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.
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.
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. 🙃
I was going to say this, that is my method of abbreviation (moreso than math sorcerer's logic notation). What's weird is the simple act of say drawing a box around big U big X and a little x , an arrow back to U, just taking each element of the statement with circles around it and connecting them in some way makes me digest the sentence much easier., I think it's the borders around things that does it. It also kind of lays bare just how fundamental / broad the statement actually is. Sometimes when you are simply reading lines of text your mind kind of fills in all these extraneous ideas and expectations, but when I draw it out with these simple shapes around objects it makes it much more obvious that say they REALLY DO just mean 'any set x', just a bag of elements, which is obvious maybe to experienced math people. I find any time I use text abbreviations it doesn't have the same impact, I'm glad to hear others are similar
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 😅
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.
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
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!
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
Read 5% of the book, then go back to the beginning and start again. Read 10% of the book, then go back to the beginning and start again. Read 15% of the book, then go back to the beginning and start again. Read 20% of the book, then go back to the beginning and start again. Etc. It takes a long time, and you can skim things you're sure you're 100% familiar with, but its sure to help you master a book. And of course do all the problems!
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!
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!
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.
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.
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
I’m starting on his Introduction to Topological Manifolds. Smooth Manifolds is on my list. I’m aiming to get really good at Algebraic Topology and Differential Geometry.
Of all these old "Dover Books", I think Advanced Trigonometry by Durell and Robson is hardest to follow. Just soul crushing. "Here's two sentences on the material and now do these 135 exercises..."
Always try writing down your solutions to problems and *come back to them.* Reviewing your solutions, in case of a book with no solutions, will allow you to proof-check them. More importantly, it will provide you much-needed insights pertaining to what you studied earlier.
If memory serves, saw most of these definitions in 2nd trimester of 1st year of grad school, Functional Analysis at Berkeley in 1975 taught by the brilliant and very nice guy Professor Paul Chernoff.
my opinion is there should be an organization which checks maths books and checks if they are well written. If they are , they would get a badge. I hate the fact that there are so many bad-written books that i have to go through in order to find a good one. Thank god channels that review books (like yours) exists
en mi opinión, copiar lo que está el libro no es siempre suficiente, pues realmente tenemos la misma información en en libro ( but I know that it can help us to go more slowly and focus better on every part of the definition). Lo que yo he hecho (no sé si correctamente) ha sido tratar de visualizar qué significa la definición. He hecho un dibujo del espacio X y de su subconjunto U. He dibujado un punto que es el vector cero, el cual debe estar dentro de U y de X. He dibujado más cosas... lo que está claro es que para poder dibujar o visualizar o entender, hay primero que comprender bien qué es un espacio vectorial ¿no? Porque para entender bien una definición, necesitamos comprender bien los objetos matemáticos que entran en la definición.
What I got was, "Paris is beautiful, here's how to build an airplane." I keep coming back to this channel because so many people have told me Paris is beautiful, and I really want to get there. Or at least close enough to get a sense of it's beauty.
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. 😄
You'd be surprised. It's a first for me, but quite intuitive. The symbols were precisely why I never looked for direction in textbooks. I followed professors lecture notes and just worked textbook examples and problems. Now that I've seen this video it's made me think maybe I should give studying from those textbooks another shot learning the notation en route.
I wish there was a You Tube site where people would post old exams from different colleges.I study maths on my own, and I am not truly convinced that end-of-section textbook problems are sufficient. I'm only at the Calculus stage, and it would be nice to see ,for example, how an exam on sequences and series written by various different college professors would compare to the the material in the textbook.I get the feeling I would realize very quickly that I didn't know as much as I thought I did!
In response to your question, I have seen stuff like that. I am studying linear algebra on my own and Halmos looks like that to me. All higher math looks like this to me actually. This is my second pass at linear algebra. The first pass was a great survey of the subject with a 40 hour online Udemy class, which involved some light proofs but mostly computation and concepts. On my second pass, I am digging more into application/problem solving and proofs. Reading 'Finite Dimensional Vector Spaces' is my goal. But for me it is dense. I am not yet accustomed to proof based math, and the notation is still not second nature to me.
I came across these definitions studying locally convex spaces and the strict inductive limit topology of test functions in a course on distribution theory, I'm pretty sure Rudin's real and complex analysis has these definitions
hii math sorcerer. first of all thanks for making such amazing helpful content for us. i am from india. when i was in my 9th and 10th class, i used to memorise the math problems solutions and could do only simillar pattern of questions. In short my math was not as good as others. but your videos changed my mindset towards math. i started to understand math deeply then devloped my brain to do some of the problems by my own. i am aspiring a engineering entrance [JEE] . The math problems of jee is quite different from the standard maths books available in market. Actually my problem is sometimes i am unable to understand a question and how to start the solution. the questions seems little bit complex. so can suggest me any tips to develop my problem solving skills. My maths background is not so strong and learning maths deeply takes a lot of time, our syllabus is literaly very huge. Learning maths deeply is interesting but when i see the syllabus, i get a headech. i have to finish the syllabus with in time. so plz suggest me some tips that i can gain problem solving skills in limited time. A lot of people suffer in this problem in india , so plz tell where i am lacking? if you read my comment till now, thanks.
hey sorcerer, could you do a review on a good differential geometry/tensor calculus book? I’ve been trying to learn these subjects, but its tough to get a good introduction. thank!!
@@abdalrahmanmuni5560 no engineers don't need abstract algebra to that extent, engineers use linear algebra. Electrical, mechanical, civil, computer science engineers need linear algebra in there work.
@@abdalrahmanmuni5560 I am also mechanical engineer I study pure maths for fun and to acquire knowledge. Engineers should study partial differential equations, that will prove to be beneficial for you. Well which degree do you have?
According to Sheldon Axler “if you zip through a page in less than an hour you’re probably going too fast”. I think it’s a bit of a hyperbole, if you had to spend 1 hour in every page + working the exercises, there would be no physical time when you have 3-4 courses in a semester! I prefer spending more time working on exercises, that usually leads you to re-read parts where you’re missing out. Let’s not forget! What’s written in a text is there to be applied in problems, not to be spelled out.
When reading a definition, I come up with a visualization. I imagine like all possible solid "circle"s of numbers of a center x with the radius alpha, that is up to r. Thanks 3blue1brown for training my intuition to allow this. I haven't read adv undergrad pure math yet, given that I need to be extra careful when it comes to using intuition, but it does help.
Dearest Sorcerer, I love functional analysis. I cannot believe there is a section in this Excellent book on Hilbert spaces. Hilbert was the first person to come up with functions (Hilbert functions), series (Hilbert series), and polynomials (Hilbert polynomials). Those are my three favorite topics in Calculus 2 that I am taking right now. I just love reading math books and seeing names of famous mathematicians from a long time ago. It just gives me this warm and fuzzy feeling inside, like I am one with them for that moment only. It is unreal. Do you feel the same way? I mean, it's even cooler when it's the guy that invented functions. Peace and love. Thanks for all the laughs. Yours truly, LP
Hilbert was the first to come up with functions, series and polynomials? Certainly these concepts were know long before Hilbert! The particular item known as _Hilbert series_ or _Hilbert polynomials_ are subjects of advanced algebra, and should not appear in Calculus 2.
I always thought symbol or no symbol, what’s the difference. Why can’t I concentrate. Very often I get caught up on notations. I think it’s psychological.
oh god functional analysis is so incredibly boring, even though it has more applications than most higher level maths beyond analysis, stochastics and linear algebra. I just cant imagine finding it more interesting than topology or algebra.
Sir my question is, I want to learn science and maths. But in what sequence should I go, i mean what are the basic topics that i should start with specially in maths.
have you done calculus? if not you should probably start there. after calc 1 you could start to learn linear algebra. I would advise choosing a more pure math/proof based course for linear algebra though, you will probably see if you like pure maths then or if you prefer the engineering side with more calculations and less proofs. After that you can basically learn every topic you want.
Hey math sorcerer can you review functional analysis by Peter lax and infinite dimensional analysis by charamlobus d aliprantis as well as functional analysis by erdogan suhubi. These books 📚 can be treasured in your library 😁
Will propse the following book, which helped me understand set theory among other key things back in college: Elements of Set Theory. Herbert B. Enderton
I was doing this my last year in math. There was abstract algerbra, non euclidian geometry, etc... can spend 2hrs on a page just decifering what they are saying. No solutions of course. As a proof and that is all there are, can run into page or 2. Authors and pubishers dont want all that extra work and expense of publishing soutions.
Hey Math Sorcerer😸 Could you possibly do an overview of what math people are required to do/what courses to take in Architecture degree programs and how proficient one is required to be as an Architecture major compared to a Math major? How would you rate the difficulty of the degree math-wise for somebody more attracted to the creative side of Architecture as a field? Thanks a lot. Love your content. 🤗
I have enough math background to immediately understand what you wrote on the paper, yet I still have no idea what the book definition is on about. LOL
@Richard Farmbrough Thank you for trying to explain it. You know it's a hard book when you instantly go mentally completely blank at the very first page. :)
What about just ask ChatGPT what it thinks about the paragraph? and to explain it further? this sounds like a good space where parametric books could be awesome...
Your videos are very helpful, thanks a lot for this! 😊 I want to ask question . I belongs to India and my one year college fee is 37 dollars. Springer books are so costly that I can't afford to buy Springer books and most standard text in mathematics are published by Springer. Is there any way that I can buy springer's (used and paperback ) books around 20 dollars ? Please reply !
people here talking about their experiences with functional analysis and advanced calculus meanwhile im just here trying to decode halmos' naive set theory. 100ish pages only and yet im reading at a pace like it's 1000 pages shit is weird man. once it gets to complements and whatnot i'm lost 👍 luckily i somehow found some solution manual to the exercises (never sure if there was an official one) made by some dude on github for whatevere reason. thanks george mplikas
Most calculus books have decent problems, however here is one that is free and in the public domain, and it has answers to all of the problems. archive.org/details/cu31924031254042/mode/2up
Sir i need a book of an elementary introduction to number theory by calvin t. Long , but i have no money so that i can buy it can you help , please sir , i want to study number theory through this book
You just wrote the definition in a different way but you didn't explain anything.. The one who didn't understand it from the book directly still didn't understand it after you wrote it on paper. The difficulty is understanding, grasping and visualising what these definitions or formulas mean. I don't see how what you did is helpful in any way?
To be honest, I am a tiny bit disappointed with this video. I hoped to get a bit more of practical advice from somebody, who is so experienced with reading really hard math books. Though, what do I know? I am not a mathematician. In my personal (rather tiny compared to video author's) experience writing out definitions and theorem statements in symbols before doing a proof is immensely useful. But not so much for my understanding. May be that is why I think coming up with examples should be rather helpful for interpreting hard math definitions in a way that could be useful for both interpreting all of those definitions and theorems and may be even guide one's intuition when thinking on a hard proof. For example, if we consider X=R^2 as our vector space over R. If we consider definition of a point- absorbing subset U, then It seems to me that any set U, that contains a neighborhood of the origin should qualify. I feel like notion of a "Balanced subset" is not as strong. For example, in R^2 set U={(x,y) in R^2 : |x|
To make my point a bit more precise, it is easily possible to know definitions and theorem statements by heart and have some proof-writing skill to be able to prove a range of moderately-complex statements and yet have little idea what those statements actually mean. Intuitive understanding is at least equally important as knowledge of definitions and ability to write proofs. And one gains intuition ( I believe) by re-interpreting complex definitions, coming up with examples that are easy to comprehend and imagine. I would imagine, that for anybody, who is trying to understand some hard mathematics for sake of its applications, it is even more important to understand intuitively what each statement means.
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.
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.
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. 🙃
I was going to say this, that is my method of abbreviation (moreso than math sorcerer's logic notation). What's weird is the simple act of say drawing a box around big U big X and a little x , an arrow back to U, just taking each element of the statement with circles around it and connecting them in some way makes me digest the sentence much easier., I think it's the borders around things that does it. It also kind of lays bare just how fundamental / broad the statement actually is. Sometimes when you are simply reading lines of text your mind kind of fills in all these extraneous ideas and expectations, but when I draw it out with these simple shapes around objects it makes it much more obvious that say they REALLY DO just mean 'any set x', just a bag of elements, which is obvious maybe to experienced math people. I find any time I use text abbreviations it doesn't have the same impact, I'm glad to hear others are similar
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 😅
We appreciate your videos. Happy Holidays. God bless you.
You are such a king, love your videos. Thank you for inspiring me on my Maths' journey!
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.
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
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!
Thank you so much for the motivation!
keep it up
Don't let anything get in the way of your math grind, keep it up.
Nice.
Do your best!
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
Read 5% of the book, then go back to the beginning and start again.
Read 10% of the book, then go back to the beginning and start again.
Read 15% of the book, then go back to the beginning and start again.
Read 20% of the book, then go back to the beginning and start again. Etc.
It takes a long time, and you can skim things you're sure you're 100% familiar with, but its sure to help you master a book. And of course do all the problems!
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!
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!
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.
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.
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
I’m starting on his Introduction to Topological Manifolds. Smooth Manifolds is on my list. I’m aiming to get really good at Algebraic Topology and Differential Geometry.
If there are no solutions in the textbook, then how do we check our work if we do the exercises?
I an a chemistry PhD student trying to get better at math ( for quantum mechanics lol!) - your videos are so helpful and encouraging!
Of all these old "Dover Books", I think Advanced Trigonometry by Durell and Robson is hardest to follow. Just soul crushing. "Here's two sentences on the material and now do these 135 exercises..."
Always try writing down your solutions to problems and *come back to them.* Reviewing your solutions, in case of a book with no solutions, will allow you to proof-check them. More importantly, it will provide you much-needed insights pertaining to what you studied earlier.
It might seem like a waste of time at first but when you lose a book you’ll understand the pain of lost knowledge.
If memory serves, saw most of these definitions in 2nd trimester of 1st year of grad school, Functional Analysis at Berkeley in 1975 taught by the brilliant and very nice guy Professor Paul Chernoff.
my opinion is there should be an organization which checks maths books and checks if they are well written. If they are , they would get a badge. I hate the fact that there are so many bad-written books that i have to go through in order to find a good one. Thank god channels that review books (like yours) exists
Oh that would be so interesting! What an amazing idea:)
At least someone should grade the difficulty of exercises, like Donald Knuth does.
en mi opinión, copiar lo que está el libro no es siempre suficiente, pues realmente tenemos la misma información en en libro ( but I know that it can help us to go more slowly and focus better on every part of the definition).
Lo que yo he hecho (no sé si correctamente) ha sido tratar de visualizar qué significa la definición. He hecho un dibujo del espacio X y de su subconjunto U. He dibujado un punto que es el vector cero, el cual debe estar dentro de U y de X. He dibujado más cosas... lo que está claro es que para poder dibujar o visualizar o entender, hay primero que comprender bien qué es un espacio vectorial ¿no? Porque para entender bien una definición, necesitamos comprender bien los objetos matemáticos que entran en la definición.
What I got was, "Paris is beautiful, here's how to build an airplane."
I keep coming back to this channel because so many people have told me Paris is beautiful, and I really want to get there. Or at least close enough to get a sense of it's beauty.
"Introductory functional analysis with applications" by Kreyszig. If you want to learn.
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. 😄
those are really common symbols. I should hope you know what they are
You'd be surprised. It's a first for me, but quite intuitive. The symbols were precisely why I never looked for direction in textbooks. I followed professors lecture notes and just worked textbook examples and problems. Now that I've seen this video it's made me think maybe I should give studying from those textbooks another shot learning the notation en route.
I wish there was a You Tube site where people would post old exams from different colleges.I study maths on my own, and I am not truly convinced that end-of-section textbook problems are sufficient. I'm only at the Calculus stage, and it would be nice to see ,for example, how an exam on sequences and series written by various different college professors would compare to the the material in the textbook.I get the feeling I would realize very quickly that I didn't know as much as I thought I did!
I also find quantification really useful when trying to gulp a definition down, it's just more concise
In response to your question, I have seen stuff like that. I am studying linear algebra on my own and Halmos looks like that to me. All higher math looks like this to me actually. This is my second pass at linear algebra. The first pass was a great survey of the subject with a 40 hour online Udemy class, which involved some light proofs but mostly computation and concepts. On my second pass, I am digging more into application/problem solving and proofs. Reading 'Finite Dimensional Vector Spaces' is my goal. But for me it is dense. I am not yet accustomed to proof based math, and the notation is still not second nature to me.
Thank you very much sir. Useful video.
"Reading without a pencil is daydreaming." (From an unknown math book.)
Oh dear, starting with topology in functional analysis. It took me a semester to feel comfortable with topology
would love more videos like this
I think I might even do some maths on Christmas Eve.
A sentence past-me would never have even imagined this time last year...
Lots of love and respect. Thanks for the review and information. Peace!
I came across these definitions studying locally convex spaces and the strict inductive limit topology of test functions in a course on distribution theory, I'm pretty sure Rudin's real and complex analysis has these definitions
By 6:30 I am usually reduced to trying to draw a picture faintly resembling a flowchart.
(:
Perfect timing, thanks MS!
hii math sorcerer. first of all thanks for making such amazing helpful content for us. i am from india. when i was in my 9th and 10th class, i used to memorise the math problems solutions and could do only simillar pattern of questions. In short my math was not as good as others. but your videos changed my mindset towards math. i started to understand math deeply then devloped my brain to do some of the problems by my own.
i am aspiring a engineering entrance [JEE] . The math problems of jee is quite different from the standard maths books available in market. Actually my problem is sometimes i am unable to understand a question and how to start the solution. the questions seems little bit complex. so can suggest me any tips to develop my problem solving skills. My maths background is not so strong and learning maths deeply takes a lot of time, our syllabus is literaly very huge. Learning maths deeply is interesting but when i see the syllabus, i get a headech. i have to finish the syllabus with in time. so plz suggest me some tips that i can gain problem solving skills in limited time. A lot of people suffer in this problem in india , so plz tell where i am lacking? if you read my comment till now, thanks.
hey sorcerer, could you do a review on a good differential geometry/tensor calculus book? I’ve been trying to learn these subjects, but its tough to get a good introduction. thank!!
Yes, this is one difficult part of mathematics. The other different part is the Abstract Algebra.
Is abstract algebra harder than calculus ?
@@abdalrahmanmuni5560 it can be
@@Fekuchand_ do engineers have to do abstract algebra ?
@@abdalrahmanmuni5560 no engineers don't need abstract algebra to that extent, engineers use linear algebra. Electrical, mechanical, civil, computer science engineers need linear algebra in there work.
@@abdalrahmanmuni5560 I am also mechanical engineer I study pure maths for fun and to acquire knowledge. Engineers should study partial differential equations, that will prove to be beneficial for you. Well which degree do you have?
According to Sheldon Axler “if you zip through a page in less than an hour you’re probably going too fast”. I think it’s a bit of a hyperbole, if you had to spend 1 hour in every page + working the exercises, there would be no physical time when you have 3-4 courses in a semester! I prefer spending more time working on exercises, that usually leads you to re-read parts where you’re missing out. Let’s not forget! What’s written in a text is there to be applied in problems, not to be spelled out.
When reading a definition, I come up with a visualization. I imagine like all possible solid "circle"s of numbers of a center x with the radius alpha, that is up to r.
Thanks 3blue1brown for training my intuition to allow this.
I haven't read adv undergrad pure math yet, given that I need to be extra careful when it comes to using intuition, but it does help.
Dearest Sorcerer,
I love functional analysis. I cannot believe there is a section in this Excellent book on Hilbert spaces. Hilbert was the first person to come up with functions (Hilbert functions), series (Hilbert series), and polynomials (Hilbert polynomials). Those are my three favorite topics in Calculus 2 that I am taking right now. I just love reading math books and seeing names of famous mathematicians from a long time ago. It just gives me this warm and fuzzy feeling inside, like I am one with them for that moment only. It is unreal. Do you feel the same way? I mean, it's even cooler when it's the guy that invented functions.
Peace and love. Thanks for all the laughs.
Yours truly,
LP
Hilbert was the first to come up with functions, series and polynomials? Certainly these concepts were know long before Hilbert!
The particular item known as _Hilbert series_ or _Hilbert polynomials_ are subjects of advanced algebra, and should not appear in Calculus 2.
I always thought symbol or no symbol, what’s the difference. Why can’t I concentrate. Very often I get caught up on notations. I think it’s psychological.
oh god functional analysis is so incredibly boring, even though it has more applications than most higher level maths beyond analysis, stochastics and linear algebra. I just cant imagine finding it more interesting than topology or algebra.
Perhaps dynamical systems would be an area of math that is high level, widely applied, and interesting in its own right.
You can read the whole book, do all the demonstrations, do all the exercises and yet do not understand what this subject is all about.
Sir my question is, I want to learn science and maths. But in what sequence should I go, i mean what are the basic topics that i should start with specially in maths.
Up
have you done calculus? if not you should probably start there. after calc 1 you could start to learn linear algebra. I would advise choosing a more pure math/proof based course for linear algebra though, you will probably see if you like pure maths then or if you prefer the engineering side with more calculations and less proofs. After that you can basically learn every topic you want.
Calculus and physics
An Introduction to Functional Analysis by Robinson, please review it
Hey math sorcerer can you review functional analysis by Peter lax and infinite dimensional analysis by charamlobus d aliprantis as well as functional analysis by erdogan suhubi. These books 📚 can be treasured in your library 😁
Will propse the following book, which helped me understand set theory among other key things back in college: Elements of Set Theory. Herbert B. Enderton
Precocious = mature for one's age; unusually well-developed
Sorry the comment is off topic. But is calculus required before Linear Algebra?
What are the applications?
Hello sir !!! Could you please tell me a good book to practice recursion (basic to intermediate )??
I was doing this my last year in math. There was abstract algerbra, non euclidian geometry, etc... can spend 2hrs on a page just decifering what they are saying. No solutions of course. As a proof and that is all there are, can run into page or 2. Authors and pubishers dont want all that extra work and expense of publishing soutions.
Hey Math Sorcerer😸
Could you possibly do an overview of what math people are required to do/what courses to take in Architecture degree programs and how proficient one is required to be as an Architecture major compared to a Math major? How would you rate the difficulty of the degree math-wise for somebody more attracted to the creative side of Architecture as a field?
Thanks a lot. Love your content. 🤗
I have enough math background to immediately understand what you wrote on the paper, yet I still have no idea what the book definition is on about. LOL
@Richard Farmbrough Thank you for trying to explain it. You know it's a hard book when you instantly go mentally completely blank at the very first page. :)
Any recommendations for dense papers?
Greetings Mr.Professor,
Thoughts :
- Rare, Interesting Engineering Mathematics Textbook in Advanced, Applied Level.
Take Care,Professor. Thanks..
With regards,
RanjithJoseph (R.J)
What about just ask ChatGPT what it thinks about the paragraph? and to explain it further? this sounds like a good space where parametric books could be awesome...
Your videos are very helpful, thanks a lot for this! 😊
I want to ask question . I belongs to India and my one year college fee is 37 dollars. Springer books are so costly that I can't afford to buy Springer books and most standard text in mathematics are published by Springer.
Is there any way that I can buy springer's (used and paperback ) books around 20 dollars ?
Please reply !
Get them online for free from ZLibrary or other book pirate websites
@@Omar_Ebrahim thank you very much !
But this is last option .
Hmmm I don't really know. Springer books tend to be fairly expensive I think. Some are cheaper, but many do cost more it seems.
@@TheMathSorcerer ok sir, thank you !
Actually sir I am not obsessed with springer but I love reading Conway, Lang , atiyah et cetera , and having a original copy gives a feel good factor.
writing simplified definition.... that's what i do too! :)
people here talking about their experiences with functional analysis and advanced calculus meanwhile im just here trying to decode halmos' naive set theory. 100ish pages only and yet im reading at a pace like it's 1000 pages shit is weird man. once it gets to complements and whatnot i'm lost 👍
luckily i somehow found some solution manual to the exercises (never sure if there was an official one) made by some dude on github for whatevere reason. thanks george mplikas
Am I the only one that tried to remove from the screen the small dot on the paper at 2:32? 😄
Another same size Functional Analysis book - Steven G. Krantz “A Guide to Functional Analysis”.
Man, I think we could be true friends in real life, I also have a fetish for vintage math books😜
SL Loney plane trigonometry
Hola hechicero buen libro el mostraste
for me, the first requirement is coffee
Hi math sorcerer :)
Hi😊
these symbols are there to reduce the burden of wording. imagine discriminant represented by Delta (∆ )
I learn by doing
:)
Sir,could you plzz suggest best practice book for differentiation.
Most calculus books have decent problems, however here is one that is free and in the public domain, and it has answers to all of the problems.
archive.org/details/cu31924031254042/mode/2up
@@TheMathSorcerer thank u sir
Wait! You’re not going to sniff it?????
Sir i need a book of an elementary introduction to number theory by calvin t. Long , but i have no money so that i can buy it can you help , please sir , i want to study number theory through this book
Libgen
Look at PDFs online
Bro , i didn't find this book on any website please can you send me its pdf ,i can give you my WhatsApp number
Is this real analysis?
A mixture of real analysis, topology and linear algebra
You just wrote the definition in a different way but you didn't explain anything.. The one who didn't understand it from the book directly still didn't understand it after you wrote it on paper. The difficulty is understanding, grasping and visualising what these definitions or formulas mean. I don't see how what you did is helpful in any way?
First🎉
1st like
1st view
You cant read if you havent learnt the pre-requisites
To be honest, I am a tiny bit disappointed with this video. I hoped to get a bit more of practical advice from somebody, who is so experienced with reading really hard math books. Though, what do I know? I am not a mathematician. In my personal (rather tiny compared to video author's) experience writing out definitions and theorem statements in symbols before doing a proof is immensely useful. But not so much for my understanding.
May be that is why I think coming up with examples should be rather helpful for interpreting hard math definitions in a way that could be useful for both interpreting all of those definitions and theorems and may be even guide one's intuition when thinking on a hard proof.
For example, if we consider X=R^2 as our vector space over R.
If we consider definition of a point- absorbing subset U, then It seems to me that any set U, that contains a neighborhood of the origin should qualify.
I feel like notion of a "Balanced subset" is not as strong. For example, in R^2 set U={(x,y) in R^2 : |x|
To make my point a bit more precise, it is easily possible to know definitions and theorem statements by heart and have some proof-writing skill to be able to prove
a range of moderately-complex statements and yet have little idea what those statements actually mean.
Intuitive understanding is at least equally important as knowledge of definitions and ability to write proofs. And one gains intuition ( I believe) by re-interpreting complex definitions, coming up with examples that are easy to comprehend and imagine.
I would imagine, that for anybody, who is trying to understand some hard mathematics for sake of its applications, it is even more important to understand intuitively what each statement means.
What is functional analysis? complex analysis? real analysis?
alien languages.