I Just Want to Be a Quiet Top Student

Before studying Banach spaces, we need to fully understand the differences and connections between Banach spaces, Hilbert inner product spaces, and normed linear spaces.

A normed linear space is a distance space, a Hilbert inner product space must be a normed linear space, and a Banach space is a complete normed linear space. This is the basic relationship among the three.

As senior experts and master-level mathematics researchers, Muller and Shen Qi also need to rely on the most basic theories to prove the theorems in the system.

The inner product in the inner product space can define the norm, but the norm does not necessarily have to be defined by the inner product. A Hilbert space is a special case of a Banach space, and a Banach space is a special case of a complete distance space.

Therefore, Shen Qi redefines it based on a piece of proof Muller gave in 1982 as follows:

"A non-empty subset C of Banach space X is called approximately compact, which means that for any {xn}∞n=1∈C and any y∈X, if such that

‖xn-y‖→dist(y, C) = inf {‖xn-y‖: x∈C},

Then {xn}∞n=1, there is a Cauchy column, which says that X is approximately compact, and every closed convex subset of X is approximately compact. "

"The thinking is gradually clear. Shen Qi, you think that a Banach space X is approximately compact if and only if it has the property of drop." Professor Muller checked the preconditions set by Shen Qi again.

Banach space integrates functional analysis, topology, space geometry and many other branches. It is a difficult field and is not suitable for beginners.

"That's right." Shen Qi and Mueller communicated very smoothly. Smart people don't talk nonsense, and mathematicians don't talk nonsense.

"What do I need to do?" Professor Muller asked, without putting on the airs of a professor or a mentor. He regarded Shen Qi as an equal academic partner, just like Hardy and Ramanujan, Gauss and Riemann, Euler and Lagrange, Euclid and Archimedes.

A master who treats his apprentices equally academically can often be recorded in the annals of history with his disciples. Pythagoras, who loves to destroy his apprentices, has a negative reputation in the history of mathematics.

Shen Qi said: "The RNP properties of Banach spaces and the analysis of martingale theory are entrusted to Professor Muller, you are an expert in this field."

"OK, then what are we waiting for? Let's work separately and try to finish the manuscript in a month." Professor Muller did it himself. After all, this was the first theorem he proposed. Spring and Autumn, a topic that permeates complex emotions.

"OK." Shen Qi packed up the materials and was about to leave Muller's office to complete the first mathematical theorem in his life named after his surname.

"By the way, you are going to lead the tutoring class tomorrow. Here is my lesson plan backup and student information. You should familiarize yourself with it." Mu Le handed Shen Qi a thick and heavy file bag.

"It seems that Professor Muller's one hundred and twenty dollars is not easy to earn." Shen Qi smiled, took the file bag and left.

In the evening and the next morning, Shen Qi began to write the proof part of the "Muller-Shen Theorem" which he was responsible for.

Shen Qi insisted on putting his surname behind to show respect for his mentor.

All afternoon, Shen Qi pored over Muller's lecture notes on non-Euclidean geometry as part of his job, and he was one of Muller's teaching assistants.

Muller is in charge of a course, Non-Euclidean Geometry, for undergraduates in the Mathematics Department of Universities this semester.

There are two forms of teaching, lectures and tutorials.

Large classes are usually arranged in a large classroom that can accommodate more than one hundred students, and the lecturer is a professor.

When former U.S. President Woodrow Wilson was the principal of Puda, he added a "preceptorial" course, that is, a tutorial course. It was in 1905, and this course has continued to this day.

Tutorial classes are implemented in small classes, usually led by teaching assistants. There are 10-12 students in a tutorial class, and no more than 15 students.

In the tutorial class, the teaching assistants lead the undergraduates to review the last week's work. Most of the tutorial classes in the Department of Mathematics consist of problem solving and discussion.

Princeton attaches great importance to undergraduate education. Undergraduates have to take tutorial courses after major courses. Each core professional course has tutorial courses.

An undergraduate student in a general university has a heavy academic load. If he or she gets A's in all subjects, can meet girls or handsome guys, and can take care of club activities and sports, then he or she must be the best minority elite.

At 8 o'clock in the evening, Shen Qi arrived in the small classroom. This was the first time he was in charge of a tutorial class as a teaching assistant.

A tutorial class usually lasts about 2 hours, and Shen Qi can receive 120 US dollars in remuneration.

An hourly salary of US$60 is not low in the United States. After all, Shen Qi is engaged in mental work with high professional quality.

In addition to the tuition fee for the tutorial class, Shen Qi also has a fixed salary, with a monthly income of several thousand dollars, which is similar to that of white-collar workers outside.

"Good evening, gentlemen. I'm Shen Qi, from China, the person in charge of your tutorial. You can call me Shen, or Qi, or Shen Qi. Come on, follow me, Shiyi um Shen, Wife Yiqi, Shen~Qi~" Shen Qi said to 12 second-year undergraduate students in the classroom, and wrote his name "Shen-Qi" on the blackboard.

"Sion~Ki~" The twelve second-year students are all boys, with an average age of 20 years old, a little younger than Shen Qi.

"Key, you are so young, you are the youngest teaching assistant I have ever seen."

"Hey Key, have you got your PhD yet?"

"Key, I guess you're a Pisces."

The juniors were very interested in Shen Qi's age and identity, and the person in charge of such a young tutoring course had never seen him before.

"Shen" is difficult for Wai Guoren to master the standard pronunciation, so the juniors call Shen Qi "Qi".

Because of the habit of pronunciation, "Qi" was pronounced "Ki" by the juniors, which is consistent with the pronunciation of the word "key". In American English, the key does not make a long sound, and the phonetic symbol is [ki].

"Actually, I am studying for a Ph. D. As for my age, I keep it a secret. And my zodiac sign is also kept a secret." Shen Qi laughed. These young people were full of vigor, enthusiasm, curiosity and excitement, which reminded him of his youth .

"If you answer one question correctly, I will unlock one of my secrets." Shen Qi said.

"Come on, Key, when you walk out of the classroom in two hours, there will be no more secrets." A white student said confidently, he has a strong desire to explore.

"Although it is a tutorial course on non-Euclidean geometry, I will ask a few questions about the history of mathematics before starting the tutoring of non-Euclidean geometry. I always believe that only by understanding the history of mathematics can I learn mathematics well." Shen Qi said, "I passed The roster has already remembered the names of twelve handsome guys, and I will roll them one by one, and I want to know each of you."

"Stephen Curry, how many three-pointers did you score yesterday?" Shen Qi focused on several white boys, and he thought Stephen Curry should be born among them.

"Unfortunately, I play center forward." A white boy about 1.8 meters tall raised his hand, indicating that he is Stephen Curry. The pronunciation of this name is exactly the same as that of NBA star Stephen Curry. The only difference The most important thing is the spelling method.

"Okay, our powerful insider Stephen, please tell me, who rescued European mathematics from the long darkness of the Middle Ages?" Shen Qi asked.

"There is no doubt that this savior is Cardan." Stephen replied without hesitation, and then immediately asked: "Key, please tell me, what is your age? If you are younger than mine, I will not be able to accept this reality !"

"I am the same age as Fibonacci." Shen Qi wrote a line of information on the blackboard, Leonardo Fibonacci, 1175-1250.

"No-Way!" Stephen made an exaggerated look. Of course he knew that Shen Qi could not be the same age as Fibonacci. He also realized that he answered the first question wrong. It was Fibonacci who saved European mathematics. Not Cardan.