Chapter 52 Chapter 52 Four Proof Methods of Banach’s Fixed Point Theorem
Qiu Chengtong was still asking Ning Chen questions in the form of chat.
Ning Chen simply organized the language and answered truthfully:
"Mainly because of my love for mathematics. After I completed the advanced mathematics course by myself, I found a teacher who asked me to I read the course list for the mathematics major and borrowed a lot of relevant textbooks from the library. There is still a lot of time in my freshman year, which is enough for me to complete many self-study courses..."
Listening to Ning Chen's answer, Qiu Chengtong nodded repeatedly.
This is basically consistent with Qiu Chengtong's previous guess about Ning Chen. Qiu Chengtong thought that Ning Chen should be a very special kind of student. He relied on his strong talent and perseverance to find himself in an environment where the teaching level was not that good. , make yourself stand out.
"Ning Chen, tell me about your favorite mathematical theorem."
This time, Qiu Chengtong finally asked Ning Chen a professional question, but this question had a A lot of openness.
"What is your favorite mathematical theorem... it should be Banach's fixed point theorem."
Hearing Ning Chen's answer, Qiu Chengtong was also very interested.
Banach's fixed point theorem is not a particularly advanced theorem. As long as you have completed a one-year course in mathematics, you will basically understand this theorem and complete the proof of this theorem.
“Why do you like Banach’s fixed point theorem?” Qiu Chengtong asked.
“Actually, at the beginning, I didn’t think this theorem was that special. But as I continued to learn mathematical knowledge, I found several methods that proved Banach’s fixed point theorem from different angles. This method is very interesting to me.”
“I’m all ears,” Qiu Chengtong said with a smile.
Ning Chen didn’t hesitate, picked up the chalk and started writing on the blackboard.
It is no exaggeration to say that with the knowledge Ning Chen has mastered, even if Ning Chen is asked to name a hundred of his favorite mathematical theorems, he will not get stuck at all.
Ning Chen knew that Qiu Chengtong wanted to use this question to gain a more comprehensive understanding of his mathematical abilities and attitude towards mathematics, so he chose a theorem that he thought could best demonstrate these things.
“When I first started self-study of mathematics textbooks, I learned about Banach’s fixed point theorem and its proof method for the first time...”
Ning Chen wrote on the blackboard , while explaining to several teachers under the podium.
Because the teachers below are all top teachers in the field of mathematics, Ning Chen explained very quickly to avoid making the oral examination take too long.
This first proof method is derived from the textbook. Apart from showing that Ning Chen's mathematical foundation is very solid, there is nothing special about it.
"The second proof method came to my mind when I was learning differential manifolds..."
When Ning Chen explained the second proof method, the expressions of several teachers started to change. There have been some changes.
Speaking of which, this proof method was not difficult for them to understand, but before Ning Chen explained it, they had never thought that this method could also prove Banach's fixed point theorem.
After explaining the two proof methods, Ning Chen suddenly realized that his oral exam seemed to be a bit long.
"Teachers, do you want to continue listening to the next two methods?"
A teacher next to Qiu Chengtong said to Qiu Chengtong:
"Professor Qiu, there are many more to come. As for the students, let’s forget it.”
Qiu Chengtong didn’t pay any attention to the teacher’s words.
“It doesn’t matter, keep talking, I’m very interested.”
Qiu Chengtong will naturally not care about these rules. Anyway, this is a competition held in his own name, and he must have the final say in everything.
At worst, we can just let the students behind us postpone it. Those students who are not so satisfied with themselves can completely shorten the oral examination time.
After receiving Qiu Chengtong’s consent, Ning Chen continued:
“The third proof method was something I thought of when I was learning the calculus of variations…”
While Ning Chen was explaining these things to several professors, several students were waiting anxiously outside the examination room. "Isn't my oral exam at 7:15? Why is it almost 7:30 now and it's not my turn yet?"
Tang Ding kept looking at the time on his watch and pacing outside the examination room.
In anxiety, Tang Ding found the staff and asked.
"Teacher, yesterday I was notified that I will have an oral exam at 7:15, why haven't I arrived yet?"
"Student, this is just a reference time, the specific time will be subject to the on-site notification. .”
"Okay, then I'll wait."
A student next to Tang Ding saw this and said to Tang Ding:
"This is just like seeing a doctor in the hospital, don't worry about your death What time is it? The last patient didn’t finish reading, so he couldn’t go in even after it was called.”
“Hey, that’s true. But who was the last one? Why did it take so long? Why don’t you come out?”
In the examination room, Ning Chen continued his explanation.
"The last method came to my mind when I came into contact with the knowledge of cohomological groups..."
Finally, after writing several more blackboards full of content, Ning Chen completed his explain.
After listening to Ning Chen’s entire explanation, Qiu Chengtong was completely impressed by Ning Chen.
“I really didn’t expect that you could give so many different proof methods for a simple Banach fixed point theorem. And each of these different proof methods has its own significance. ”
Qiu Chengtong had to admit that when he first learned this knowledge, he didn’t think as much as Ning Chen did.
Now it seems that there is indeed a reason why Ning Chen was able to achieve excellent results in the written examination.
"Ning Chen, I see that you just explained it in a thorough and orderly manner, just like a teacher. Do you usually lecture like this to others?"
Qiu Chengtong was a little surprised. Ning Chen's explanation ability, among the outstanding students Qiu Chengtong has come into contact with, few are as clear as Ning Chen.
Even in terms of expressive ability alone, many university teachers may not be able to compare with Ning Chen.
“I take some tutoring classes during my free time. While exercising my abilities, I also earn some living expenses.”
Of course, Ning Chen's answer was just an excuse, which was mainly due to Ning Chen's experience as a teacher in his previous life.
Ning Chen came into contact with a variety of students in education and training institutions, and he was able to train himself to teach in different situations.
"Well, it would be good to be a tutor."
Qiu Chengtong glanced at the time and found that it was already past 7:30.
If this continues, I won’t be able to complete the oral exam on this day, and there will be an awards ceremony in the afternoon.
"Okay, let's stop here."
"Okay, thank you teachers."
After the oral exam, Ning Chen could finally relax completely. He had finished it. All tasks for this competition.
The next step is to wait for the results to come out.
Tang Ding, who had been waiting outside the examination room for a long time, saw that the door of the examination room finally opened and hurriedly walked over.
(End of this chapter)