From 65438 to 0950, he taught many courses, such as linear transformation in analysis, modern algebra, real variable function, functional analysis and so on. In teaching, he can combine China's achievements in ancient mathematics, inspire students' patriotic thoughts, and often warn students not to despise themselves. For example, he often talks about China's achievements in ancient times in Pi, seeking a skill through big deduction, and the theorem of quotient height, which advocates naming after China; When discussing the numerical solutions of higher-order equations, Lin Shi-Ming method is especially introduced. He is very interested in the history of Chinese mathematics.
He also actively recommended Soviet works to students. When the Chinese version was not published, he chose Soviet textbooks as the teaching content. What impressed the students was that he recommended two textbooks from the Soviet Union: The Theory of Real Variable Functions by Ha ип. Natansohn and ллксч. Lu Stanica (лкс). On the one hand, he explains theorems, on the other hand, he talks about mysteries, but what he writes on the blackboard is very concise. As soon as the theorem was proved, he patted the chalk on his hand and said, "Great, great!" " Many students only take notes and have no time to think. How can we know beauty? For example, the construction of Lebesgue measure, Vitaly covering lemma and its significance, and the method of selecting interval in proof. He points out all the nuances and often gives amazing comments in class.
His strict attitude is not only in teaching methods, but also requires students not to be vague when studying mathematics, paying special attention to some vague statements in mathematics. For example, he said that the word almost everywhere is not good, it should be said that the discontinuous point set is zero set; Students should pay attention to the proof of the extension theorem of linear functional, because some documents are wrong. He stressed that even people's names were misspelled, such as mispronouncing Lebesgue as Lebesque and Hausdorff as Housodoff, which left a deep impression on people.
In 1950s, new courses were often offered in the Department of Mathematics, with the aim of enabling students to acquire new knowledge and keep up with the times. Because there were very few Chinese textbooks at that time, he wrote his own book and taught it, fearing that the content was not mature, he declared that no one else would come to the lecture. Once, a review teacher sat down to listen without asking for details. Suddenly, Professor Zeng found out and asked him which unit he was from, which made him feel very nervous. In the mid-1950s, because of studying the Soviet Union, an oral examination was conducted, and each person was limited to no more than 30 minutes. In an exam of real variable function theory, whenever a student's answer is not satisfied, he is asked to reconsider, but the student still can't figure it out after thinking for a while, so he is asked to reconsider. Even if the questions on the examination paper are answered well, he will ask supplementary questions, as if to let the students find out all the relevant questions. In this way, there are more and more people next to the preparation classroom, but few people leave the examination room. There are many people who take exams from morning till night and can't even eat. In his view, this is an exercise in learning mathematics and it is essential to become a mathematician.
Professor Zeng is a veteran in the field of functional analysis in China and the first scholar engaged in functional analysis research in China. As early as 1930s, Professor Zeng made many important contributions. Starting from 1932, he introduced the infinite dimensional linear space on real, complex or quaternion body, and defined the inner product-Hermite symmetric bilinear universal function (f, g) on it. He has made a series of studies on this kind of space, including the representation of bounded universal functions, the eigenvalue and spectral representation of unbounded self-adjoint operators (he got some results earlier than some famous foreign scholars, such as F. Riesz, F. Rellich, Lowig and o. teichüller). In mathematical literature, operator spectrum theory is called "mathematical masterpiece", which mainly refers to linear operator spectrum theory in inner product space. Zeng's doctoral thesis (published in 1936) was an important breakthrough in the development of spectrum theory at that time. In the inseparable quaternion inner product space, the eigenvalue problem of unbounded self-adjoint operator is studied, and even the only three-part solution of this operator is given: (a) absolute continuous operator, (b) singular continuous operator and (c) point spectrum operator. In addition, the corresponding inherent extension has been made. In particular, Hellinger integral is used as the projection operator of two kinds of continuous spectral operators. Before that, there was no three-part solution even when we studied the bounded Hermite transform in separable Hilbert space. In 1942, he introduced the generalized biorthogonal system in Banach space and inner product space, which extended the problems raised by foreign workers and obtained good results. The theory of Hilbert space and its linear operator is the oldest branch of functional analysis. He has been engaged in this research, and he introduced the concepts of real solution and generalized inverse. He used modern operator theory to study a wide range of linear equations.
x’A 12 = G2，x∈D 1(A)。 (*)
Where A 12 is a closed operator from dense set D 1(A) in the inner product space m 1 to the inner product space m2, and g2 is a known element in m2. If an equation has no solution, it is called a contradiction equation. He introduced the "degree of contradiction" ρ(0≤ρ≤ 1) of the contradictory equation, and determined the specific expression of ρ. He introduced the basic concept of "extremely realistic solution". The element x' δ is called the actual solution of equation (*), which means that
The x'* with the smallest module in the real solution is called the extremely real solution of (*). When the equation (*) has a solution, the real solution is the (true) solution. He proved the uniqueness of the extremely realistic solution, obtained the necessary and sufficient conditions for the existence of the extremely realistic solution, and estimated its norm. If g2 belongs to D2(A*), then the real solution of the original equation is the same as the normal equation.
x’a 12A * 2 1 = g2A * 2 1
The real solution to coincidence, and
x′* = g2A * 2 1(q 1 1)- 1，
Here q11= a12a * 21.
Let x ′ m be an extremely realistic solution of x ′ a12 = G2, and for any u2 in D2(A*) (similar to D 1(Q)), the sequence (u2,) converges to (u2, G2), then: ① For the weak convergence of x ′ m, ② It is necessary and only necessary for the strong convergence of x ′ m..
In every convergence situation, the limit of x ′ m is an extremely realistic solution of x ′ a12 = G2.
He combined the method here with spectral theory to solve quadratic general function.
F(x)=Q(x)+λ‖x‖2+L(x)+C
Simplify the problem (Q(x) is an unbounded closed quadratic homogeneous functional and L(x) is a bounded functional), and obtain the necessary and sufficient conditions and formulas of the solution. If m 1=m2 and operator A is self-adjoint (or normal), then the extremely realistic solution also has an internal expansion of Hilbert-Schmidt)- Carleman type.
Until the 1940s, the main work on the inverse operator problem in inner product space was Toeplitz classification of bounded infinite matrices, improvement of G Julia (only seven classes were proposed) and Moore's generalized inverse matrix. The systematic study of inverse operators (divided into 16 categories) was completed along a fundamentally different train of thought.
Let m 1, m2 be the inner product space, A 12 be the (unbounded) linear operator from D 1≡D 1(A) to m2, and R2 1 be the (unbounded) of m 1. Let P 1 and P2 represent orthogonal projection operators on D2R2 1 and D 12, respectively, and R2 1 is called the generalized inverse operator of A 12, that is,
A 12R2 1=P 1，R2 1 12 = P2。 He proposed the necessary and sufficient conditions for the existence of generalized inverse operators, proved that A 12 has a unique maximal generalized inverse operator, and determined its definition domain. In particular, it happens to be an extremely realistic solution of the equation x'A 12=g2. Any closed operator A 12 has a unique closed generalized inverse operator R2 1, and the expression R2 1 is obtained. In order to make A 12 have a bounded generalized inverse operator, it is necessary and unique that the equation X'a 12 = Y2 has a real solution for any element y2 in m2. He divided closed operators (bounded or unbounded) into four categories from the perspective of geometry, and each category was further divided into four subclasses, and obtained their characteristics for three of them and their subclasses.
Zeng Rongyuan proposed and applied realistic solution and generalized inverse operator to solve the problem of L.O. Hesse canonical form: the Hesse canonical form of any functional equation x'A 12=g2 is, here is the generalized inverse operator of B2, and W 12 and B2 are the only polar coordinate operators of A: A12 = W1. Actually, for any point H' in m 1, the norm is only the distance between the "hyperplane" formed by all the real solutions of the equation X'a 12 = G2 and H'.
Professor Zeng is recognized as the founder of generalized inverse, and people call him Zeng generalized inverse, which has a wide influence in the world. Generalized inverse also permeates other branches of computational mathematics and becomes an important content of computational mathematics.
Zeng Rongyuan also continued his work on generalized biorthogonal systems. He added H.кки Barry (бибби AP и) and A.T. Taldykin (талдык) of 1850.
E * ≦( E *(P′，P ″) | P′，P″∈P)
It is a (semidefinite) positive Hermite matrix. For a generalized biorthogonal system (gp) (for E*), the Gram matrix Eg has a lower module M * (EG) >: 0, and it is necessary and only necessary for Eh to have an EG module. At this time, there is a unique linear closure operator B in the linear closure of G system, which makes hp=gpB, where (hp) is the adjoint system of G' system (for E*) and B is the bounded positive definite Hermite operator. The author also gives an explicit formula of B. Let two elements (gp) and (hp) satisfy (gp ′, gp″)= E *(P' (p ′, P ″, P' ′, P″∈P, and the closure of one system is contained in the closure of another system. If the Gram matrices of G system and H system have mutual modulus, then
On the basis of spectrum theory, it is of great significance to use the inherent expansion of Heeringer type integral. Has made an important exposition in this regard. He started with three inherent expansions of normal operators in the inner product space of complex numbers.
In this paper, F(ω) is a continuous function, which belongs to
Fα is the intrinsic element of A, gβ(w) is the concrete intrinsic differential element, hγ(w) is the absolutely continuous intrinsic differential element, and all the index sets [α], [β] and [γ] are not necessarily countable.
10, he gave a report entitled "the function and trend of functional analysis" at the second national academic exchange meeting of functional analysis held in Jinan. It is the first time to put forward "universal functionalism" as a new branch of mathematics. What is important here is that it is not a "mixture" of several projects, but an organic whole formed by the integration of various disciplines. For example, the report emphasizes algebraic topology, algebraic geometry, differential geometry and differential topology of infinite dimensional space (especially inseparable space).
Since the early 1930s, Professor Zeng has devoted himself to the teaching and research of functional analysis for 60 years. He worked meticulously and conscientiously, and trained and brought up a large number of mathematical talents.
As early as in Tsinghua University, he enrolled Xu Xianxiu as a graduate student. Dr Yang Zhenning, an internationally renowned physicist, attended his lectures when he was working in The National SouthWest Associated University. Professor Guan, the late famous mathematician, former director of the Institute of Systems Science of Chinese Academy of Sciences and member of the department, came from his school. Before liberation, as his outstanding students, there were also famous mathematicians Professor Tian Fangzeng, Professor Jiang Zejian and Professor Xu Lizhi. After liberation, he actively cultivated new forces, especially trained graduate students many times and instructed other teachers in the functional theory teaching and research department of Nanjing University to actively engage in research work. His students are deeply inspired by their thinking methods and understanding of the essence of mathematics. Under his guidance and leadership, most of his students became associate professors and professors, became the backbone of teaching and research in Nanjing University and other universities (such as Zhejiang University), and several of them were rated as doctoral supervisors.
Died in1February, 994. Shortly before his death, although he was 89 years old, he still frequented the library of the Department of Mathematics of Nanjing University, looking for and reading materials, actively engaged in research and mastering new academic trends. Often put forward suggestions on the research direction to young and middle-aged teachers, and put forward opinions on the reform of mathematics education to leaders. Ziyun: Although I am retired, I still have to work hard and contribute my evening heat. He disagreed with the idea of "waste heat", saying that the heat in the later period was sometimes very strong, and this spirit of devoting one's life to the scientific cause was admirable.