Compact sets are my go to example when I'm explaining to math students that math is all about getting the "right" definitions. Definitions can turn someone's PhD dissertation into an undergraduate homework exercise. As usual you somehow manage to explain it so well in such a short video. Great work man.
This video is great. Your reference to Hausdorff spaces was, in my opinion, brilliant. I had one of my mayor WTF moments when I learn that. I wish my teacher made more of those kind of comments, so one could learn which properties are "more fundamental".
I'm glad you liked it! Yeah, when I was reading through Rudin's proof, it struck me that this isn't how I'd prove this in a metric space (I was going to include that but the audio was really bad). When I thought about it, I realized all we were doing was separating points with open neighborhoods, and there is exactly a space defined to do that.
It's a great day when you see this channel has uploaded, I see the algorithm has blessed this video more than the last hopefully it continues to do that for future ones this channel really deserves more attention
I'm glad you like it! Hopefully, this trend will continue. It's half craft, and half luck. We will see where it goes. Truly, I'm happy to have you guys here at all. This is some esoteric stuff!
I’m glad it’s helping you! It’s a series I’ve been wanting to do for a while now. Have a good excuse, since I’m teaching Intro Analysis 1 this semester. Tell your friends! Lol
I'm glad you liked it! I'm glad the animations came through ok. It's what goes through my mind when I think about these things, and so it's not always clear what will communicate well.
I think that at 4:10 it refers to [0,1) as a topological subspaces of the reals with the induced topology, ergo the whole space [0,1) is a clopen set, so in particular it is closed. Is it plausible?
I'm like 90% sure that you are talking about how I speak with my hands. But if you are speaking about "handwaving" in a mathematical sense, then this is a hilarious dig on a mathematician.
Hello I have a question : Im a math undergrad and I saw those analysis concepts in my analysis 1 course, so it seems to be introductory stuff. Can you tell me how the concept of compact sets is used by professional mathematicians like you in your work ?
Compact sets are at the core of things like functional analysis and measure theory. The critical feature that they exhibit is that every continuous function attains its maximum on that set. This means that we can define a metric to talk about the distance between functions by taking the absolute value of their difference and looking at the maximum of that function. Once you have a metric on functions, that leads to all sorts of interesting questions. Can you take derivatives of functions of functions (this is called a variation), and this comes up in optimal control theory. You can talk about how to minimize the difference between an approximation and a function you are approximating. We leverage concepts like orthogonal polynomials or even Taylor series, depending how strict we are about it. This is the field of approximation theory, and it's also the field that machine learning is based on. The existence of a maximum value over a compact set is also critical to optimization theory, where it essentially justifies the search for maximum values over compact domains. In the study of fractals, we actually define a distance on compact sets themselves. And questions about iterations of operations on these sets are very interesting, and lead to questions about what dimension means, where you can sometimes have non integer dimensions for some shapes. All of this is built upon the concept of compactness and compact sets. Does that answer your question?
Can you do some videos on real variable harmonic analysis? Something at the level of Stein's monograph on real variable methods, orthogonality, and oscillatory integrals?
no the video is very good, I was just casually watching then I found myself having to think pretty hard and rewatch after 6:40. But this is definitely necessary in any explanation of a complex topic. I have not yet taken real analysis or topology, so some of this video's content is new to me. @@JoelRosenfeld
It helps keep retention a good deal. There is a distinct difference in how long the audience sticks around when I have the music and when I do not. I have toned it down considerably in the more recent videos though, as a compromise to some who do not like it.
Surface(cos(u/2)cos(v/2),cos(u/2)sin(v/2),sin(u)/2),u,0,2pi,v,0,4pi The radially symmetric Klein bottle. Notice the 4pi needed to complete the surface. This is half a sphere caught over its own phase inversion. Being a single sided form once you so the outside then you do the inside. Like renormalization. Once you pass infinite you are on the other side but never changed direction exactly. 🖖 Time is a compactified dimension one single Planck second in size Neutron decay cosmology. Inevitable.
@@JoelRosenfeld I seriously cannot wait for that! I often think false-starts are at least as interesting as the correct result. They give a much better sense of the finer-grained steps that one should (in a psychological sense) be doing, both by attempting things and then realizing what part was wrong and then realizing how to fix it. Super pumped about this idea!
@@JoelRosenfeld some may find your explanation great for them, so this is only my opinion. You spent quite some time commenting on chatgpt explanation on compact sets, that is not helping for me to understand the topic, and quite wasting time. When you actually got into topic, i found that it was unnecessary complicated, compared to most other videos on this topic.
@@weisanpang7173 not really concerned with taking off. I have a great career already. But I do put a lot of effort into these videos, so I am open to outside input. I appreciate your critique.
Compact sets are my go to example when I'm explaining to math students that math is all about getting the "right" definitions. Definitions can turn someone's PhD dissertation into an undergraduate homework exercise. As usual you somehow manage to explain it so well in such a short video. Great work man.
This video is great. Your reference to Hausdorff spaces was, in my opinion, brilliant. I had one of my mayor WTF moments when I learn that. I wish my teacher made more of those kind of comments, so one could learn which properties are "more fundamental".
I'm glad you liked it! Yeah, when I was reading through Rudin's proof, it struck me that this isn't how I'd prove this in a metric space (I was going to include that but the audio was really bad). When I thought about it, I realized all we were doing was separating points with open neighborhoods, and there is exactly a space defined to do that.
Sets are way cooler than the maths in my school “sets” them out to be. Thanks for the video, I never knew there was a symbol for multiple unions!
They really are. I had no idea until I started digging into it.
It's a great day when you see this channel has uploaded, I see the algorithm has blessed this video more than the last hopefully it continues to do that for future ones this channel really deserves more attention
I'm glad you like it! Hopefully, this trend will continue. It's half craft, and half luck. We will see where it goes.
Truly, I'm happy to have you guys here at all. This is some esoteric stuff!
your channel started to took off just around the time i started learning real analysis haha (lucky for me), awesome work!
I’m glad it’s helping you! It’s a series I’ve been wanting to do for a while now. Have a good excuse, since I’m teaching Intro Analysis 1 this semester.
Tell your friends! Lol
Me too lol
Great video as always, thats a lot of information contained in a few minutes. The animations also were on point!
I'm glad you liked it! I'm glad the animations came through ok. It's what goes through my mind when I think about these things, and so it's not always clear what will communicate well.
I think that at 4:10 it refers to [0,1) as a topological subspaces of the reals with the induced topology, ergo the whole space [0,1) is a clopen set, so in particular it is closed. Is it plausible?
I think it’s highly unlikely. At best it’s still misleading since it does say anything about the subspace topology being used.
Continue offering educational content
❤with all that hand-waving your step counter must go through the roof while you are sitting still in a compact and bounded subset of your space.
I'm like 90% sure that you are talking about how I speak with my hands. But if you are speaking about "handwaving" in a mathematical sense, then this is a hilarious dig on a mathematician.
Obviously your explanation about any topic is awesome.
Sir please give lecture series on Real Analysis and metric space.
Is Frechet’s definition equivalent?
Very good video. The only problem is, the background music is too offensive.
Hello I have a question : Im a math undergrad and I saw those analysis concepts in my analysis 1 course, so it seems to be introductory stuff. Can you tell me how the concept of compact sets is used by professional mathematicians like you in your work ?
Compact sets are at the core of things like functional analysis and measure theory. The critical feature that they exhibit is that every continuous function attains its maximum on that set. This means that we can define a metric to talk about the distance between functions by taking the absolute value of their difference and looking at the maximum of that function.
Once you have a metric on functions, that leads to all sorts of interesting questions. Can you take derivatives of functions of functions (this is called a variation), and this comes up in optimal control theory.
You can talk about how to minimize the difference between an approximation and a function you are approximating. We leverage concepts like orthogonal polynomials or even Taylor series, depending how strict we are about it. This is the field of approximation theory, and it's also the field that machine learning is based on.
The existence of a maximum value over a compact set is also critical to optimization theory, where it essentially justifies the search for maximum values over compact domains.
In the study of fractals, we actually define a distance on compact sets themselves. And questions about iterations of operations on these sets are very interesting, and lead to questions about what dimension means, where you can sometimes have non integer dimensions for some shapes.
All of this is built upon the concept of compactness and compact sets.
Does that answer your question?
@@JoelRosenfeld Thank you for taking the time to give me all those details ! Looking forward to your next video
@@voroldrwarfff8858 You’re welcome! Happy to have you here!
@@voroldrwarfff8858In topology, especially manifold theory, compactness provides an easier way to gain global information from local information.
Very good video, but the music doesn't fit at all😊
And also sometimes it's hard to hear you over the music
@@incredulity I make sure to cut the music at important technical parts of the video
@@JoelRosenfeld it's very distracting throughout the video. EDIT: OK I made it to the quiet bit, what a relief!
Can you do some videos on real variable harmonic analysis? Something at the level of Stein's monograph on real variable methods, orthogonality, and oscillatory integrals?
I’ll give it some thought. Might take a while to get to it
video gets really dense at 6:40 haha
Yeah this video is building on the series on real analysis. Can I help clear up anything?
no the video is very good, I was just casually watching then I found myself having to think pretty hard and rewatch after 6:40. But this is definitely necessary in any explanation of a complex topic. I have not yet taken real analysis or topology, so some of this video's content is new to me. @@JoelRosenfeld
@@tighemcasey7589 ok! I’m glad you are finding it educational! Happy to clear anything up, if you want more info :)
One could even call it *compact*
Can you turn off the lousy music? How does that help with anything? A distraction.
It helps keep retention a good deal. There is a distinct difference in how long the audience sticks around when I have the music and when I do not. I have toned it down considerably in the more recent videos though, as a compromise to some who do not like it.
Surface(cos(u/2)cos(v/2),cos(u/2)sin(v/2),sin(u)/2),u,0,2pi,v,0,4pi
The radially symmetric Klein bottle.
Notice the 4pi needed to complete the surface. This is half a sphere caught over its own phase inversion.
Being a single sided form once you so the outside then you do the inside. Like renormalization. Once you pass infinite you are on the other side but never changed direction exactly. 🖖
Time is a compactified dimension one single Planck second in size
Neutron decay cosmology. Inevitable.
"Like" isn't enough! ❤️
Cheers!
Neat bit of history I never knew!
I have a more dedicated video about Frechet coming up in the next month or so. It’s really interesting to see his false starts in defining metrics.
@@JoelRosenfeld I seriously cannot wait for that! I often think false-starts are at least as interesting as the correct result. They give a much better sense of the finer-grained steps that one should (in a psychological sense) be doing, both by attempting things and then realizing what part was wrong and then realizing how to fix it. Super pumped about this idea!
Lower the background music.
Can’t do anything for this video. More recent videos have much less music
Americans pronounce French words with a final é sound as being accented on the final é sound. Even words such as Fréchet.
Please use gpt-4 and not bare chat-gpt
When I made this video, gpt 4 was not created yet. It’s in the plans though
Worst explanation.
@@weisanpang7173 wow ok. How do you think it can be improved?
@@JoelRosenfeld some may find your explanation great for them, so this is only my opinion. You spent quite some time commenting on chatgpt explanation on compact sets, that is not helping for me to understand the topic, and quite wasting time. When you actually got into topic, i found that it was unnecessary complicated, compared to most other videos on this topic.
You responded to comment for a video that's already 1 year old, i believe your channel will take off just fine.
@@weisanpang7173 not really concerned with taking off. I have a great career already. But I do put a lot of effort into these videos, so I am open to outside input. I appreciate your critique.