Before the founding of the People's Republic of China, functional analysis was only a scientific research topic of individual mathematicians in China. As a secondary branch of mathematics, it developed in a planned way in the mid-1950s, and the Institute of Mathematics of Chinese Academy of Sciences is one of the development centers. At that time, the publication of Fu Lun by Tian Fangzeng and Guan He and Generalized Function Theory by Feng Kang marked the beginning of planned and systematic academic research activities on functional analysis in Institute of Mathematics. Fan Huan theory has four chapters and two appendices. Tian Fangzeng continued his research on harmonic analysis of groups while studying in France, and wrote Chapter 4: Group Algebra, Appendix 2: Measure on Locally Compact Spaces-Hal Measure. According to his academic point of view, he discussed some concepts of N. bourbaki. Tian Fangzeng, the earliest pioneer of functional analysis in China, returned from the all-Soviet academic conference on functional analysis with Professor Zeng of Nanjing University, and published the article 1956 in advances in mathematics, which systematically reviewed the academic achievements of the Soviet Union in the theory and application of functional analysis at that time and played an academic role in the early development of functional analysis in China. With the cooperation of Tian Fangzeng, Guan and Feng Kang, the Institute of Mathematics of China Academy of Sciences recruited the first batch of functional analysis research interns in 1956, and then received a large number of advanced students from universities. In a series of academic discussion classes, such as topological vector space, normed ring, measurement and integration, linear operator theory and generalized function theory, they systematically taught young people the academic achievements, latest academic progress and problems of the international functional analysis community at that time (mainly the academic circles of the Soviet Union, France and Eastern Europe). Tian Fangzeng is also concerned about the understanding of mathematics. He once translated an article by A. Mostowski on the current situation of basic research in mathematics into Chinese and introduced it to the mathematics community in China.

Almost from the beginning, the functional analysis discipline of Institute of Mathematics of China Academy of Sciences developed on the basis of paying equal attention to basic theory and application. Early research on numerical methods. According to the spirit of scientific planning, from 65438 to 0958, the functional analysis discipline of the Institute of Mathematics emphasized that the development should focus on the relationship with equations, physics, advanced science and technology and national economic construction. Therefore, Tian Fangzeng and Guan often cooperate with Wu Xinmou and Zhang Zongsui, so that the development of functional analysis in Institute of Mathematics always pays attention to the connection with differential equations and modern mathematical physics. They contacted physicists in some units in Beijing and organized academic discussion classes on mathematical problems in quantum field theory, particle migration theory and electromagnetic wave theory. In the early 1960s, Tian Fangzeng set up a specialized course of "Mathematical Problems in Particle Migration Theory" in the Department of Mathematics of the Chinese University of Science and Technology. From then on, he focused on the research of the mathematical basic theory of "particle migration theory" until the middle and late 1970s. During this period, his academic papers have made great contributions to the development of mathematics research in this field in China. Together with Guan, Tian Fangzeng successfully opened up an important field of applied functional analysis in China-the mathematical basis and problem research of particle migration theory.

Starting from the early 1970s, Tian Fangzeng chose nonlinear functional analysis to explore the academic direction in China after investigating the international academic trends at that time, especially those in Western Europe, the Soviet Union and the United States. At the group meeting of the 3rd National Mathematics Congress in Chengdu from 65438 to 0978, Tian Fangzeng gave a report entitled "A Brief Introduction to Recent Situation of Nonlinear Functional Analysis Abroad", expounding and analyzing the emergence and development of nonlinear functional analysis, the main research directions in the world at present, its position and role in nonlinear analysis, and its application in boundary value problems and numerical analysis of partial differential equations. Subsequently, he gave an academic report on "bifurcation theory analysis" including "stability theory" at the second national academic conference on functional analysis in Jinan from 65438 to 0979. At this Jinan meeting, chinese mathematical society organized the participants to negotiate and set up a "National Leading Group for Functional Analysis" composed of Guan, Tian Fangzeng, Jiang Zejian and four people, which consists of three academic groups: linear operator theory, spatial theory, applied functional analysis and nonlinear functional analysis. Tian Fangzeng was in charge of the academic group of nonlinear functional analysis until 1990. During this period, Tian Fangzeng opened the course of nonlinear functional analysis in the Graduate School of Chinese Academy of Sciences, directed the graduate students in the direction of nonlinear functional analysis in the research room, organized and led six national academic conferences on nonlinear functional analysis, and wrote special reports such as Bifurcation Theory, Types and Properties of Nonlinear Operators, and Several Aspects of Fixed Point Theory. It has played a positive role in introducing and guiding the further development of nonlinear functional analysis in China.

1In July, 1985, Tian Fangzeng, one of the pioneers of China's famous functional analysis discipline, was invited to attend the "Regional Analysis Conference of Southeast Asian Mathematical Federation" held in Hongkong, and delivered an academic report entitled "Some Progress of Beijing Nonlinear Functional Analysis" on behalf of China.

fundamental research

During World War II, the appearance of atomic weapons stimulated the vigorous development of neutron physics and nuclear reactor physics. An integral-differential neutron transport equation that describes the law of neutron movement in nuclear matter has become a frontier national defense research topic. It is a special linearization form of Boltzmann equation which describes the theory of molecular distribution dynamics. In China, since 1960s, this kind of equation has changed from quantitative research to basic mathematical qualitative research, and Tian Fangzeng is one of the pioneers who initiated this kind of qualitative research. 1960, the "125 Task" group, which was set up by the Institute of Mathematics of China Academy of Sciences and No.2 Machinery Department 40 1, was the first scientific research group in China to qualitatively study the basic theory of particle (neutron) transport equation (Tian Fangzeng was one of the group leaders). It is not difficult to start from scratch. In order to quickly master and learn the mathematical ideas, theories and methods of studying neutron transport equations in the United States, Europe and the Soviet Union, Tian Fangzeng systematically explained and analyzed international academic achievements and existing problems to young people in the discussion class, and organized and guided them to tackle key problems. Tian Fangzeng offered a special course on "Mathematical Problems in Particle Migration Theory" for the 59th senior three students of the Department of Mathematics at the University of Science and Technology of China 1962- 1964, including radiation migration and neutron migration. This is the first time that this kind of specialized course has been offered in colleges and universities in China. Tian Fangzeng wrote a lecture with more than100000 words for this purpose, and instructed students to take this as the direction in their graduation thesis. The Invariant Embedding Principle and Migration Problems, the Properties of Problem Spectrum of Neutron Migration Equation in Sphere Geometry and the Asymptotic Properties of Solutions of Homogeneous Initial Value Problems published by him in 1963- 1964 are the first two academic papers on qualitative mathematical research of particle migration theory in China. The first article discusses how to develop the invariant embedding principle in astrophysics to solve the special transfer problem in mathematics. The latter paper unifies the linear operator semigroup theory method of finite offset medium and the characteristic line method of infinite offset medium, which just appeared in the United States and Europe, into the demonstration of spherical offset medium. This academic paper has a great influence on the research direction of qualitative theory of neutron transport equation in early China.

The structure of the migration equation is complex, so it is difficult to solve it strictly in general. In addition, due to the need of numerical calculation, various approximate solutions are important means to discuss the problem from scratch in theory and application. However, for a long time, most effective approximate solutions have not established reasonable mathematical arguments. In 1970s, Tian Fangzeng successively published academic papers "Time Discrete Solution of Non-homogeneous Offset Equation" and "Weak Solution and Finite Element Method of Unsteady Offset Equation", which demonstrated the rationality of the time discrete solution of non-homogeneous equation, took the lead in developing J. Lions' concept of weak solution of offset equation in China, and discussed the feasibility of finite element approximation of unsteady equation with finite element method.

Today, the research on the mathematical basis and problems of migration theory in China has gone deep into the research on the constructive theory of an unbounded asymmetric linear operator in Banach space, and has developed from application to the integration of various dynamic or steady-state phenomena (such as particle migration, ecological balance, population and economic problems, etc.). ) conforms to some conservation laws and is determined by statistical (probability) methods. Although Tian Fangzeng has been focusing on nonlinear functional analysis since the middle and late 1970s, he still insists on devoting part of his energy to transfer equations. The functional-analytical method of transfer equation problem is about 40 thousand words, which has been completed long ago. In the framework of the basic theory and method of functional analysis, this paper includes the study of boundary value problems of linear and nonlinear offset equations, the solution of boundary initial value problems and the properties of solutions. He kept this article for a long time, hoping to make it more perfect, so that this article can reflect China's academic achievements and thoughts in this academic field.