"Functional analysis at the undergraduate level is mainly based on linear functional analysis, and nonlinear functional analysis is generally not studied until the graduate level. The main content of linear functional analysis, to sum up, is what we often call one king and one queen and the four heavenly kings. One king and one queen are the Bell theorem and the Hahn-Banach theorem, and the four kings are the open mapping theorem, the closed image theorem, the Banach-Steinhaus theorem, the closed value range, and the Mazur theorem......"

This is a review class, and the teacher didn't say much, so he started directly in class: "Today, we will start with one king and one queen and the four heavenly kings, and sort out the relevant knowledge points learned this semester, including the spectral analysis of closed operators, the self-supporting extension of symmetric operators, the semi-group theory of operators, linear monotonic operators, and the algebra ...... operators."

Pang Xuelin yawned in his seat, the guest house where he lived had average sound insulation, he didn't sleep well last night, and he had been on the road for the past few days, he was really tired. In addition, the teacher also brought some dialect accents when he gave a lecture, and the tone was tepid, although there was nothing wrong with the content of the lecture, but it still made people feel sleepy.

Pang Xuelin closed his eyes unconsciously.

I don't know how long it took, Pang Xuelin was suddenly woken up by a loud noise.

He opened his eyes a little dazed, and found that many students in the classroom were looking at him with playful eyes.

On the podium, the bald professor was speaking in the direction where he was: "That classmate in the army green coat, you sleep so soundly during class time, it seems that you should know everything, come over and help me solve this problem......"

Pang Xuelin looked around, and then pointed to himself a little blankly: "Teacher, me?" ”

"Yes, that's you! Come up and help me solve this problem on the blackboard......"

"Teacher, I'm not ......"

"You can't get up? If you don't come up, you'll let me out, but don't even think about taking this class. ”

"This ...... All right! ”

Pang Xuelin hesitated for a moment, but still chose to get up.

He originally thought that he would wait until after class to talk to the professor and ask for the addresses of several major journals.

If you go out directly now, you will really offend people.

Pang Xuelin walked up to the podium, and everyone saw his outfit clearly.

A shabby military coat, a black cotton jacket can be faintly seen inside, and the thread of the collar of the cloth jacket has fallen off, revealing a small half of cotton.

He had a pair of black cloth shoes on his feet, but after a few days of traveling, the cloth shoes that had been washed had already become dirty and unpresentable.

In this era, college students are still the pride of the sky, and no matter how poor the family is, there are a few decent clothes to change into.

If it weren't for Pang Xuelin's own good temperament and image, he would have been regarded as a blind stream from the countryside long ago.

"What class is this person from? I haven't seen it before, this dress is enough! ”

"Lao Wang dares to sleep in class, this guy is really bold."

"I guess he has given up the idea of cramming for a while, functional analysis is too difficult, Lao Wang's lecture accent is heavy, I am sleepy and want to sleep......"

……

The students in the audience were discussing, and they looked at Pang Xuelin curiously.

Wang Chongqing looked at Pang Xuelin's dress, frowned, and said, "Classmate, you can solve this problem!" ”

Pang Xuelin nodded and looked at the question on the blackboard.

[Let E and F be two Banach spaces, so that A:D(A)?? E→F is a closed operator, and D(A)?? =E。 Verification: D(A??)?? σ(F′，F)=F′D(A??)?? σ(F′，F)=F′。

where A?? is the adjoint operator of A, F′ is the dual space of F, σ(F′,F) is the weak* topology on F′, D(A??)?? σ(F′,F) for D(A??) Closures under weak* topological σ (F′, F). 】

After skimming through the questions, Pang Xuelin almost didn't think much about it, and directly began to write down the answers below.

[Conclusion 1: Let F be a subvector space of E satisfying F?? ≠E. then there is f∈E' is not 0 such that (f,x)=0,?? x∈F。

Conclusion 2: Let ??:E′→R be a linear map, and the topological σ(E′,E) is continuous, then there is x∈E such that ?? (f)=(f，x)，?? f∈E′。

Proof: Set?? is a continuous linear functional on F′ to the topological σ(F′,F), at D(A??) The value in the upper parameter is 0. From conclusion 1, in order to prove the density of the weak* topology, it is only necessary to prove ?? ≡0。

From conclusion 2, the existence of x∈F makes ......]

Pang Xuelin's writing speed was very fast, and the whole proof process hardly paused, and it took less than two minutes to complete the answering work.

"Teacher, after answering, there should be no problem, right?"

Wang Chongqing was a little distracted, this question is the finale in the functional analysis, and it is difficult for undergraduates.

He was originally going to wait for Pang Xuelin to teach him a good lesson when he couldn't answer, but he didn't expect that this guy's foundation seemed to be good, and he gave a proof in the blink of an eye.

Both the proof idea and the process are concise and almost flawless.

In the audience, the discussion of students also sounded.

"Who the hell is this guy, it's hidden!"

"I didn't have any ideas for this problem, but I didn't expect him to solve it so quickly."

"It seems that we have a lot of good people in the mathematics department."

……

Many people have focused their attention on Pang Xuelin.

Wang Chongqing's face was slightly gloomy, he slept in class, no matter how good his grades were, he didn't want to let go of this guy easily.

He thought for a while and said, "This classmate, it seems that you have a good foundation, so you can tell you about your understanding of the course of functional analysis." ”

Functional analysis is inherently a highly abstract course, which is why it is difficult to learn, and even if a Ph.D. is put on stage, he may not be able to fully describe his understanding of the course.

The corners of Wang Chongqing's mouth curled slightly, he didn't believe that an undergraduate student had such ability.

"Teacher, are you sure...... Let me give you a lecture? ”

Pang Xuelin laughed.

"Sure!"

Wang Chongqing faintly felt that the other party's smile was a little weird, but he still nodded.

Pang Xuelin said: "In that case, then I will start with the history of the course of functional analysis. ”

"As we all know, the discipline of functional analysis was born at the beginning of the 20th century and is itself a consequence of axiomatic development in mathematics. In other words, mathematicians want to achieve axiomatization of analytics. The same axiomatic motion appears in geometry and algebra. Functional analysis has become a behemoth, especially when it is put together with harmonic analysis, and it is difficult to distinguish what is called harmonic analysis and what is called functional analysis. But what I'm going to talk about next is not to figure out what it is, but to focus on its foundation and future trends. ”

"We start by discussing some of the early abstract analyses, especially how mathematicians expanded a particular example into a theorem in a general sense. Our discussion mainly covers the following: 1. The work of Fredholm, Hilbert on integral equations; 2. Volterra and Hadamard's work on the problem of momentum; III. The work of Lebesgue, Frechet and Riesz on abstract space and, finally, Hahn and Banach's research on the concept of duality......"

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