Pang Xuelin's tone is unhurried, he speaks fluent Mandarin, like a TV announcer, and when he speaks English names, he also speaks authentic American English.

The tattered clothes did not hide his temperament at all, but in his every move, there was a faint demeanor of a top scholar that all the audience was impressed.

The classroom gradually fell silent, leaving only Pang Xuelin's voice echoing in the air.

"Fredholm, Hilbert's work on integral equations, we can start with the following two specific examples. The earliest integral equations came from Fourier's study of thermal problems. In 1822, Fourier discussed how to find g if we were to reverse the following equation: f(x)=∫Re^itxg(t)dt, that is, f is known. In modern language, this is actually the inverse of the Fourier transform......"

Second, when Liouville studied second-order ordinary differential equations, he discovered that they were equivalent to a class of integral equations. For example, the solution of the equation f"(x)+g(x)=f(x). If the boundary condition f(a)=1 and f'(a)=0 are satisfied, the solution of the equation satisfies ......"

"The integral equations in the above examples can be classified into the following two forms: ∫K(x,y)f(y)dy=g(x); f(x)-K(x，y)f(y)dy=g(x)。 The first person to give a strict treatment of these two types of equations was the Swedish mathematician Fredholm, who published a series of articles between 1900 and 1903 to give a solution. I will briefly discuss the modern version of his ideas that eventually evolved: the Fredholm principle of two rules and one principle. If K(x,y) is a kind of regular symmetric kernel ......"

……

Wang Chongqing on the side was dumbfounded, he never thought that the course of functional analysis could be told like this.

In particular, in the course of his lecture, Pang Xuelin combined the development history of functional analysis with various theorems and concepts, which did not sound boring at all, but gave people a refreshing feeling.

Even many of the examples mentioned by Pang Xuelin are not even clear to Wang Chongqing himself.

In this era, the students who can be admitted to Lincheng University have a good foundation.

In addition, they have already taken a semester of basic functional analysis courses, and they are no strangers to various concepts and theorems.

Pang Xuelin deduced some concepts by explaining the history of mathematics, so that those originally abstract concepts and theorems suddenly became vivid and interesting.

"Damn, that's how it ......"

"I feel that he is even better than Professor Wang, I have always been a little ignorant about the Hahn-Banach theorem that Professor Wang said, but now I understand it all at once when I hear him say how this theorem came about......"

"What class is this guy from? I didn't expect there to be such a great person in our school......"

"You say, this guy doesn't look like a sweeping monk in the Eight Divisions of the Heavenly Dragon, he doesn't look good, he is no different from ordinary people, but in fact he is a real master......"

……

The students in the audience were full of discussions.

If you are not afraid of not knowing the goods, you are afraid of comparing goods.

Pang Xuelin's explanation was like a catalyst, which straightened out everyone's thinking at once.

Even Wang Chongqing himself, because of Pang Xuelin's explanation, has a deeper understanding of the course of functional analysis.

Time passed minute by minute, and unconsciously, the bell rang for the end of class, and at this time, Pang Xuelin had spoken less than a third.

Pang Xuelin turned his gaze to Wang Chongqing and said, "It's time for class, why don't you get here here today." ”

Wang Chongqing smiled, looked at the watch on his wrist, and said, "Classmates, is this classmate speaking well?" ”

"Good!"

In the classroom, everyone shouted in unison.

Wang Chongqing said: "It's half past eleven o'clock at noon, the morning class has ended, this classmate, you see if this will work, you continue to lecture, finish what you want to talk about, as for the students here, if you want to continue to listen to the class, you can stay and continue to listen, if you don't want to listen, you can also leave, but please try to keep quiet, don't affect this classmate's lecture." ”

"Great, you can keep listening!"

"I agree!"

"Professor Wang's proposal is good! I didn't have a good grasp of functional analysis, but after listening to this student's explanation, I think I am at least 70% sure that I will pass the final exam. ”

"Yes, this student spoke very well, and I feel that my understanding of most of the functional analysis has deepened."

……

Pang Xuelin didn't expect that the bald professor's proposal would trigger the consent of so many people, so he had to smile and say, "Okay, then I'll continue to talk." ”

The classroom fell silent again, and most of the students remained, except for a few people.

It wasn't until one o'clock in the afternoon that Pang Xuelin finished a rough explanation of the linear functional analysis part.

"Okay, I'll talk about the whole linear functional analysis part first, and finally I will talk about the learning method of functional analysis, and I will give you the following two suggestions.

First, the idea must be concrete.

The more advanced the analysis becomes, the more abstract it becomes, and many people save the method of images, but the disadvantage of this method is that it is easy to lead yourself into the ditch, because there are a lot of counter-intuitive examples in analysis, and relying too much on images will make you overturn when you do the problem later: because you will mistakenly think that a certain result is right, and then you will get stuck there. The best way to do this is with concrete examples. Memorizing and understanding theorems with concrete examples is a very important step, but it is clear that many people overlook this.

Second, it is necessary to integrate theory with practice. The theory of analytics, especially functional analysis, is essentially a summary that draws from many specific applications. The largest piece is the partial differential equation, so almost all the functionals of the partial differential equation are not badly learned. It is not an exaggeration to say that every functional concept in the textbook corresponds to an application. It is natural for many people to have an empty mind after learning functional theory, because you don't know what these concepts are for, the application of finite dimensions is not enough, only infinite dimensions are meaningful. ”

"Okay, that's all for today's explanation of functional analysis, thank you!"

The voice fell, and the classroom was quiet for a moment.

Bang Bang Bang ......

Wang Chongqing, who was sitting in a corner, took the lead in standing up and applauding, and then the applause swept through the entire lecture hall like thunder.

Wang Chongqing stepped forward and said with a smile: "This classmate, which class are you in?" I didn't expect that there are still talents like you hidden in our school. ”

Pang Xuelin said with a smile: "Professor Wang, I am not a student of the Department of Mathematics, if you really count it, we are peers ......."

Just now he already knew the professor's surname from the student's address.

"Oh, are you the new math teacher in our department?"

Wang Chongqing was slightly stunned.

Pang Xuelin shook his head, smiled and stretched out his hand and said, "No, Professor Wang, reacquaint yourself, I am a primary school teacher in Lingziding Village, Dayu Township, Anhe County, G Province, Pang Xuelin!" ”

In an instant, the originally noisy classroom was quietly audible.

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