The Science Fiction World of Xueba

"For functional analysis at the undergraduate level, we mainly study linear functional analysis and analysis. Non-linear functional analysis generally does not start until the graduate level. The main content of linear functional analysis, summed up, is what we often say. The queen and the four heavenly kings. One king and one queen are Bell's Gang Theorem and Hahnbanach Theorem, and the four heavenly kings are the open mapping theorem, the closed image theorem, the banachsteha theorem, the closed value range, and the azur theorem..."

This is a review class, and the teacher didn't talk too much, and started to talk directly in class, "Today, we will start with one king, one queen and the four heavenly kings, and sort out the relevant knowledge points learned this semester, including the closed operator. Spectral analysis, self-adjoint extension of symmetric operators, semigroup theory of operators, linear monotone operators, operator algebra..."

Pang Xuelin yawned in his seat. The sound insulation of the hostel he lived in was mediocre, he didn't sleep well last night, and he had been on the road for the past few days, so he was really tired. In addition, the teacher also used some dialect accents during the lectures, and his tone was tepid. Although there was nothing wrong with the content of the lectures, it still made people feel sleepy.

Pang Xuelin unconsciously closed his eyes.

After an unknown amount of time, Pang Xuelin was suddenly awakened by a loud noise.

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

On the podium, the bald professor was talking in his direction, "That student in the army green coat sleeps so soundly during class, it looks like you can do everything, come here and help me solve this problem." question……"

Pang Xuelin looked around, then pointed to himself in a daze, "Teacher, me?"

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

"Teacher, I'm not..."

"Can't you get in? If you don't get in, you can get out, but you don't even want to take this class."

"This... alright!"

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

He originally wanted to talk to the professor after class and ask for the addresses of several major journals.

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

When Pang Xuelin stepped onto the podium, everyone saw his attire clearly.

He was wearing a shabby military overcoat, and a black cotton jacket could be vaguely seen inside. The thread at the collar of the jacket had fallen off, exposing a small half of the cotton.

He had a pair of black cloth shoes on his feet, but after a few days of driving, the cloth shoes, which were originally starched and washed, had long since become dirty and out of shape.

In this day and age, college students are still the pride of heaven, no matter how poor the family is, they still have a few decent clothes to exchange for.

If it weren't for Pang Xuelin's good temperament and image, he would have long been regarded as a blind person from the countryside with his clothes.

"Which class is this guy from? Why haven't I seen it before, this outfit is good enough!"

"Old Wang dares to sleep in class,

What a guts this guy is. "

"I guess he has given up the idea of ​​cramming. Functional analysis is too difficult. Lao Wang's lecture accent is so heavy that I feel sleepy and want to sleep..."

...

The students in the audience were discussing and looking at Pang Xuelin curiously.

Wang Chongqing looked at Pang Xuelin's attire, frowned, and said, "Student, you can solve this problem!"

Pang Xuelin nodded, looking at the topic on the blackboard.

Let e, f be two banach spaces, let a:da??e→f be a closed operator, and da??e. Prove that da????σf', ff'da????σf', ff'.

where a?? is the adjoint operator of a, f′ is the dual space of f, σf′, f is the weak topology on f′, da???σf′, f means da?? is in the weak topology σf′, The closure under f.

After browsing the questions, Pang Xuelin hardly thought about it, and directly started to write down the answers below.

Conclusion 1 Let f be the subvector space of e and satisfy f??≠e, then there exists f∈e'not 0, so that f, x0,??x∈f.

Conclusion 2 Suppose ??: e′→r is a linear mapping, and for topology σe′, e is continuous, then there exists x∈e such that ?ff, x, ?f∈e′.

Proof Let ?? be a continuous linear functional for topology σf', f on f', and take the value 0 on da??. From conclusion 1, in order to prove the density under the weak topology, it is only necessary to prove that ??≡0.

From conclusion 2, there exists x∈f such that...

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

"Teacher, after finishing the answer, there should be no questions, right?"

Wang Chongqing was a little preoccupied. This question was regarded as the final question in functional analysis, and it was quite difficult for undergraduates.

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

Whether it is to prove the idea or the process, it is concise and clear, almost impeccable.

In the audience, there were also discussions among the students.

"Who is this guy? It's hidden!"

"I haven't had any ideas for this question. I didn't expect him to solve it so quickly."

"It seems that there are quite a lot of talented people in our mathematics department."

...

Many people focused their attention on Pang Xuelin.

Wang Chongqing's face darkened, he fell asleep in class, no matter how good his grades were, he didn't want to let this guy go easily.

He thought for a while and said, "Student, it seems that your foundation is good, so please tell everyone about your understanding of the course of functional analysis."

Functional analysis is essentially a highly abstract course, which is why it is difficult to learn. Even if a doctor is on the stage, he may not be able to fully describe his understanding of this course.

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

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

Pang Xuelin laughed.

"Sure!"

Wang Chongqing faintly felt the strangeness in the other party's smile, but he still nodded.

Pang Xuelin said, "In this case, let me start with the history of the functional analysis course."

"As we all know, the subject of functional analysis was born in the early 20th century, which itself is a result of axiomatic development in mathematics. That is to say, mathematicians hope to achieve axiomatic analysis. The same axiomatic movement also appeared in geometry and Algebra. The current functional analysis has become a behemoth, especially when it is put together with harmonic analysis, it is difficult to distinguish what is called harmonic analysis and what is called functional analysis. But I will What I want to talk about is not to clarify its definition, but to focus on its foundation and future development trend.”

"We begin by discussing some early abstract analysis, in particular how mathematicians can extend a particular case into a theorem in a general sense. Our discussion mainly covers the following. 1. Fredhol, Hilbert's work on integral equations ; two, volterra and hadaue, frechet and riesz's work on abstract spaces and finally, hahn and banach's work on the concept of duality..."