Before studying Banach space, it is necessary to completely 'understand' the difference and connection between Banach space, Hilbert's inner product space, and the norm line 'sex' space. шщш.(vii) (ix) x.S.сом The update is so fast.

The space of the norm line 'sex' is the space of distance, the space of Hilbert's inner product is necessarily the space of the norm line 'sex', and the Barnach space is the complete space of the norm line 'sex'. This is the basic relationship between the three.

As senior experts and master-level mathematical researchers, Mill 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, and the norm does not necessarily have to be the inner product to be defined. Hilbert space is a special case of Banach space, and Banach space is a special case of perfect distance space.

Therefore, Shen Qi redefined it as follows based on a proof by Mill in 1982:

"A non-empty subset c of Barnach space x is called 'approximated' compact, and refers to any {xn}∞n=1∈c and any y∈x, if such that ‖xn-y‖→dist（y，f{‖xn-y‖：x∈c}，

Then there is a Cauchy column for {xn}∞n=1, saying that x is 'approximate' and each closed convex subset of x is 'approximated' compact. ”

"The idea is gradually becoming clear, Shen Qi, you think that a Banach space x is 'pressing' close only if it has the quality of dropping 'sex'. Professor Müller checked again the preconditions set by Shen Qi.

Banach space is a difficult field that is not suitable for beginners, integrating many branches such as functional analysis, topology, and spatial geometry.

"That's right. "Shen Qi and Muller's 'communication' flow is very smooth, smart people don't talk nonsense, and mathematicians don't talk long-winded.

"What do you need me to do?" asked Professor Müller, not as a professor and mentor, but as an equal academic partner, like Hardy and Ramanujan, Gauss and Riemann, Euler and Lagrange, Euclid and Archimedes.

Masters who treat their apprentices equally in academics often go down in history with their disciples. Pythagoras, who was fond of exterminating his apprentices, had a negative rating of prestige in the history of mathematics.

Shen Qi said: "The analysis of the 'sexual' nature of the RNP and martingale theory of the Banach space is entrusted to Professor Müller, you are an expert in this area. ”

"Okay, so what are we waiting for? Let's split up and try to finish the manuscript in a month. "Professor Müller did it himself, and this was after all one of his first theorems, and he wanted to complete this project with his own hands, which spanned more than 30 'spring' and autumn and penetrated complex feelings.

“ok。 Shen Qi packed up his materials and prepared to leave Mueller's office to complete the first mathematical theorem named after his own surname.

"By the way, tomorrow you go to lead a tutorial, this is my lesson plan backup and student information, you can familiarize yourself. Mueller handed Shen Qi a thick, heavy file bag.

"It seems that Professor Mueller'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 morning of the next day, Shen Qi began to write the proof part of the "Muller-Shen theorem" for which he was responsible.

Shen Qi insisted on putting his last name at the back to show respect for his mentor.

Throughout the afternoon, Shen Qi pored over Müller's lesson plans on non-Euclidean geometry as part of his work, and he was one of Müller's teaching assistants.

Muller is in charge of a 'course' in non-Euclidean geometry for undergraduate students in the Department of Mathematics at the University of Mathematics this semester.

There are two types of classes, large classes and tutorials.

Large classes are usually held in large classrooms with a capacity of more than a hundred students, and the lecturers are professors.

When former U.S. President Woodrow Wilson was president of the University, he added a "preceptorial" course, or a tutorial, in 1905, and this "course" continues to this day.

Tutorials are conducted in small classes, usually led by teaching assistants, with a maximum of 10-12 students and a maximum of 15 students.

In the tutorial class, the teaching assistant led the undergraduate students to review the last week's work, and most of the tutorials in the mathematics department consisted of problem solving and discussion.

Princeton attaches great importance to undergraduate education, and undergraduate students are required to take tutorials after completing large classes, and each 'course' of core majors has a tutorial.

Undergraduate students of the University have a heavy academic load, get an A in 'door' and 'door' homework, can get a girl or a handsome guy, 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 at the small classroom, which was his first time as a teaching assistant in charge of the tutorial class.

A tutorial class usually lasts about 2 hours, and Shen Qi can receive a reward of $120.

An hourly wage of $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, Shen Qi also has a fixed salary, earning thousands of dollars a month, which is about the same as that of white-collar workers outside.

"Good evening gentlemen, I'm Shen Qi, from China, and you're in charge of the tutorial. You can call me Shen, or Qi, or Shen Qi. Come, follow me, wet clothes mmmm, wives Yiqi, Shen ~ Qi ~" Shen Qi said to the 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're so young, you're the youngest teaching assistant I've ever met. ”

"Hey Key, have you got your PhD?"

"Key, I'm guessing you're a Pisces. ”

The little juniors were very interested in Shen Qi's age and identity, and they had never seen such a young person in charge of the tutorial class.

"shen" is more difficult to master the standard pronunciation for crooked nuts, so the disciples call Shen Qi "qi".

Because of the pronunciation habits, "qi" was pronounced as "ki" by the disciples and disciples, which is consistent with the pronunciation of the word "key". In American English, key is not pronounced as a long sound, and the phonetic transcription is ki.

"Actually I'm doing a PhD, and as for my age, keep it a secret. And my horoscope, too, is kept secret. Shen Qi said with a smile, these young people were vibrant, enthusiastic, curious and excited, which reminded him of his youthful 'spring' years.

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

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

"Although it is a tutorial in non-Euclidean geometry, I ask a few questions about the history of mathematics before I start tutoring in non-Euclidean geometry. I have always believed that only by understanding the history of mathematics can we learn mathematics well. Shen Qi said, "I have memorized the names of twelve handsome guys through the 'flower' roster, and I will name them one by one, I want to know each of you." ”

"Stephen Corey, how many three-pointers did you make yesterday?" Shen Qi locked on a few white boys, and he felt that Stephen Corey should be produced among several people.

"It's a shame that I play as a centre-forward. A white boy about 1.8 meters tall raised his hand to indicate that he was Stephen Corey, a name that is pronounced exactly the same as NBA star Stephen Curry, except for the spelling.

"Well, our strong 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 Cardin. Stephen replied without thinking, and then immediately asked, "Key, please tell me, what is your age? If you are younger than me, I will not be able to accept this reality!"

"I was the same year as Fi'bonaci. Shen Qi writes a line on the blackboard, Leonardo Fi'bonaci, 1175-1250.

"no-ay!" Stephen exaggerated, knowing that Shen Qi could not be the same year as Fi'bo, and he realized that he had answered the wrong first question, that it was Fi-Bo Naci who saved European mathematics and not Cardin.