
longer black.", Shen Yang of Ming Dynasty "Green hills and wide rivers remain the same, how many times has the setting sun spread its flame!", Hu Cui of Tang Dynasty "Now her face is nowhere to be seen at all.Only the peach blossoms still smile in the spring breeze" .
Chapter3Bounded linear operator（12 classhours）
Understand the definition of bounded linear operator and bounded linear function, master Banach Steinhaus theorem and some applications, open mapping theorem and closed image theorem, Hahn Banach theorem and Its Applications. Understanding the general forms of bounded linear functional on some normed Spaces, master reflexivity and weak convergence properties of normed spaces, and the definition and property of compact operator.
Deological and Political Course:“Political Education”introduced into Resonance Theorem: the uniform boundedness of operator’s family W can be obtained by the pointwise boundedness of operator’s family W. When explaining the point of "consistency", we should not only tell students the necessary conditions for finding the uniformly bounded operator’s family W. At the same time, it can also guide students to understand the truth. Life is like a continuous point, which is the only way to go. It is the need for growth. Failure is not discouraged, willing to be plain and not indulgent, and success is not publicized, which is called broad-minded. To learn to look at problems from the perspective of sports, success and failure is just a turning point in our life. In order to understand the essence of things, we must go beyond the narrow scope and get rid of the subjective prejudice.
“Political Education”introduced into Banach Steinhaus theorem: the boundedness of linear functional on subspace can be extended to the whole space. When explaining the knowledge of "extension property", we should not only teach students to master the extension conditions of bounded linear function, but also let students understand it. Things are developing and changing. A single spark can start a prairie fire. Every success or failure in life is developed from little to little. For example, the novel coronavirus epidemic occurred in China in 2020, which first occurred in Wuhan. The Central Committee and the State Council made decisive decisions and arrangements, launched the people's war epidemic disease, controlled the epidemic and infected persons, and did not spread the epidemic to the whole country. But the epidemic has developed to Europe and America, especially Italy, which shows the superiority of our national system.
When learning the relation between the inverse operator theorem and the inverse function theorem, teachers can start from the details and emphasize the rigorous and realistic scientific attitude.Inverse operator theorem and inverse function theorem are very similar in form. One is expressed by function, the other is expressed by operator, which is easy to be confused at the beginning. As a teacher, we should emphasize their differences in form and essence, and can't forgive the students' mistakes in details. Let the students establish a "serious" and "rigorous" learning habit at the very beginning, and cultivate the students to have a rigorous and realistic scientific attitude.
Chapter4Hilbert space（12 classhours）
Understanding the basic concept of inner product space, mastering orthogonality and the structure of orthogonal system, and Riesz representation theorem and the structure and analytical properties of conjugate space of Hilbert space. And can apply these theories to solve some optimal control problems.
Chapter5Operator algebras（2 classhours）
Understanding Banach algebra, multiplicative linear function, maximal ideal space, and the definition and property of spectral decomposition theorem of normal operators.
Chapter6Operator semigroup（2 classhours）
Understandthe vector valued functions, concepts of Bochner integral and Pettis integral, master the concept of operator semigroup and the structure of infinitesimal generator.
Chapter7Nonlinear functional analysis（2 classhours）
Understand themeaningof differentialof nonlinear functions, master the implicit function theorem, the existence condition of function’s extremum, and master the fixed point principle and its application.
V. Reference Books, Reference Literatures, and Reference Materials
A. Text Books, Monographs and References
1. K.C.Chang, Q.Y.Lin. Lecture Notes on Functional Analysis (Volume 1)(Third Edition). Peking University Press，2015.
2.J.Sun,W.Y.Wang, J.W.He.Functional Analysis.Higher Education Press, 1998.
3.X.P.Xue, G.J.Zhang, L.M.Sun, L.Z.Wu. Applied Functional Analysis(Third Edition). Harbin Institute of Technology Press,2012.
4.JohnB. Conway.A course in functional analysis(Second Edition).Springer-Verlag,2019.
B. Learning Resources (Time New Rome 12 points)
1.http://jiangsu.icourses.cn/jpk/getCourseDetail.action?courseId=7021
2.http://www.1ketang.com/course/1432.html
Outline Writer (Signature):Ge Bin
Leader in charge of teaching at the College (Signature)：
Date：2020/6/22
中文
