Chapter 97
Creak-.
I opened the classroom door and walked in.
[Functional Analysis: Professor Yoo A-yeon]
Thirteen students were seated in the classroom.
Unlike classes that undergraduates were required to take.
Graduate classes that weren't required courses tended to be this small.
The more specialized and obscure the subject, the more common it was for only five students to take it.
The atmosphere was markedly different from that of the bewildered freshmen in Monday's Calculus lecture.
Everyone wore the seasoned expressions of people who had been through all sorts of hardship.
After all, the graduate-student unnis and oppas who had made it this far were elite scholars who had already climbed a long way up mathematics' treacherous mountain path.
They might find the fact that I was a seven-year-old professor fascinating, but in the end, their main concern would be whether I could teach the class well.
Mom, who had come in holding my hand, chose a seat and quietly sat down.
This time, Mom wasn't here as a teaching assistant, but as a student.
Because this was a graduate class, not an undergraduate class.
Well, well.
A super-genius daughter teaching her graduate-student mother? This was a full-on fire-element filial daughter, with flames blazing away-!
But thinking of myself as a fire-element filial daughter made me feel strangely like a mage, and it was so cool that I felt great.
That made fourteen students in the class.
I climbed onto the footstool and confidently looked around the classroom.
“Hello, graduate-student unnis and oppas! I’m Yoo A-yeon, the professor teaching Functional Analysis!”
There were a few students whom I somehow felt I ought to call uncles and aunties.
But they had devoted their youth to graduate school, so hearing that would make them sad. I decided to call them unni and oppa even if they looked older.
I’m Yoo A-yeon, considerate even in the little things!
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The first Functional Analysis class.
I began with an orientation.
Click-clack-.
I climbed onto the footstool and wrote the title in large letters on the blackboard.
[Functional Analysis (Functional Analysis)]
“First, let’s go over the prerequisites.”
I briskly wrote a list on the blackboard.
- Real Analysis
- Measure Theory (Lebesgue Integration)
- Linear Algebra
- Topology
- Complex Analysis (Good to know!)
“I’ll assume that everyone already knows this much and proceed from there. You all signed up for this class yourselves, so of course you know it, right?”
Everyone silently nodded.
There were no reactions like the freshmen’s Gasp-!, Waaah-!, or I-don’t-wanna-take-class-.
It was the calm composure of elite scholars who had chosen graduate school of their own accord.
“Now, what exactly is Functional Analysis a field of study about? In one sentence.”
I wrote a sentence on the blackboard.
“It’s doing linear algebra in infinite-dimensional spaces.”
“Would it be fair to interpret that as moving from finite-dimensional spaces to infinite-dimensional ones?”
One of the oppas asked.
“Exactly. And to deal with infinite dimensions, we need a tool called a norm instead of distance. That’s where we’ll begin.”
(*Norm: a function that measures the size of a vector. It is a different concept from distance, which measures the gap between points.)
Click-clack-.
I began organizing the overall flow of the lecture on the blackboard by part.
[Part 1. Normed Spaces (Normed Spaces)]
“This is the first part. We’ll study spaces equipped with a norm instead of distance.”
I wrote the subtopics below it.
“Norms, the distances they induce, and Banach spaces with completeness. As examples, I’ll briefly introduce ℓᵖ, Lᵖ, C(K), and Sobolev spaces as well.”
I pointed to a theorem with a piece of chalk.
“And here’s the most famous theorem you’ll learn in this part: the Hahn–Banach theorem. Mathematicians sometimes call it the starting point of Functional Analysis.”
[Part 2. Banach Spaces (Banach Spaces)]
“When a normed space has completeness, it’s called a Banach space.”
(*Completeness: the property of being densely filled without gaps. The real numbers are a representative number system with completeness because they are packed tightly with no empty spaces.)
I wrote four theorems side by side.
- Baire Category Theorem (Baire Category Theorem)
- Uniform Boundedness Principle (Uniform Boundedness Principle)
- Open Mapping Theorem (Open Mapping Theorem)
- Closed Graph Theorem (Closed Graph Theorem)
“These four are commonly called the ‘Big Four.’ They’re the most important theorems in Functional Analysis.”
At that moment, Mom in the front row quietly raised her hand.
“Professor. Can the Uniform Boundedness Principle and the Banach–Steinhaus theorem essentially be regarded as the same thing?”
Oh, that’s my Mom.
That was a sharp question.
“That’s a good question. Yes, that’s right. You can regard the Banach–Steinhaus theorem as another name for the Uniform Boundedness Principle. The terminology hasn’t been standardized, so different sources use different names. Don’t get confused when you read papers later, unnis and oppas!”
Mom nodded, looking satisfied, and wrote something in her notebook.
[Part 3. Dual Space (Dual Space)]
“The difficulty level rises sharply from here.”
I wrote down the list.
“Continuous linear functionals, dual spaces, and biduals. We’ll also cover the weak topology and the weak-* topology. The representative theorems are the Banach–Alaoglu theorem and Goldstine’s theorem.”
“Professor, would it be fair to say that the core motivation for introducing the weak-* topology is ultimately the compactness of the unit ball?”
One of the unnis in the back asked.
Is this unni a doctoral student? Sharp.
“Exactly. The Banach–Alaoglu theorem guarantees that the unit ball is compact in the weak-* topology. What’s obvious in finite dimensions only becomes true in infinite dimensions when we move to the weak topology.”
[Part 4. Hilbert Spaces (Hilbert Spaces)]
“This is the part many people find the most interesting.”
I raised my voice, getting a little excited.
“It’s a space equipped with an inner product, a tool even stronger than a norm!”
I wrote down the topics.
“Inner products, orthogonality, the projection theorem, the Riesz representation theorem, and orthonormal bases. The Fourier series familiar to us also appears very naturally here.”
[Part 5. Bounded Operators (Bounded Operators)]
“From here on, we won’t study functions themselves, but rather ‘functions that act on functions.’”
“So they’re functions of functions.”
“That’s right. We’ll cover linear operators, operator norms, compact operators, and adjoint operators.”
I stopped writing on the blackboard and turned to face everyone.
Then I struck a V sign and proudly showed off.
“Heheh! And this is another topic that I frequently covered in the paper where I proved the Riemann Hypothesis. When I solved the Riemann Hypothesis, I found a particular operator H on a Hilbert space and used it in the proof!”
The moment I said that.
I remembered the incident at the French Millennium Awards ceremony when mathematicians had attached my name to things, and my face grew hot.
“But then the mathematicians started calling the operator I found the Ayeon operator, and they called the theorem establishing its self-adjointness the Ayeon Self-Adjointness Theorem… I was so embarrassed….”
Ha-ha-ha-.
The classroom erupted in laughter.
At the same time, they looked at me with eyes filled with awe.
In those eyes, I could see the desire to discover a mathematical theorem worthy of bearing their own name someday.
Hmm, good, good.
A scholar ought to have eyes like those!
[Part 6. Spectral Theory (Spectral Theory)]
As I wrote this part down, I smiled faintly.
“This part is actually a field I struggled with quite a bit when proving the Riemann Hypothesis.”
At those words, tension appeared in the students’ eyes.
As expected, the words ‘the Riemann Hypothesis,’ humanity’s greatest mathematical challenge, carried a lot of weight.
“Ah! Of course, don’t worry. I’ll only teach it at a level that graduate-student unnis and oppas can properly understand.”
I waved my hands back and forth to reassure them.
“We’ll study spectra, eigenvalues, resolvents, and spectral radii. And compact self-adjoint operators. The representative theorem is the spectral theorem.”
I tapped the blackboard with my chalk.
“The interesting thing is that modern quantum mechanics also begins right here. It’s the intersection of physics and mathematics. That’s why I’m good at quantum mechanics too! Ahem!”
I puffed out my chest and thumped it twice.
As a bonus, the corners of the unnis’ and oppas’ mouths lifted at that adorable moment.
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After that, I quickly went over the remaining parts.
“That covers the basics. If we have time, I might briefly introduce C*-algebras (C*-algebra) or cover Fredholm theory. This is connected to PDEs (partial differential equations).”
I wrote a list of partial differential equations on the blackboard.
- Laplace equation
- Heat equation
- Wave equation
- Schrödinger equation
“In fact, you can think of Functional Analysis as a tool for solving partial differential equations like these. We interpret all of these equations within the framework of Functional Analysis. That’s why it’s also closely connected to physics.”
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“Lastly, let me tell you about the difficulty level of this course.”
I set down the chalk and looked around at the students.
“This isn’t a class with lots of calculations, like undergraduate Calculus or Linear Algebra. Rather, most of the lectures consist of proving theorems.”
I wrote some example questions on the blackboard.
- Why does every bounded linear operator have this property?
- Why is completeness necessary for this theorem to hold?
- What exactly happens in the weak topology?
“In other words, the focus is on abstract structures and logical proofs rather than calculations. It’s a course where you dig endlessly into the question, ‘Why?’”
I grinned.
“So keep thinking, thinking, and thinking without end! All right, then, let’s begin the class!”
I climbed onto the footstool and scribbled out equations.
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Of course, this time too, I bulldozed my way through the material.
I’m Yoo A-yeon, the runaway locomotive who shows no mercy to undergrads or grad students alike!
When it was time to end the class.
I looked around the classroom and saw that everyone was completely exhausted.
One person was pressing firmly on their temple, and another was massaging their eyes.
It’s tough, but it can’t be helped.
The path of scholarship is supposed to be difficult…
I decided to call attendance, which I hadn’t called earlier.
“Before we end class, I’ll take attendance!”
I examined the attendance sheet from every angle.
Ahem, ahem.
Looking over the list, I saw that the composition was quite diverse.
Of the fourteen students, five were doctoral students, seven were master’s students, and two were postdocs, or postdoctoral researchers.
Besides that.
Ten of them were from our Department of Mathematical Sciences. Two were from the Department of Physics and Astronomy, one was from the Department of Statistics, and one was from the Department of Computer Engineering.
People had come from all sorts of places to take my class.
Well, that made sense when I thought about it.
Functional Analysis was used across the board, whether in theoretical physics, statistics, or theoretical computer science.
And because of the administration’s leisurely pace, the name of the professor in charge had remained hidden behind question marks until right before the semester began.
So without a doubt, Mom was the only person who had known in advance that I would be teaching Functional Analysis.
The graduate-student unnis and oppas here hadn’t come to see a seven-year-old professor. They had come simply to learn Functional Analysis.
Then I had to teach them properly more than ever!
“You unnis and oppas are lucky.”
I said with a grin.
“You’re taking Functional Analysis directly from the mathematician who proved the Riemann Hypothesis! Hilbert spaces, spectral theory, and so on. You’ll be able to study in great detail the theories actually used in the proof of the Riemann Hypothesis!”
At that, the unnis’ and oppas’ eyes sparkled.
A theory lecture served up personally by a master?
How could anyone pass that up!
If I were in their shoes, and there were an elliptic-curve lecture personally taught by Professor Wiles, I’d book a plane immediately and zoom across the continent.
I called the names in the order they appeared on the attendance sheet.
“Kim Seung-hyeon oppa?”
“Yes.”
“Lee Ji-yeon unni?”
“Yes.”
“Im Seong-min oppa?”
“Present.”
Each time I called a name.
I fixed each face firmly in my memory.
The doctoral students all looked haggard, as if they had already been through every kind of hardship, and their dark circles were deep.
The newly arrived master’s students had tense expressions in their eyes.
Will my pretty Mom end up with huge dark circles too someday?
…Thinking about it that way, I seemed to be becoming more and more of a fire-element filial daughter.
It can’t be helped.
I’m not someone who compromises when it comes to the path of learning.
If an advisee says they want to earn a PhD, the professor has no choice but to give them a tearful smack-smack-! Sob.
After calling thirteen names.
One last name remained.
But then.
Hmm.
There was no way I could call this last person unni.
“Um… the last person is….”
I said with a slightly troubled expression.
“Since I absolutely can’t call her unni, please forgive me if I just call her Mom, okay?”
The graduate-student unnis and oppas chuckled and nodded as if they understood.
Good.
Then, without holding back.
“Mom!”
“Yes!”
Mom raised her hand high and smiled elegantly.
A professor calling out “Mom!” from the attendance sheet.
I’m probably the only person in South Korea—or in the entire world—who does that.
“Including Mom, all fourteen students are present!”
I sharply closed the attendance sheet.