How to Read Hard Math Books

In this video I will show you a more advanced math book and I will show you how to read it, or at least how to start. The book is called Functional Analysis and it was written by Edward W. Packel. Do you have any advice for people? If so, please leave a comment below.

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[Jim-besj]

I remember the first time I looked at a really hard math book. I found some Dover book called "Advanced Calculus" at a thrift store and brought it home. I had finished a couple of semesters of calculus at the time and figured I could easily augment what I knew. When I opened the book and looked at the first page, I didn't know what was going on. There were all kinds of unfamiliar symbols on the page and the the first sentence was something like "A sigma-algebra has the following properties..." I have a PhD now but would still have a hard time with that book I am sure. :)

I recently began to study math again after a long break. I have been taking notes in a notebook as I read, which is something that I never did before. I don't know why I began to do this, but it does seem to be working. I go to a library or a study room without my phone or computer -- just a notebook, a pencil, and a few math books. I stay there for however much time I have that day, and I learn math.

One of the books that I'm learning from now is one I learned about from your channel. It's "All the Math You Missed" by Thomas Garrity. I like it. I'm getting a lot of knowledge into my brain by reading a chapter in this book along with a textbook on the subject of the chapter to work through or review from.

I used to be afraid of math that was beyond my current level of knowledge. Now I think I was overestimating how "big" math is. No one has time or memory capacity to learn all of math, but every area of math is learnable. Math is wide, but it isn't really very tall. All those abstract mathematical structures built out of other structures are mysterious and weird only until you just take some time to patiently work through a few books on a subject. Then those concepts become meaningful, familiar, and useful.

[ielZxro]

These videos are worth their weight in gold, please continue this type. There is tons of material on what to do to learn, but nothing actually teaching people how to learn as you said in the video. Can sit and watch someone problem solve, and do things on paper all day; but the real magic happens in their thought process' behind the problem solving techniques they implement after years of hard work and study. Impossible to read minds, so unless they explain their thought process as they are doing it to someone else; their personal technique will be lost forever. Between the comments, and the video it just goes to reaffirm all the different ways people come to same conclusion with different routes. Please keep up these type of videos. I can personally say I have benefitted from the videos, and comments in improving technique and method of thinking.

[sophiaisabelle]

We appreciate your videos. Happy Holidays. God bless you.

[itswazowski]

You are such a king, love your videos. Thank you for inspiring me on my Maths' journey!

[coffeeconfessor]

I've learned with super hard theory books that if I make something more mentally "crunchy" by drawing pictures of what things are saying or finding a way to model the topic I'm working on. It makes the material feel less like slogging through mental pudding. I used my collection of dice (I'm a dice goblin) to learn how groups worked (this only works for the Euclidean plane, but it's a good start to be able to understand the groups). Currently trying to figure out how to model a Fuchsian group in the hyperbolic plane. 🙃

[kapoioBCS]

Functional analysis was both my nightmare and an amazing journey at the same time when I took the course this year. We study mainly for the book Einsieler and Ward. Fantastic subject but very scary when it comes to examinations, homework and generally evaluations with deadlines 😅

[callmedeno]

Also, I just realised you have nearly half a million subscribers! Early congrats! Plus, it is great to think that there are so many people out there tuned in. I imagine a good portion are people like me who discovered mathematics later in life after really believing they were too dumb for it. It's such a positive experience to let go of that bs and just go down an open highway in a new direction

[argonwheatbelly]

Lighting candles and making a circle of salt helps. That's not merely a definition, but an incantation. 🧙‍♂️ I tend to use '|' for "such that, " and ', ' for "with." But that's me. 😊 Excellent video!

[nucreation]

I think it's more than just writing it down. When I come across a new definition, I have to pause and reflect on it for a good long while before I feel comfortable moving on. For example with the definition you used in the video of a "point absorbing" set, I would try to find examples that fit that definition in R2 (eg. ask myself: do any open balls satisfy this definition?). Then once I found a few examples, I would try to make a conjecture about these examples and try to break my understanding by looking for counterexamples (eg. ask myself: If U is point absorbing and an open ball in R2, must it be centered at the origin?) Then I would try to find other examples and so on. Also, if it's a definition I can represent with a drawing, I try to do that because I prefer visuals.

[InoceramusGigas]

Day 71 of an hour of math a day. Hard to believe I haven't missed one yet! A combination of your videos, and Dr Lex fridman outlining his protocol for acquiring any skill, are what motivated me to begin my challenge! Looking forward to day 100, 200 and 365! Thank you, math sorcerer and friends!

[ryanang]

Would highly appreciate if you could do more videos on graduate/advanced undergraduate level math from time to time, be it book reviews, motivational videos or generic etc. Certainly helps graduate students like me and many more in the future. Keep em rolling!

Speaking of functional analysis, my lecturer based the course off a combination of Reed/Simon and Stein/Shakarchi. Was by far the most fun course in analysis at least for me thus far. Kreyszig is indeed a gentler introduction in my opinion. But the impression is that research in this area seem niche to applications in quantum mechanics (rather unsurprisingly since functional analysis arguably grew out of quantum mechanics). Research in other areas seem to be much more happening in comparison

[skateforlife]

I an a chemistry PhD student trying to get better at math ( for quantum mechanics lol!) - your videos are so helpful and encouraging!

[swordofstrife]

Thanks for this! I'm on my work term/break from University rn and so am planning on self-studying Lee's Smooth Manifolds (I eventually want to go on to study differential topology). This was very informative :)

Planning on 2 hours a day, I think that's realistic

[satyavivekanandbattula]

Thank you very much sir. Useful video.

[ThinkForYourself]

I'm in online college, and I'm learning math from a book like this. No help, no assignments with walk-throughs that aren't in the book. They also need to teach you how to learn from a book, but don't. So, you just have to figure out how to teach yourself very complex ideas with the benefit of scholarly discussion, and it isn't easy. Many people drop out of the science courses because they can't learn the math. I've been struggling to keep my 3.9 GPA as I reach the midpoint of my sophomore year. I NEED this lesson.

[BorisTreukhov]

When I first saw this definition (I believe, it was W.Rudin book) this balanced set did not make any sense, it took me several minutes to realize that the absolute value or modulus of alpha is the key, so it's balanced in this sense for example if alpha is say "0.1" than "- 0.1" times element also exist and they "balance" each other around zero element. Also not every symmetric shape count for example not all shuriken knife shapes will work, only the star shaped because you take scaling into consideration(if the set includes +/-0.1 times element it should include +/- 0.01 +/-0.001 etc). When I got it(this exactly same definition) I started think to how important are small details in the definitions and how easy to miss them. Thank you very much for this inspiring video it's time for me to finally return to Rudin and try to read it once again.

[MiketheCoder]

If there are no solutions in the textbook, then how do we check our work if we do the exercises?

[deanmiller]

I have seen those definitions before while studying functional analysis. The main idea of an absorbing set is that you are able to scale vectors so that if you are scaling a given vector by a small enough quantity, the scaled version of that vector ends up in that set. For example, a ball around the origin on the plane has this property. If you scale the vector down enough so that it is close to the origin, it will be inside the ball. This provides some intuition as to how an absorbing set behaves.

Later on, sets which are absorbing, balanced and convex play an important role in creating objects such as seminorms and sublinear functionals on that vector space, which provide some notion of measuring distance on that vector space. The connection between absorbing, balanced, convex sets and seminorms is an important one which plays a significant role in developing the theory of locally convex spaces. Locally convex spaces provide a setting to study distributions, which are frequently used in the study of partial differential equations. So although this concept is not typically used that much, it is an important building block in developing a good setting to study distributions for partial differential equations!

[TDG]

I have actually seen those type of definitions, and understood the symbols you wrote for "for all", "exists", etc. I learned those in my math courses. Good times. 😄