Recitation Class
Before studying Banach space, it is necessary to fully understand the differences and relationships between Banach space, Hilbert Inner Product Space, and Normed Linear Space.
Normed Linear Space are metric spaces, Hilbert Inner Product Space are necessarily Normed Linear Space, Banach space are complete Normed Linear Space. These are the fundamental relationships among the three.
As seasoned experts and master-level Mathematical Research, Muller and Shen Qi also needed to rely on the most fundamental theories to prove theorems within their systems.
The inner product in an inner product space can define a norm, but the norm is not necessarily defined by the inner product. Hilbert space are special cases of Banach space, and Banach space are special cases of complete metric spaces.
Therefore, Shen Qi redefines the definition based on a proof by Muller in 1982 as follows:
"A nonempty subset C of Banach space X is called approximate compact if for any {xn}∞n=1∈C and any y∈X, such that..."
‖xn-y‖→dist(y, C)=inf{‖xn-y‖: x∈C},
Then there exists a Cauchy sequence for {xn}∞n=1, which states that X is approximatingly compact, and every closed convex subset of X is approximatingly compact.
"Your thinking is becoming clearer. Shen Qi, do you believe that a Banach space X is approximately tight if and only if it has the drop property?" Professor Muller checked the preconditions set by Shen Qi again.
Banach space integrate many branches of Functional Analysis, topology, and spatial geometry, making them a challenging field that is not suitable for beginners.
"That's right." Shen Qi and Muller communicated very smoothly; intelligent people don't waste words, and mathematicians don't mince words.
"What do you need me to do?" Professor Muller asked, without putting on airs of a professor or mentor. He regarded Shen Qi as an equal academic partner, just like Hardy and Ramanujan, Gauss and Riemann, Euler and Lagrange, Euclid and Archimedes in the past.
Masters who treat their students as equals in academic matters are often recorded in history alongside their students. Pythagoras, who was fond of eliminating his students, has a negative reputation in the History of Mathematics.
Shen Qi said, "I'll leave the RNP property of Banach space and the Martingale Theory analysis to Professor Muller; you're an expert in this area."
"OK, then what are we waiting for? Let's split up and get to work, and try to finalize the manuscript in a month." Professor Muller was personally involved, after all, this was a theorem that he had first proposed, and he wanted to complete this project that spanned more than thirty years and was permeated with complex emotions.
"OK." Shen Qi packed up his documents and prepared to leave Muller's office to complete the first mathematical theorem named after himself.
"By the way, you'll be teaching the Recitation Class tomorrow. Here's a backup of my lesson plans and student information. Familiarize yourself with them." Muller handed Shen Qi a thick, heavy file folder.
"It seems Professor Muller's $120 isn't easy to earn." Shen Qi smiled, took the file bag, and left.
That evening and the following morning, Shen Qi began writing the proof of the "Muller -Shen Theorem," which he was responsible for.
Shen Qi insisted on putting his surname last to show respect for his mentor.
Shen Qi spent the entire afternoon carefully studying Muller's lesson plan on Non-Euclidean Geometry, as part of his work; he was one of Muller's teaching assistants.
This semester, Muller is teaching a course on Non-Euclidean Geometry to undergraduate students in the Mathematics Department at Princeton University.
There are two forms of instruction: large lectures and Recitation Class.
Large lectures are usually held 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 president of Princeton University, he added a "preceptorial" Recitation Class in 1905, which continues to this day.
Recitation Class are conducted in small classes, usually led by teaching assistants. Each tutorial class has 10-12 students, with a maximum of 15 students.
In the tutorial classes, teaching assistants led undergraduates in reviewing the coursework from the previous week. Most Recitation Class in the Mathematics Department consisted of problem-solving and discussions.
Princeton places great emphasis on undergraduate education. After attending large lectures, undergraduates are required to Recitation Class. Recitation Class are offered for every core major course.
Undergraduates at Princeton University have heavy academic workloads, get A's in every course, can still date girls or handsome guys, and can also balance club activities and sports. They must be the most outstanding minority of elites.
At 8 p. m., Shen Qi arrived at the small classroom. This was his first time being in charge of Recitation Class as a teaching assistant.
A Recitation Class typically lasts about two hours, and Shen Qi receives a reward of $120.
$60 an hour is not low in the United States, after all, Shen Qi is engaged in mental labor that requires a high level of expertise.
In addition to the tuition fees for the Recitation Class, Shen Qi also has a fixed salary, which is several thousand US dollars a month, similar to that of white-collar workers.
"Good evening, gentlemen. I am Shen Qi, from China, and I am in charge of your Recitation Class. You can call me Shen, or Qi, or Shen Qi. Come on, repeat after me, 'Shen Qi, 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 were all boys, with an average age of 20, a little younger than Shen Qi.
"Key, you're so young! You're the youngest teaching assistant I've ever seen."
"Hey Key, have you gotten your PhD yet?"
"Key, I guess you're a Pisces."
The junior students were very interested in Shen Qi's age and identity; they had never seen such a young person in charge of the Recitation Class before.
The pronunciation of "Shen" is difficult for foreigners to master, so his juniors call Shen Qi "Qi".
Because of pronunciation habits, the younger students started pronouncing "Qi" as "Ki," which sounds the same as the word "key." In American English, "key" is not pronounced with a long vowel; its phonetic symbol is [ki].
"Actually, I'm currently pursuing a doctorate. As for my age, that's a secret. And my zodiac sign, that's a secret too," Shen Qi said with a smile. These young people were energetic, enthusiastic, curious, and excited, reminding him of his own youth.
"For every question you answer correctly, I will unlock one of my secrets," Shen Qi said.
"Come on, Key, when you walk out of this classroom in two hours, there will be no secrets left," a white student said confidently, his curiosity palpable.
"Although it's Recitation Class on Non-Euclidean Geometry, I'll ask a few questions about the History of Mathematics before I start tutoring on Non-Euclidean Geometry. I've always believed that understanding the History of Mathematics is essential to learning mathematics well," Shen Qi said. "I've already memorized the names of twelve handsome guys from the roster, and I'll be calling them out one by one. I want to get to know each and every one of you."
"Stephen Corey, how many three-pointers did you make yesterday?" Shen Qi locked onto a few white male students, thinking that Stephen Corey should be among them.
"Unfortunately, I play center." A white male who was about 1.8 meters tall raised his hand to indicate that he was Stephen Curry. The name was pronounced exactly the same as NBA star Stephen Curry, the only difference being the spelling.
"Alright, our Strong Force inside player Stephen, tell me, who rescued European mathematics from the long darkness of the Middle Ages?" Shen Qi asked.
"Without a doubt, this savior is Cardano," Stephen answered without hesitation, then immediately asked, "Key, please tell me, how old are you? If you are younger than me, I will not be able to accept this reality!"
"I was born in the same Fibonacci." Shen Qi wrote a line of information on the blackboard: Leonardo Fibonacci, 1175-1250.
"No-Way!" Stephen exclaimed dramatically. He knew, of course, that Shen Qi couldn't possibly be the same Fibonacci, and he also realized that he had answered his first question wrong. It was Fibonacci, not Cardano, who saved European mathematics.