I.The Basic Information of the Course(Time New Rome, 12 points, bold font)
Course Number:202012420004
The EnglishName of theCourse：Functional Analysis
The ChineseName of theCourse：泛函分析
In-classHours andAllocation
total class hours:48, classroom teaching:48classhours,Credit(s):
Semester:Spring semester
AppliedDiscipline(ProfessionalDegreeCategory)：
CourseObjectOriented：Sample:AcademicDoctor
EvaluationMode:Closed-bookExamination.
TeachingMethod:Seminar-styleTeaching
CourseOpening Department:Science and Engineering Discipline
Prerequisite CourseCalculus, linear algebra
III. The Objectives and Requirements of the Course(Time New Rome 12 points, total words: about 200 words)
This course is a required course for mathematics and applied mathematics major. The purpose of the course is to cultivate students' theoretical thinking ability, master the basic concepts and important theorems of functional analysis 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 theoretical foundation for further learning and teaching of mathematics.
By studying this course, students should meet 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) Be able to consult relevant literature and understand relevant conclusions after class.
IV. The Content of the Course(Time New Rome, 12 points, bold font, 1000-2000 words)
Description:Other course should reflect the content and cases of “Courses for Ideological and Political education”, besides the Course for Ideological and Political Education.
First, learn the set theory to understand the concept of set cardinality. Learn the basic theory of real number (including closed interval cover theorem, Cauchy convergence principle, finite cover theorem, etc.) and learn the basic theory of leberg integral (including control convergence theorem, Riess theorem, lemma of method, etc.). To learn metric spaces (including the definition and examples of metric spaces, we need to master some known metric spaces, such as continuous function spaces C [a, b], metric spaces composed of all bounded real number columns, discrete metric spaces and finite dimensional Euclidean spaces. Learn the basic concepts of metric spaces, such as Separability, and know some separable metric spaces, such as the continuous function space C [a, b] finite dimensional Euclidean space, etc. Know some indivisible metric spaces. For example: all the bounded real number columns constitute the metric space. Learning the basic concept of Cauchy Series in metric space, we know that Cauchy Series in metric space is not necessarily a convergent point series. Then we learn the length concept of complete metric space. Some examples of complete metric spaces are known, such as the metric spaces composed of continuous function spaces C [a, b] and all bounded real number columns. Some examples of incomplete metric spaces are known; for example, metric spaces composed of all rational numbers. Learn the most basic theorem in metric space - contraction mapping principle, and can use contraction mapping principle to deal with some specific problems, for example, can use contraction mapping principle to prove the existence and uniqueness theorem of solutions in ordinary differential equations. Learn the concept of set compactness in metric space, know some specific concepts of compact set in metric space, and know some specific discriminant methods of compact set in metric space, such as continuous function space C [a, b] and finite dimensional Euclidean space. Learning linear normed space and linear operators (learning the definition of linear normed space, we know that linear normed space is a special metric space. It is known that the complete linear normed space is Banach space and the finite dimensional linear normed space is Banach space. Learning the concept of equivalence of norms in linear normed spaces, we know that all norms in finite dimensional linear normed spaces are equivalent. Learning the concept of bounded linear operator on linear normed space, we know that a linear operator on linear normed space is a bounded linear operator if and only if it is a continuous operator. We know the concept of bounded linear function and know that a linear function is bounded if and only if its zero space is a closed subspace. Learn the concept of conjugate space and know some conjugate spaces of Banach space. For example, know that the conjugate space is a space, know that the conjugate space of continuous function space C [a, b] is a bounded variation function space. Learning Hahn Banach theorem, we know that for an infinite dimensional Banach space, there are enough bounded linear functional analysis on it. Learn the resonance theorem, the concept of reflexive Banach space, the concept of weakly compact set on Banach space and the concept of sequence weakly convergent. It is known that the bounded set on reflexive Banach space is weakly compact. Learn inner product space (including the definition and basic properties of inner product space, orthogonal and orthogonal basis, and the concept of orthogonalization). It is known that the inner product space is a special linear normed space and the complete inner product space is Hilbert space. We know that Hilbert space is a special Banach space. The isomorphism of Hilbert space and the application of Hilbert space theory in the optimal approximation problem. We know that the conjugate space of real Hilbert space is itself. We know that real Hilbert space is reflexive space.
Ideological and political course: briefly describe the famous Chinese mathematics experts and deeds in the development history of this course. Combined with the actual classroom teaching, the teachers choose to teach according to the materials. By analyzing the patriotic feelings and scientific spirit of the experts, the patriotic education and cultural self-confidence of the students are promoted imperceptibly.
V. Reference Books, Reference Literatures, and Reference Materials
A. Text Books, Monographs and References
1. Yao Zeqing, Su Xiaobing. Applied functional analysis. Science Press, 2002.
2. Zheng Weixing. Real variable function and functional analysis. Higher education press, 1990.
3. Li Yongjin. Functional analysis. Science Press, 2011
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