And basically, I think I want to bring us back to a little bit of the philosophy why I think most or just about all of us are probably here. And I think Brian Armstrong, actually, the CEO of Coinbase, summed it up very, very well with the quote right here. And he talks about how digital currency will change the world. And he says that digital currency may be the most effective way the world has ever seen to increase economic freedom. If this happened, the implications are profound. It could lift many countries out of poverty, improve the lives of billions of people, and accelerate the pace of innovation around the world. And I want to take special emphasis on digital currencies. And we've heard a lot of other people that were involved with something that used to be a digital currency, mocked the idea of using it as a currency. But I think the people in this room are pretty excited about digital currencies, not digital stores of value that don't work very well as a store of value if they're not usable as a currency. And so let's talk a little bit about what economic freedom is, right? Economic freedom is a measure of how easy it is for members of a society to participate in the economy. Has a number of factors such as how easy it is to start a business. Can I see here, with a show of hands, how many people have started a big business based on cryptocurrencies, right? That's 80% of the room has their hand up. Who here has been paid for doing some sort of a business of some sort in cryptocurrencies? I think that's everybody here. And boy, does that make it easy. So actually, I'll tell you a little bit of a story. There was a little bit of a snag with actually paying for the venue because a traditional bank account closed the bank account that was going to be used to pay for the venue. I tried my best to get them to accept Bitcoin cash this afternoon. I succeeded in getting them to accept a personal wallet with a little tip that I think that he's gonna buy some cookies with later. But I think next year when we have the next Satoshi Vision Conference, we'll be able to pay for the entire venue in Bitcoin cash. And we might not even necessarily have a business license for it. And once again, property rights being enforced. A favorite quote that's been floating around in Bitcoin and cryptocurrency ecosystem for a while is that no amount of violence can solve a math problem. And that's kind of just a fancy way of saying that property rights are secure when it comes to Bitcoin because if you own the private keys, you own the Bitcoin, just like that. And when I say Bitcoin, of course, I'm referring to the digital currency, Bitcoin Cash. And let's talk about free trade with people in other nations. Who here had to travel from a different nation to Japan for this conference? Probably 95% of the people in this room. So thank you all for coming from around the world. Who here paid for their travel in a cryptocurrency? That's a good chunk of the room there too. Who used a credit card that's paid with digital currency? So there's another good chunk of the room there. And again, regulation of labor and business, right? I've already set up a couple of people that weren't here for the conference that are working for the building. I set them up with Bitcoin Cash wallets today, right? And when I did that, I told them that their eyes got big. I said, you didn't give your name, you didn't give your email, you didn't give your tax ID number. You didn't apply for some sort of business permission slip to do it. And against the ability of the currency, it's very important when it comes to economic freedom as well. So all these things, Bitcoin helps improve. And here's a really interesting chart. We have a list of the countries that have the most economic freedom around the world. And the ones with the least economic freedom around the world. And if you look at the ones with more economic freedom, we have places like Hong Kong, Singapore, New Zealand, Switzerland, Australia, Canada, on and on. They all sound like pretty nice places to live. And the countries with the least amount of economic freedom are North Korea, Cuba, Venezuela, Zimbabwe, and on and on, and a bunch of places that probably aren't nearly as pleasant to live as the places with more economic freedom. And we have some photos here. You can see the difference of Hong Kong from then to now and Cuba from then to now. And it's pretty apparent, right? And these are two little islands with just two different levels of economic freedom there. And so that leads us to why economic freedom is important. It leads to a higher per capita income, a higher life expectancy. Who here wants to live longer? One of the very best tools we have to make that happen is economic freedom. Who here likes to read, or at least likes to know how to read? Economic freedom helps with that as well. Who here wants to see the poorest 10% around the world have a higher standard of living? Economic freedom helps that, too. Who wants to have clean air and water, right? Everybody likes that. Economic freedom helps that, too. Who wants to see lower amounts of wars and conflicts around the world, right? Economic freedom helps that, too. Who wants to be happy, right? Put both hands up if you want to be happy. People in countries with more economic freedom are happier as well. And of course, nobody likes corruption or bribery either. And so here's some actual statistics, right? Where you have the income level of the poorest 10% around the world. The dark green bar there are the most economically free countries. And the red bar, and in order from dark green to red, it's so apparent. The more economically free a country is, the higher the income level of the poorest 10% of that country are. And again, per capita income, the more economic freedom a country has, the more income per capita. In adult literacy, the more economic freedom a country has, the more people know how to read. Life expectancy, the more economic freedom a country has, the longer people live. Unemployment, right? The more economic freedom a country has, the lower the rate of unemployment in that country. Who loves babies? Right? The more economic freedom a country has, the lower the rate of infant mortality. So if you want to see fewer babies die at childbirth, you should be supporting economic freedom.  And if you hate babies, support economic intervention by the state. And if you love babies, support free trade and economic freedom. And again, the same is true when it comes to child labor and kids working in the workforce. The more economic freedom a country has, the lower the amount of kids working in the workforce is. And it's not because a bunch of laws were passed, it's because they're able to afford not to work. And it's correlation doesn't prove causation, but you see example and example after country after country and society after society around the world, you can draw some really, really strong inferences between those things. One fantastic example that a history teacher I had in high school had is he had a picture of, said East Germany and West Germany. So you have the same ethnic culture, basically the same geography, same natural resources, just two different forms of government. And in West Germany, you had them producing porches and Mercedes and BMWs and lots of wonderful things and people had a relatively high standard of living because they had more economic freedom. And in East Germany where they had less economic freedom about the only thing they were able to produce over there was a great big giant wall to prevent people from escaping because things were so bad there. And you have example after example of that around the world. So let's talk about bitcoins, economic, the impact on economic freedom, right? It makes it easier to start a business, it enforces property rights, it promotes free trade, it enables freedom of contract, it enables people to opt out of corrupt systems, it enables all of these benefits of economic freedom that raise the standards of livings of everybody all over the planet. And so again, I'll quote Brian Armstrong, the CEO of Coinbase, one of the biggest cryptocurrency businesses in the entire world at this point. And again, all these slides that I just showed are actually taken from a blog post by Brian Armstrong. So full credit to Brian for that. I think he has his head in the right place by that. And Brian says, if we can create more economic freedom in the world it will serve as a giant economic stimulus package for the world, accelerate innovation, reduce wars, make the poorest 10% better off, overthrow corrupt governments and raise happiness. And I say bitcoin and digital currencies are the best tools the world has ever seen to accomplish these goals. So that brings us to the next question. If you were a government, how would you stop all of this from happening? Because if you look around the world that's what governments do is they encroach upon the economic freedom of the people. They retard the rate of economic growth around the world and they reduce the standard of living that people have around the world. So here's how I think that it could be done. And I'm not saying that they did this out of malice. Maybe it was just ignorance. But if you stop and think for a moment, if you wanted to stop or delay all of the benefits that are brought by economic freedom around the world that are brought by bitcoin and digital currencies, how would you do it? Well, I think you would advocate for one megabyte blocks to intentionally create high fees, less privacy, slow confirmations and unreliable transactions. And that would end up resulting in fewer users, fewer nodes, lower adoption, less privacy, a lower bitcoin market cap, a lower bitcoin market share, and less censorship resistance. And we can see bitcoin cores market share in the last year and this is about a month out of date though, but in the last year it went from about 85% to about 45% today in just one year. That's how much market share it lost. And we saw it happen. I think just about everybody in this room is aware that people were worried about the blocks becoming full and when the blocks became full, altcoins started getting all sorts of adoption around the world and bitcoin's market share plummeted. We have a clear graph here showing that happened. And we can see that there was directly some correlation there between bitcoin losing market share and altcoins gaining market share and that happened at the exact time that the blocks became full, causing all these negative effects on bitcoin. And we can see the delayed adoption around the world. Bitcoin was clearly growing and more and more people were using it around the world and more and more transactions were happening around the world until we hit that block size limit which caused the transactions to be slow, expensive and unreliable and delayed adoption of bitcoin around the world. So that brings us to what has the market done to solve this problem. Here's where we are today. The market has moved to altcoins. I think our favorite altcoin in this room is probably bitcoin cash because it stays true to the spirit of bitcoin that most of us got involved with in the first place. But there's a thousand and one competitors out there. So if we want bitcoin cash to be the winner, we have to make it the most useful of digital currency out there in the world which means fast, cheap, reliable transactions for everybody, permissionless innovation, censorship resistance, private transactions, all of these things need to be built. And again, I'm not saying that blockstream and core are intentionally a government attack on the adoption of cryptocurrencies around the world but if you stop and think about it, I can't really think of a more effective way to do it. They literally shattered what used to be this unified bitcoin ecosystem. They've now shattered it into a thousand and one altcoins around the world in fighting within the communities and people competing on this and that and all sorts of hostility. And they did it through primarily censorship and propaganda and lies. If you're in favor of a censorship resistant currency for the entire world to use to gain more economic freedom, you shouldn't be using censorship to promote it. And that's exactly what the bitcoin core supporters have been doing, which should be condemned by anybody who supports free speech. And again, that's why I'm here, supporting bitcoin cash before you because I want to use it to bring more economic freedom to the world. And whatever tool does it, I'm on board with that. I think bitcoin cash has the biggest chance of doing that because it has the bitcoin name. It has almost all the early adopters of bitcoin, all the people that made bitcoin useful as currency around the world are supporting it. It has the most stuff going on. And if you are excited about that and agree with that, tell your friends, set them up with a bitcoin wallet, tell them about bitcoin.com. I have an announcement I guess I'll make here that I've told just a few people in private, but in the next update to the bitcoin.com wallet coming very, very soon, we will be dropping bitcoin core wallet creation by default. And it will only be making bitcoin cash wallets by default. And if you're a bitcoin core supporter and upset by that, stop and ask yourself, well, if bitcoin core is just supposed to be digital gold and a store of value that you don't transact with, why the heck would you need a wallet on your phone? Bitcoin cash is to be used for buying digital beanie babies and physical beanie babies. And there's some lovely young ladies right out this door selling cookies for bitcoin cash. And so I implore you all to use bitcoin cash as cash anytime you possibly have the opportunity.  There's lots of fantastic options now. We have coin base, we have bit pay, we have more and more options around the world to allow businesses to easily accept bitcoin cash as payment. So I feel like I've given this speech and helped with this once before several years ago, but I'm not old yet. I'm gonna do it again. We're gonna do it with bitcoin cash. It's gonna become the currency of the entire world for everyone to use worldwide. But I need all of your help to do it. But the fact that all of you have traveled from around the world to be here today shows that you're excited by that too and that you're willing to do that. So let's all go spread the word. When you go back home to wherever country you came from or wherever you're going next, tell your friends, tell your family, set them up with bitcoin cash wallets, right? On their bitcoin.com wallet. Tell them don't even worry about bitcoin core. That's just a digital beanie baby thing that's going in the wrong direction. And it's a little bit frustrating. I'm sure all of us feel frustrated by that. But at the end of the day, we can wish them all good luck. We can be upset that they've tried to steal the bitcoin brand name, but anybody that's been in bitcoin for any amount of time realizes that bitcoin cash is the real bitcoin. And if they want to build some sort of fancy settlement layer, we can wish them good luck. But that's not what's described in the bitcoin white paper. And if it's not described in the bitcoin white paper, then it's not bitcoin. Bitcoin cash is bitcoin. And today there's some metrics where maybe bitcoin cash doesn't have as much of a claim to bitcoin as bitcoin core. But there's probably seven or eight metrics out there. And I think bitcoin cash has the majority of those metrics on its side.  If your user start using it, just tell your friends, tell everybody. I got a little bit too excited and went through my slide so fast. So it looks like I have about 10 minutes for questions though. I'm more than happy to answer them because what an exciting thing to do. So I guess we have such a wonderful MC. I'll open it up to you to find some questions. I noticed earlier a question or two was actually decided to be off topic earlier for an earlier speaker. I'm happy to answer any question about any topic at all. So I'll get that right out here. And even if it's about something called Bcash, I'm not really sure what that is. So maybe I won't be answering questions about that. But anything else is on topic. Well played. All right. Thank you, Roger. Let's give Roger one more hand of applause for that awesome presentation. Thank you. All right, as always, the rules of the road. Let us ask a question that is worthy of the speaker. Do we have a question? You implied that governments may have sought to disrupt Bitcoin becoming a global currency in the past. If that's true, it seems logical that they would continue to do that now. How do you think they would continue to do that in our community? So to put on my conspiracy theory hat again, one of the tactics that's used very commonly to disrupt movements is to try and splinter those movements into different competing factions or warring factions. And unfortunately, we saw that happen with cryptocurrencies. All these businesses used to be called Bitcoin businesses and Bitcoin exchanges. And now they're not called that any longer. They're called cryptocurrency exchanges and crypto coin businesses. And crypto this and crypto that. It's not just Bitcoin anymore. So they've succeeded on that front. And there's quite a few anonymous figures that seem to be really doing a good job of stirring up trouble and causing fractures in fighting. One of which who I will name by name very clearly is Cobra. I don't know exactly who this person is or where he is, but recently he's been saying some things that are much more in line with Bitcoin Cash in support of Bitcoin Cash. But now seems a strange time to do it, right? We've already had this split. He could have been advocating for stuff that makes sense before the split. And the part that really makes me go, is he's advocating for a proof of work change on Bitcoin Core at this point. And that's a guaranteed way to split Bitcoin Core again. So then you'd have two different competing warring factions of Bitcoin Core. And so if you were a government agent wanting to disrupt Bitcoin Core even more, that seems like the most effective way to do it. And you can hide the fact that you're actually up to these things. So I don't have any evidence that he's some sort of government sabotage here. But it sure, so thank you. My English has faded a little bit. For those of you that don't know, I've been living in Japan as home base for about 12 years now. And sometimes I'll type up an email and I'll read what I just wrote. And I think there is no way a native English speaker just typed that. So thank you for the English help there. But it seems awfully suspicious there and there's a lot of it going on. And then the censorship going on and the all sorts of attacks from sock puppets. And another strategy too, if you read the CIA handbook on disrupting movements is they say always accuse your opponent of doing the thing that you're doing. So you see them accusing me all the time of paying for all sorts of sock puppets and things like that. I've never, ever, ever done anything like that ever. But somebody certainly has been paying for sock puppets supporting Bitcoin Core stuff. They've been caught red handed doing it. We don't know if it's Bitcoin Core supporters paying for Bitcoin Core support via sock puppets. Or perhaps it's Bitcoin cash supporters paying for sock puppets to support Bitcoin Core as a false flag attack. We don't know. So put on double conspiracy hats there. But I'll tell you straight ahead, I've never, ever done anything like that. So it really seems strange that they accuse me of it over and over and over again. So, but at the end of the day, there's nothing, what's the saying? That there's nothing more powerful than an idea whose time has come. And they've delayed our adoption by a year or two. But this idea is coming and it's coming around the world and it's not going to be stopped. But it's our job to work quickly to make all these positive benefits that are gonna come to the world because of digital currencies and Bitcoin cash. We need to bring them to the world sooner rather than later. And I see Simon Dixon, someone with his hand up very eagerly. Maybe if someone doesn't already have the microphone, I apologize. And just for those that don't know, I'm expecting a very tough question because I know Simon's been a Bitcoin Core supporter for a while, but Simon and I have known each other for a while, so how about it, Simon? Okay, no, Roger, I feel that the strategy of optimizing for digital gold and optimizing for digital cash is a very healthy thing. And I feel that if we advocate for digital cash by saying digital gold has been disrupted, store of value has been corrupted, the end result is that people choose dollars and they say, well, that crazy crypto thing, this person's saying it's corrupted and that could happen to Bitcoin cash. And so what I'm saying is can we imagine a world of more harmony here where it's okay for one chain to optimize for digital gold and another to optimize for digital cash? And if the first thing a new person that comes to the industry sees is, this is what this chain is about, and by the way, I say the same thing at a Bitcoin Core conference as well. I've tried to remain neutral here, but I feel that a better strategy would be rather than saying this has been completely corrupted, this is a CIA, and whether that's actually happened or not, and by the way, the censorship thing is completely unforgivable. That's something that no one can support, that's not for them. But I feel that if we could return to an environment of this is trying to optimize for savings, this is trying to optimize for spending, and the two I feel are different strategies, I think we could get more adoption and less people saying that crypto stuff's crazy. So if you, thank you for the great statement. And let's get one more round of applause against censorship, right? That should be unforgivable.  And the reason the dollar is used as a store of value is because you can spend it everywhere as cash. And if you look at the history of gold, the way gold came to be used as a store of value is because it was spendable everywhere as cash. And the same is true of digital currencies. If you want them to be usable as a store of value, they have to be spendable as cash. And I think part of the reason why there's so much animosity between the Bitcoin Core Crowd and the Bitcoin Cash Crowd is because the Bitcoin Core Crowd successfully brought the Bitcoin name for the most part with them and the original Bitcoin ecosystem with them while completely deviating from the actual title of the Bitcoin White Paper. It's a peer-to-peer electronic cash system. And yet the Bitcoin Core people are mocking the very idea of using Bitcoin as cash. And so if they want to build something that's only used as a digital store of value, that's fine, have at it. But please don't hijack and sabotage and censor our project and do it through censorship and propaganda. And that's exactly what has happened. And it's really, really frustrating as somebody who's poured the last seven years of my heart and soul into this to see it and then have these same people. And I've been promoting the exact same Bitcoin for seven years. I have the exact same speech I've been giving about Bitcoin that allows anybody anywhere in the world to send and receive any amount of money with anyone anywhere instantly, basically for free. And it's impossible for your transaction to be blocked or censored or controlled by any bank or government or corporation. That's the pitch I've been giving for seven years now. It's still completely true of Bitcoin cash. It's no longer true about Bitcoin Core. And so it's not me that's changed. It's the Bitcoin Core roadmap that's completely changed to something that's not even described in the original Bitcoin White Paper. And so I wish them good luck, but they shouldn't have taken the, they shouldn't even try to take the Bitcoin name with them because it's clearly not Bitcoin. Go and read the White Paper. It's not Bitcoin. And again, back to your point though about having separate digital cash and digital store value. I don't think those things are separate, separable. If something's usable as cash and usable as payments, by default, it becomes a store value. And that's the reason people are willing to use the dollar as a store value is because they can spend them everywhere. And if Bitcoin cash becomes spendable everywhere, people will be using it as a store value as well. Hello, Peter. Hey, Roger. Thanks for everything. So what are we gonna do about these national cryptocurrencies? These like fiat crypto hybrids? I was just at Singapore at Money 2020, now these central banker types, talking about how crypto would be a great way to force everyone into negative interest rates. What are we gonna do about these fuckers? So I'd love to thank Brian Armstrong's brother for coming. At the end of the day, if you don't like it, don't use it. That's what's so exciting about cryptocurrencies. It gives us the option to use whichever ones we like. So, and there was a heckler at a Hong Kong Bitcoin Cash Conference, I don't know, six months ago or something or recently, and he was shouting something from the audience, and I told him, I said, if you don't like it, don't use it. And it's just that simple when it comes to all of these governmental issued cryptocurrencies as well, or hybrids, it's, if you don't like it, don't use it. Now we have an option, if you don't like the dollar, don't use it. If you don't like the end, don't use it.  And that's what cryptocurrencies are all about, is giving us choice. And choice is just another word for more economic freedom. The more choices you have, the more economic freedom you have. And the more economic freedom everybody has, the higher the standard of living all of us are going to have. So it's a great thing to have those choices. Hi, Roger. Can you briefly speak to the things that influenced your personal philosophy and how important do you think it is for people involved in the crypto scene to seek a consistent, non-contradictory philosophy? So I've called myself a libertarian for a while and I didn't start out that way at all. It was kind of by, I guess, good luck. In the summertime, between maybe seventh and eighth grade or eighth grade and freshman year in high school, my mother told me no more video games today. You have to do something else. So I went and looked on the bookshelf and I found a book called Socialism by Ludwig von Mises. And who here knows who von Mises is, all right? A lot of the room here. At the time, I had no idea who he was and I thought that this was a pro-socialism book. I didn't really even know what socialism was. I didn't know what capitalism was, but I knew Americans are supposed to be, you know, generally pro-capitalist and anti-socialist, but I figured I should at least hear the other side of the argument. So I picked up this book and started reading Socialism by Ludwig von Mises. And for anybody who hasn't read it, basically it points out how prices transmit information. And prices tell you what the screen should be made out of, what this jacket should be made out of, what every single good in the economy should be made out of. And it transmits the information from people all over the world who have never met and have never interacted or sent an email or a text message to each other, but it tells you what raw resources should be used to produce what consumer goods. And it helps allocate all of these resources around the planet for the most efficient economic outcome so we can produce the most chairs and the most event centers and the most screens and cars and computers and Raspberry Pi is to run Bitcoin Core full nodes on. And without that pricing mechanism, all of this would get messed up and we wouldn't know if we should make, you know, the seats here, we wouldn't know if we should make them out of gold or cloth or wood or anything else. And we wouldn't know these things without this pricing mechanism. And you realize that the more government meddling you have in the economy, the more it disrupts the flow of information that's made by prices. And so I became pretty much an anarcho-capitalist from studying economics. And then I remember I picked up a Murray Rothbard book later. And who here knows who Murray Rothbard is? A good chunk of the room. For those of you that aren't putting your hands up, I think I cannot emphasize strongly enough, pick up anything by Murray Rothbard and it will change your entire view of the world. And I remember again picking up a book and I just filled out the selective service thing in the United States. For those that aren't Americans, when you turn maybe 16-ish in the US, you have to register with the government so that they can draft you if there's ever a war again. And I kind of didn't like it too much, but everybody in school was doing it and everybody else had to do it. And then not too long after that, I was reading the Murray Rothbard book and he pointed out that the military draft is the moral equivalent of kidnapping and slavery. And everybody knows that kidnapping is wrong. Everybody knows that slavery is wrong, but everybody grows up being taught that the draft is a good thing if there's a war so the government can defend the country. But here's this guy pointing out that they're morally the same thing. And so I had this incredibly uneasy feeling of cognitive distance where I was scared to even hold this book in my hand because my entire world was getting turned upside down because we know kidnappers and slave masters are bad people and the government's claiming the right to kidnap and enslave us in the form of the draft. So if they're claiming the right to kidnap and enslave people, that means the government people are bad people. And my entire world view was kind of shattered at that point when I was reading Rothbard. So for me, that's why I'm so excited about Bitcoin is it gives people back complete control of their own finances. And I saw it as a tool to bring more economic freedom to the world. So study economics and study philosophy as well. It wasn't until much later I started reading much more of the moral philosophy side. And so there's I guess two different directions to come to this conclusion.  And it's amazing just how well those two things interweave with each other to achieve the desired results. I think that's all the time I have but pick up some of your Rothbard and read that or anything, study economics, that's why I'm here. Thank you guys. Thank you.


Let us continue with our discussion of this Arzela Ascoli theorem okay. So you know the situation is that we are trying to understand what compactness means for functions okay. Basically we want to understand compactness for families of meromorphic functions defined on a domain in the complex plane okay or in the external complex plane okay but then we want to get some idea about what compactness means from basic topology okay. So as I was telling you suppose you are having a topological space you have the notion of compactness okay and if the topological space is a metric space then compactness is the same as it is equivalent to sequential compactness and it is also equivalent to the Bolzano-Weierstras property. The sequential compactness is the property that every sequence has a convergence of sequence okay. The Bolzano-Weierstras property is a property that every infinite subset has a limit point okay and these three are equivalent right and then there is also this implication that if you have a compact subset it is always closed and bounded if you are in a metric space okay but the converse is not true. So if you take r infinity you can easily look at the unit ball in r infinity the closed unit ball it will be closed and bounded but it will not be compact because you can write out a sequence which does not admit any convergence of sequence namely the sequence consisting of 1 at the i th place and 0 elsewhere okay. Any two terms of this sequence differ by a distance of root 2 okay are separated by a distance of root 2 and therefore this sequence cannot have a convergence of sequence okay because the terms of the sequence do not come you cannot find a subsequence whose terms whose elements come closer and closer together okay. So but of course in any metric space compactness implies closeness and boundedness the converse is not true the converse is true for Euclidean spaces and in fact I even told you that if you take a Banach space the condition that it is finite dimensional is equivalent every close subspace every close bounded subspace is compact if you want to put that condition then the Banach space has to be finite dimensional it cannot happen if it is an infinite dimensional Banach space okay. So now you see alright so the so there is a related notion there is this notion of total boundedness okay which is implied by compactness alright total boundedness is that is basically the property that you know you can cover the whole space by open balls of a fixed radius no matter how small the radius is the point is of course by finitely many such open balls okay. So given any positive radius epsilon I should be able to find just finitely many open balls of radius epsilon whose union is the whole space and I should be able to do this for every epsilon greater than 0 this is total boundedness is very strong form of boundedness this implies boundedness okay and in fact space that is totally bounded is actually even it even has finite diameter okay because the diameter of the space can be compared to the diameter of this finite collection of centers of these open disc which is called a net an epsilon net if the radius you are talking about is epsilon okay and the point is that total boundedness itself though it is a strong form of boundedness total boundedness is not enough to give compactness okay what you have to add to it is completeness so if you have something that is complete and totally bounded a metric space that is complete and totally bounded then it is compact okay and so that is the so that is another equivalent version of compactness. So you know so we check to check compactness either you check the abstract definition of compactness which is very hard you have to take an arbitrary open cover and you have to show that there is a finite sub cover that is very difficult in practice okay and probably if you are given a particular open cover a specific open cover maybe you might be able to because of your knowledge of geometry and topology you might be able to extract a finite sub cover but you cannot do this in an arbitrary way so it is abstract okay trying to get hold of a finite sub cover from an any given open cover is a very abstract thing it is not easy to check okay so we end up checking sequential compactness okay namely you take a sequence and show that there is a convergence of sequence if you check that then that is equivalent to compactness the other thing you can check is that the space has the metric space has the Bolzano-Weierstrass property show that every infinite subset has a limit point okay and of course if you are in if you are working with subsets of Euclidean space you know what to do you will just check that the subset is that you want to say compact is both close and bounded so there is not much to do alright but point is that and of course now there is also this new there is also this extra condition that you know if you want to check a space is compact you check that it is complete and it is totally bounded okay and of course completeness is the condition that every Cauchy sequence converges okay that is you know well whether that can be easily checked or not depends on the particular case you are looking at okay it is also not so easy okay and the other thing is total boundedness and that is also very abstract okay total boundedness is a very abstract thing you have to say that you know you can find an epsilon net for every epsilon greater than 0 and that is so you have to produce an epsilon net for every epsilon greater than 0 and epsilon net is those finitely many points in the space centered at which if you take open balls of radius epsilon these balls will cover the whole space okay you have to show the existence of those finitely many points that if epsilon net and you should do this for every epsilon it is not an easy thing to do alright so these are the various versions of compactness that we have but you know we are interested in compactness of functions okay now for functions what is it that what is it that is easy to you know verify normally what do you normally easily verify about functions you verify continuity okay you verify differentiability you verify boundedness of a function okay try to find a constant which bounds the modulus of the function values okay so these are the things that you can verify okay so you need something in the situation of functions if you are working with a space of functions then you need better conditions for compactness and that is where the Arzela Ascoli theorem steps in so basically you know we are looking at a compact metric space X okay so X is a compact metric space and so let me go back to what I wrote last lecture X is a compact metric space and let me use a different color now X is a compact metric space and you are looking at I am looking at this CXR which is the you know it is a Banach algebra it is a complete long linear space of continuous functions on X with values in the real numbers okay and you could also taken with values in the complex numbers you would get the other Banach algebra CXC okay and the question is that you want to and of course mind you since X is compact and a continuous function on a compact set is uniformly continuous and always attains its bounds every function that you are that we are worried about is already uniformly continuous and it is bounded okay so normally when you write CXR or CXC you mean also not only just continuous functions but also continuous bounded functions okay but here boundedness is automatic because X is compact alright. So what I want I want I want to take a close subset of I want to look at a subset of functions I want to look at a family of functions or a collection of functions okay which you think of as a subset of the space of functions CXR or CXC and I want to say that it is compact that is my aim I want to get a nice condition to say that a collection of functions is compact okay now you know since we are in a metric space something that has to be compact a subset has to be compact means that it has to be both closed and bounded. So certainly the subset to begin with should be closed okay so we put this precondition that it is already a closed set if it is not a closed set it is not going to be compact because compact implies it is closed okay and of course it has to be bounded because I told you that in any metric space a compact subset is always closed and bounded okay. So if you start with a closed subset A the condition that it is compact is equivalent to it being bounded and here is a extra condition the extra condition is equicontinuity okay so equicontinuity is something that helps us to get hold of compactness right.  So if A is totally bounded then I know it is compact okay so the only issue is with replacing this total boundedness with something else and I told you this total boundedness is not so easy to verify because you have to show the existence of an epsilon net for every epsilon it is not practical okay so whereas total boundedness being difficult to verify we have boundedness is easier to verify okay so you from the total boundedness you remove the total and you just put boundedness which is an easy thing to verify for functions okay and you because you have made the condition weaker from total boundedness to boundedness you have to put something else to make it strong enough to get compactness to give compactness and that strong enough thing is the equicontinuity okay. So I will tell you what this equicontinuity is okay so basically the idea is very very simple the idea is see if you want function a function to be continuous at a point then what you do is you say that there is a usual you know you have the usual definition of the epsilon delta definition of continuity at a point and what is that give me an epsilon greater than 0 I can find a delta such that whenever the point is within a delta of the given particular point then the function value is within an epsilon of the image of the function value at that point okay. Now so the whole issue here is that if you change the point okay so you are looking at continuity at a point for a given epsilon you have to verify it okay now the delta that you get will depend on the epsilon of course because if you make epsilon smaller you might expect that the delta must become smaller okay and so the delta depends on the epsilon the delta also depends on the point and the delta is particular to that function okay so the delta depends on three things okay delta depends on three things if you are so you know so let me write this down so what is the equicontinuity so you see what is the usual continuity usual continuity at let us say x0 for f what is this given epsilon greater than 0 there exists a delta greater than 0 such that whenever the distance between x and x0 is less than delta then we have that the distance between fx and fx0 is less than epsilon this is the usual good old definition of continuity epsilon delta definition of continuity and you see the point here is that of course this the first D that I use in D of x, x0 is the metric on the space on which the function is defined whose points are x and x0 x0 is a fixed point x is a variable point okay so basically this D x, x0 less than delta refers to you know basically it refers to all the points in a delta open ball centred at x0 in your space okay and what you are saying is that whenever point whenever you take a point in a delta open ball centred at x0 then its image is in an epsilon open ball centred at fx0 that is what you are saying okay so the image of this delta open ball under f goes into this epsilon open ball centred at fx0 okay that is what usual continuity is but the point is that this epsilon you see given an epsilon okay delta depends on what see depends on of course epsilon it depends on x0 and you know it depends on f also of course we are looking at a single function so we do we forget the function because if you are dealing with one you on function there is no confusion but if you are looking at a family of functions then you will have different functions f okay and as you change the function for the same epsilon and even for the same point okay if you have family of functions which are continuous at a point x0 okay then for that family of functions even if you take a single epsilon the same delta will not work the delta will change with the function okay and of course delta will change with the point now equicontinuity is the fact that you are is the condition that you know you are able to get a delta that works for all functions in one stroke that is equicontinuity okay so a collection a collection of functions of functions so well a so let me call it as a is called equicontinuous at x0 you know if given epsilon greater than 0 we can get a delta that works for all functions in that collection that is equicontinuity you get a common delta that is equicontinuity alright and well and of course if you say that a family is equicontinuous on a subset of the space if it is equicontinuous at every point of the space okay so you make a point wise definition and then you say that this definition holds if it holds on holds for a subset of points if it holds for every point in that subset okay so this is just like saying that function is continuous on a set if it is continuous at every point of the set it is like that so a is equicontinuous at or let me say on let me say subset y in x if it is or let me use x prime if it is equicontinuous at each point of x prime okay so this equicontinuity is basically if you want to you know think about it in a very simple way equicontinuity is given in epsilon you find a delta that works for all functions that is the point okay a single delta will work for your epsilon for every function in your collection then that collection is called equicontinuous collection or equicontinuous family of functions okay and what the Arzela Ascoli theorem says is that well the Arzela Ascoli theorem says that if you put equicontinuity together with boundedness then that is good enough to say that your collection is compact okay so compactness of a family of functions is equivalent to that family be equicontinuous and bounded mind you these are much nicer condition than total boundedness okay equicontinuity is it is a kind of continuity that you have to check it is just continuity you have to check but make sure that you get one delta for all the functions for any for a given epsilon okay that is like checking continuity and that should be far more easier checking boundedness is not a should not be a big problem okay so these are all things that we normally check for functions but given a family of functions you never it is not common that you check total boundedness total boundedness is you have to find finitely given an epsilon you have to find finitely many functions so that the distance of any function is within an epsilon to one of these finitely many functions you have to find this epsilon net that is not easy and that is not common okay it is not practical so Arzela Ascoli theorem helps you by saying that well you want to check a family of functions is compact do not do much you just check that it is bounded check it is equicontinuous and you have done okay so I just wanted to you know indicate something about the about this this equivalence of compactness with boundedness and equicontinuity because it is because it is a very very fundamental thing and you need to know how things work okay so let me do the following thing I will again change color and use different color so I will say proof of Arzela Ascoli so let me go through the proof because I want you to understand the ideas involved okay you will also get familiar with all these notions of total boundedness epsilon net and how equicontinuity is actually used okay so start with a closed start with the closed A inside C XR start with the closed set what do you have to show you have to show that A is compact if and only if it is bounded and equicontinuous okay so assume A is compact assume A is compact alright now what do you have to show so this is one way of the theorem you have to show that if it is compact it is bounded and it is equicontinuous okay I told you that in a metric space a subset is compact implies it is both closed and bounded okay so it is bounded compactness always implies closeness and boundedness in a metric space alright so and mind you that is true only for metric spaces if you go to arbitrary topological spaces things can become very bad you can have a compact subspace of a compact topological space which is not closed okay it is it can behave terribly you can have a whole space which is compact you can have a subspace which is compact but the subspace is not closed in the whole space that this can happen for an arbitrary topological space it cannot happen for a metric space for a metric space compactness always forces closeness and boundedness so how is the space is also if you want yeah so you see so assume A is compact then A is bounded bounded okay because compactness will imply closeness and boundedness it is already A is already closed okay but you want boundedness that is implied by compactness we only have to show that A is equicontinuous okay we only need to show A is equicontinuous okay that is what you have to show so you see so incidentally let me make a remark here in this case we are anyway going to the way we prove the Arzela-Ascoli theorem is by using the other version of compact characterization of compactness which is compactness is equivalent to A being totally bounded okay this total boundedness will come into the picture okay at least in the proof okay that is the key tool that helps in the proof of the Arzela-Ascoli theorem so if you think of since if you think of that given that A is compact I know it is totally bounded and I told you totally bounded also is a strong form of boundedness so it implies boundedness totally bounded is the condition that there is an epsilon net for every epsilon okay and you know that the diameter of the space can be compared to the diameter of any epsilon net okay to within an epsilon okay or 2 epsilon okay so therefore A is always bounded in fact it has finite diameter alright we only need to show that A is equicontinuous and for the equicontinuity also you use the fact that it is totally bounded okay so since A is totally bounded well A is A admits an epsilon net for every epsilon greater than 0 okay so here I am using the fact that compactness implies total boundedness right and compactness implies total boundedness is a very very it is a very simple thing there is nothing complicated about it okay basically what you are saying is that you should cover the space by finitely many open balls of a fixed radius okay that radius being epsilon and that is very that is very easy to see because if you give me if you take all open balls of radius epsilon that is an that is a cover for the space you allow every point of the space to be a centre and you take the collection of all open balls of radius epsilon that is obviously an open cover for the space and if you know it is compact it has to admit a finite cover and that finite cover if you take the centres that will give you the epsilon net that you want okay so it is that compactness implies total boundedness is absolutely easy to see okay so there is an epsilon net okay so let epsilon greater than 0 be given and let us take and so here is a you know this is the kind of thing you fiddle with to get epsilon finitely in your answer so you know usually the idea is that you use a triangle inequality you know finally all these epsilon arguments they finitely end up with a triangle inequality if you are breaking a distance into 2 pieces then you try to and you want that distance to be less than epsilon you know you try to make each piece less than epsilon by 2 if you are breaking it into 3 pieces then you know you make each piece less than epsilon by 3 so what I am going to do is I am not going to take an epsilon and I am going to take an epsilon by 3 net okay so let epsilon greater than 0 be given and let us take an epsilon by 3 net for a okay so what does this mean this means that so there exist functions f1 etc up to fm which are in a such that well the any function in a such that for any function in a there exist an i such that i with the distance between f i and f less than epsilon by 3 okay so this is what epsilon net means basically you cover all points of the space by looking at open balls of radius epsilon by 3 centered at the points which correspond to the epsilon by 3 right this is what you get now I wanted to you know at this point you know I want you to remember what is this distance here see this distance is the distance in this distance is the distance in the in the Banach algebra C X R okay it is mind you all these what is a a is a subset of C X R and C X R you know functions which are continuous functions on X real valued functions okay and so what is the and of course X is compact so these all these functions are bounded okay cunious function on a compact set is bounded and in fact it is uniformly continuous and it attains its bounds right so so what is the distance the distance is the supremum now so what is this this quantity is supremum as X belongs to capital X as X varies over capital X of you know mod of f i of X minus f of X this is what it is okay in fact it is actually it is precisely norm of f i minus f that is what the distance is the distance between f i and f is just norm of f i minus f and the norm is a sup norm so you calculate f i minus f at each point and then you take the mod of that and then you take the supremum okay and this should be less than epsilon by 3 this is what you have okay this is what you have. Now you see now we are kind of in more or less in a very good shape in good shape because see now you can get equicontinuity very easily so let us check equicontinuity at a point okay so let us check equicontinuity at X naught in X let us check equicontinuity there okay then what is it that is going to happen well what do you have to check for equicontinuity you have to find a delta such that whenever mod X minus whenever the distance between X minus X and X naught is less than delta okay the distance between f X and f X naught is less than epsilon and this should work for all f in your family a in your collection a okay so given for the given epsilon greater than 0 we seek delta greater than 0 such that mod I keep seeing mod because you know at the back of the back of one's mind one keeps thinking of Euclidean space but it is not mod you should replace it with D so the distance between X and X naught less than delta implies that mod f X the distance between f X and f X naught which is mod f X minus f X naught is less than epsilon and this should work for all f in a this is what you want alright so the so you see how do you do this the idea is very very simple there is this on the one hand you have an X which is a variable point you have this point X naught okay and you have an arbitrary f alright and you will have to now connect that arbitrary f in a with these particular f i's which are the elements of the epsilon net okay so you do that by a you know basically by using a triangle inequality to break down some distance into three pieces okay so what you do is that well you write this mod f X minus f X naught you write this as you know I have already told you that given this f there is an f i with the property that the norm of f i minus f is less than epsilon by 3 that is already there so use that f i so you write this as modulus of you know well f of X minus f i of X plus mod f i of X minus f i of X naught plus modulus of so I should put less than or equal to modulus of f i of X naught minus f i f of X naught this is what I have to do okay so I introduce this f i cleverly f i values at X and f i values at X naught to break this and then what happens is that you see the you know f for any X in X okay for any X in X the modulus of f i X minus f X is always less than epsilon by 3 so the point is that this fellow here this is less than or equal this is less than epsilon by 3 okay and so is this fellow here this is also less than epsilon by 3 okay I have to be worried only about the central term this is the only term I have to be worried about that I cannot apply that epsilon by 3 bound to that because the points are different the function is the same it is the same f i okay it is the same f i but the points are X and X naught whereas in the first term the point is X the third term the point is X naught the term in the middle has two points with the same function okay what do I do that what do I do here so here what I do is basically I use the fact that all these f i's are all uniformly continuous in fact they are all continuous they are continuous on a compact set so they are uniformly continuous so because they are uniformly continuous I can make sure that you know if I choose a delta sufficiently small whenever the distance between X and X naught is less than delta I can make the central term less than epsilon by 3 I can do this in fact I can simply do it because of continuity of f i at X naught but the point is that I can do this for all f i at the same time because there are only finitely many f i's okay there are only finitely many f i's see for if I want f i X minus f i X naught to be less than epsilon by 3 I can of course choose a delta I can find a delta for which whenever the distance between X and X naught is less than delta then mod f i X minus f i X naught is less than epsilon by 3 I can do this because of continuity of f i at X naught okay but then this delta may depend on the f i okay and if I change the i to other f i's okay then the delta will change but there are only finitely many of these f i's so I could have taken the minimum of them and that would work for all f i's all the elements of the epsilon at in one go alright and mind you you could even forget the point X naught that is because all these functions that we are studying they are all uniformly continuous okay.  Uniform continuity is that whenever the distance between source points is less than a delta then the distance between the image points is less than epsilon it does not matter what the source points are the only condition is they have to be within a delta you can find such a delta that is uniform continuity okay. So let me write here can find a delta greater than 0 such that mod f i of X minus so let me write f j of X minus f j of X naught is less than epsilon by 3 if D of X comma X naught is less than delta because and for all j so let me write for all j because of continuity of f j at X naught and the fact that there are only finitely many f j I can do this right not only that I can do more in fact because all of these are uniformly continuous I can do this irrespective of X naught okay I can even forget the X naught okay. In fact since the X is compact and f j are uniformly continuous so I am abbreviating it to u f l y for uniformly continuous we can do this for any X naught okay so you can find given an epsilon you can find a delta which neither depends on the f nor does it depend on the X naught okay so you have verified equicontinuity in a uniform way okay and that is how so you have got equicontinuity that is it you have found a delta that works for every f okay so thus A is equicontinuous okay so the moral of the story is we are able to see one way why if you have a compact space of functions then these functions must form an equicontinuous family okay so equicontinuity is something that is very very important and mind you I will have to prove the other way of the theorem but then before I do that which I will do in the next lecture what I will now say is I will tell you why this is called why as well as closed theorem is sometimes called the uniform boundedness principle it is because of the following reason you see the condition is that close subset of functions is compact if and only if it is bounded and equicontinuous okay now this bounded is actually bounded with respect to the metric on the space of functions and that is given by the soup norm okay and boundedness under the soup norm okay is actually uniform boundedness it means see what does boundedness normally mean it means that you are able to find a positive constant such as the modulus of the function values at all points is bounded above by this positive constant okay but of course this positive constant would change if you change the function so if you take different functions in a family each function may individually be bounded but you may not be able to find a common bound for all the functions in the family okay and that is what uniform boundedness does it gives you a common bound for all functions in your family or collection or subset okay so boundedness in CXR actually means uniform boundedness and Arzela Ascoli's theorem is just that if you have a uniformly bounded family which is equicontinuous then it is compact and conversely that is what Arzela Ascoli's theorem is okay and that is why it is called the uniform boundedness principle okay so I will continue in the next lecture and try to tell you the other way of the proof okay.