Lecture 6: Where We Go from Here

Key Takeaways

foreign yeah so welcome to the last lecture of the class uh today we're going to talk about where all this material particularly goes from here so for instance how this material applies to 18102 a little bit of 18101 901 for sure and a little bit of 152. if those numbers don't mean anything to you right now that's totally fair I'm going to write out what the number and the title of it is but Loosely speaking that's the general plan for today but before we do that I want to introduce just a little bit of brief history to metric spaces in the first place because I think that there's this weird notion that all those mathematical tools just pop up but in fact everyone sort of knew each other in the field which I think is really cool um so firstly the main thing to know is that in the early 1900s mathematics was far less axiomatized like it was far more localized in each field like oh in this field who's studying this type of function we can study you know what this means for the entire space what does being lip shits mean what does um what is a vector space things like that um but because of this a bunch of different people had all different types of convergence they just weren't unified and now this is fine but it makes it difficult to see if it's something that's true specifically about the space or if it's something true in higher generality so in 1906 fresh a invented metric spaces um the reason I think this is particularly interesting is because this this concept is one that he introduced in his PhD dissertation so he went throughout his PhD course load and then created metric spaces in his dissertation very very powerful tool and as we've seen in this class so far it unified all these different types of convergence at once no longer did people have to State convergence in their field and improve all the properties they can simply State the distance on it and then State what convergence means if it is in fact a metric so uh this in particular really Unified Notions of convergence and other types of like open sets and things like that um so in particular this lets you prove facts about metric spaces and then stated about your space that's the main idea and then later on in 1914 a mathematician known as housedorf which will be a familiar name if you've done topology a popularized metric spaces so in particular he wrote this book called principles of set theory um and in this book he was able to define or this book was very very popular and in it he includes metric spaces so it made it more and more popular but in fact he did quite a bit more than that in this book in this book he introduced topological spaces and topological spaces are of course what is studied in introduction to topology 8901 now throughout today I'm going to use this diagram where what I'm going to talk about is this idea that topological space is are more General than metric spaces uh let me write this down topological spaces the way to see this is that the definition of a topology which we'll get to in a moment is simply a definition of what it means to be open and of course we have a notion of what it means to be open under the metric you can just cover it with Epsilon balls and everything's fine and so topological spaces are for more General than metric spaces and there's quite a bit to study here so let me introduce what the definition is of a topological space so a topological space is simply a set that has a topology on it um and I'll Define what it means to be a topology in a moment to do a topology t on a set X is a collection of subsets of x such that I won the empty set and the entire met or not metric space the entire set is in your topology two if I let T I be a set or a collection of subsets of t then the arbitrary Union of them t i is in t and three the finite intersection of open sets TI rnt so again just reiterate a topology is just a collection of subsets of the metric Space X or not metric space the set X such that the empty set and the entirety of the set is in the topology uh arbitrary unions of points of anthropology are in the topology and finite intersections are as well and notice that these are the same as the topological properties of open sets we defined in metric spaces which is a good thing because this is what we would want to have if in fact metric spaces are more specific than topological spaces and what is the topological space well a topological space is simply a set with a topology on it so I'll just write with t right T is the topology so with these three properties we have Notions of openness in particular in fact one can define an apology this is more of a terminology thing a set is open let's say a set a inside a topological Space X is open if a is in the topology enclosed if x minus a is in the topology which implies that the complement of a closed set is open so that's an equivalent definition but yeah these are properties of what are known as open sets and this much much more general setting now you might be asking yourself like why should we particularly care like this is a very abstract definition right but I'll remind you that when we first started talking about metric spaces we had a very very general somewhat abstract definition right it's one that at first seems not too bad it's just the topological properties of open sets we've already discussed but with this generality we can prove quite a bit more so to do uh yeah I already noted the topology on a metric space is simply the simply unions of Epsilon bottles if you're in a topology class this would say this would be the same as having a basis of Epsilon balls for the topology but I'll leave it there for now and in fact when does when is the topological space in fact a metric space that is known as a mechanical space so a topological space X is betrayable if there exists if there exists a metric d inducing the topology on x typology so here the topology on a metric space is induced by these Epsilon balls in more generality a topological space is metrizable if there exists a metric inducing that topology and now this notion of inducing is one that I'll leave for 18901 but I just bring it up now because as I mentioned on our second day not every topological space is metrizable not every set that you're considering is going to be as nice as a metric space even though we can put a metric on any space it doesn't mean that it gives us actually good information so having a tradability is deeply important so let's let's take this time real quick to define the Notions of convergence and open sets and continuity in terms of the topology just like we did on our first day right we Define what a metric was and then we defined all of the terms we were used to in terms of that metric and so we're going to do the same now one a neighborhood I should write this out the first time a neighborhood of a point x inside your set X is simply an open set U containing X why do I bring this up because our Definition of convergence and of end of continuity will depend on this definition of a neighborhood so two a sequence xn converges to a point x if for every neighborhood of x all but finitely many x i many X I are not in the neighborhood now recall that this is the same definition that we had for metric spaces right we had the Dilemma that a sequence converges to a point x if and only if for every Epsilon ball around it um all but finally many terms are outside of it which was a pretty Direct blema um so this is how we Define convergence for topological space three we also state thinks about continuity if I have a function f from a topological space y to or sorry X to y and I'll say that the topologies are on that the topologies are called TX and Ty then f is continuous or I should write this out continuous if for every single open set T and Ty F inverse of T is in TX in other words the inverse images of open sets uh the inverse images of open sets remain open which is as we've shown as well the same definition of continuity we can have for metric spaces so all of this is still very very related to what we've been talking about so you might wonder why we even study metric spaces at all then right because topological spaces are much much more General if I can prove a fact about a topological space I will have proven it for my metric space as well and the short answer to that is in my opinion why would the same answer to why would you study real analysis after calculus right in theory real analysis shows all the facts you need to know about calculus it proves in more generality but but you gain a lot of intuition from dealing with Calculus in the first place right before you prove the mean value theorem abstractly it makes sense to have an idea of what's Happening um before you even do so so very similarly here having intuition about metrics and metrics bases provides a lot of intuition about topological spaces even though most topological spaces are not metric spaces right it still gives you the framework to move forward too for this reason that's that's essentially why whenever I could I would draw pictures of what's Happening as a like blob um because then you get the intuition of oh how can I start to problem solve via a diagram um it just gives you that right frame of mind to move forward okay now that being said metric spaces because they're much more specific than topological spaces are an ABS or our current area of research topological spaces are mostly done being researched I mean topologies of not a closed field but Point set topology the the basic framework is not as much of an active area of research and Metric spaces on the other hand very much so are so that is yet another answer to the question of why should I care about metric spaces if I could talk about topological spaces okay so now we're going to talk about 18102 unless you have some questions on the material I've talked about today cool so now generate down 18901 I didn't I meant to do so so this is the sort of material that's covered in one of the first lectures of 18901 introduction to topology foreign functional analysis or intro to functional analysis functional analysis is all about studying what are known as Norm spaces and we've already talked about them before Norm spaces are just a subset of metric spaces we talked about this in like lecture three it's slightly more specific than metric spaces for Norm spaces so again recall a norm space is simply a set with a norm on it or a vector space with a norm on it denoted with absolute value bars and specifically here the three properties we want again are a positive definiteness we want the norm to be bigger than zero or if it's equal to zero for the point B zero itself um positive definiteness absolute homogeneity and the triangle inequality these are the three properties we want to abnorm spaces and on the homework you've already shown that a metric induced by the norm is is in fact a metric so Norm spaces are generalization of metric spaces okay now with this definition of a norm we can yet again redefine all of our Notions of convergence in neighborhoods and continuity as we've done before so I'll quickly do so the way to do it is to just write it in terms of the metric right a sequence xn converges to X in a norm space if for all Epsilon bigger than zero zero exists in n in the natural numbers such that for all n bigger than or equal to n the distance between x n and X is less than Epsilon this is the definition in terms of the metric space but recall that we can Define this metric in terms of the norm so if you're in a class like functional analysis you'll probably just use this notation as opposed to going to the metric one um but this is the definition of convergence in an arm space and again we want all these definitions to be compatible so if it feels like repetition there's a good reason why it does it's just we want a new definition but one that still works with the broader setting of metric spaces so that we don't have to redo all of our work two um of course we can Define Koji sequences in the same way simply one such such that for all Epsilon bigger than zero there exists an n in the natural numbers such that for all n and M bigger than or equal to n is the distance between xn and x m is less than Epsilon so this is the definition of cushy sequences and finally we Define a set to be open if and only if we can find a ball of radius Epsilon around each point so three a subset a inside of x is open if for all X and A there exists an Epsilon bigger than zero such that the ball of radius Epsilon around X is contained in a and in terms of the norm this is the set of Y such that the norm distance between x minus y is less than Epsilon so precisely the same as in a metric space um now to this point as I was talking about last time okay so again a set is open if and only if for all X and A there exists a ball of radius Epsilon that's contained in a okay so just to continue off where we left off last time in terms of lecture five um we started introducing completions of metric spaces because koshy completeness is very very important so much so in the setting of functional analysis that we have a name for it um a bond like space is a norm space that's kosher complete essentially the only difference is that it's complete with respect to the metric because again we have a notion of koshy completeness in terms of metric spaces so we want it to be compatible and as we talked about last time you can take the completion of a norm space to get a buttock space so bottom spaces are slightly more General or sorry slightly more specific than a normal space but the tools that we use are very very important so in particular as an example of ebonics base you can show that c Infinity of a metric space m or even a normal space um is a bionic space we talked about this last time but the proof is essentially the same as in this sense of continuity where again this is the set of functions F that are continuous and this inbounded so the separamum over M and M of f of M sorry F of m is less than infinity so you can show that this is a buttock space there's a few other classic examples like RN CN instead of continuous spaces or C naught a b but most of them are based off of these four these are the few key ones to have in mind and these are so useful that it makes sense to use these as our main sense of note intuition so um in fact we even have weirder examples of buttock spaces though and I'll introduce one of them now where specifically you're interested in the Dual of the vector space Have you heard of the notion of dual before no worries yeah it's totally fair in the case of Norm spaces it's not too bad it's the set of functionals a functional is a linear map let's say t from your Norm Space X into the real numbers or complex numbers if you prefer so it's just a linear map from your Norm space into the real numbers of the complex numbers and in fact this is the definition of a dual of a vector space in general this replace x with the vector space um and in fact you can show that the set of functionals are robotic space but to do so we need to introduce a norm right in order to even have the possibility of defining or showing that a space is a bionic space you need to have a notion of a norm so the norm on it the norm on T known as the operator Norm is simply the supremum over X in X in capital x what's the norm of x equal to one of T of x so it's just the largest image sorry this upper bound of points on the ball of radius one around X that's what the definition of the operator Norm is and you can show the disinfectant a norm homogeneity is not too bad the triangle inequality is where things get worse as always but homogeneity and positive definiteness are nearly immediate so um yeah so with this Norm then you can show that the set of functionals is ebonic space and why is this important because it turns out that studying your Norm space is pretty much analogous to studying your dual space um so studying this at a functionals you can redefine continuity you can redefine open sets things like that you can show all of these properties but for functionals as um as a specific example and the fact that operators are bubonic space is particularly really helpful okay now I want to know one more example of a norm space before I move aurobonic yeah I want to give one more example of a norm space before I move on which is one that I briefly talked about before uh let me I'll write up here because we're done with specifically you can study what are known as inner product spaces which is something which is a title that I which is a word I introduced last time but I'll Define right now it's basically the same concept as a norm space and a metric space you just introduce an inner product on your set and your inner product you can think of like a DOT product on RN create the dot having the dot product was very very helpful in our end because that lets you define things like magnitude which is part of the reason why we introduced Norms in the first place is to introduce a notion of magnitude okay so in inner products base is a set or a vector space X with an inner product defined on it and specifically we want this inner product as usual to have three main properties the three properties are and I'm going to assume I'm going to assume that the inner product takes in two points in x and spits out a real number all right so the three properties are symmetry the inner product of X and Y should be the same as Y and X 2 you want um linearity the inner product of a X plus b y and a product with z is the same as a x inner product C plus b y inter product C so this is linearity and lastly we want positive definiteness but it's slightly weirder If X is not zero then the inner product of X with itself is bigger than zero it's not an if and only if statement because of course we have Notions of orthogonality uh from real and out sorry from RN but if it's not zero the inner product should be non-zero uh these are the three properties of an inner product space and the reason I bring this up right now is because you can show uh consider just as we did in calculus you can consider uh the inner product of X and X to the one-half this makes sense because it's positive so taking the square roots totally fine what you can show is that this induces a norm on x the way that you show this so in other words induces Norm so what you would want to show is that given an inner product space if I look at the square root of x in the product itself you would want to show that this implies you have the three properties you want for a norm space just as we did in the proof of metric spaces um and this is um really important this is this shows up all the time in fact in quantum mechanics if you're uh is that some part of math and physics that you're interested in uh and in fact uh I should say we can go should complete it as usual once we have that our inner product induces a norm we know that it then induces a metric um the one thing we don't know is if it's a bionic space or not and that's totally fine we can just coach you complete it and get what is known as a Hilbert space a Hilbert space is a koshy complete inner product space so um you can view it as a completion you can also view it as just the definition and here koshy completeness is with respect to the metric that's induced um I'm going to rearrange this diagram real quick because not every inner product space is a bionic space but it is a subset so specifically you can have something like this inner product spaces are a subset of Norm spaces and Hobart spaces are inner product spaces that are koshy complete so therefore our bionic spaces this is the end of my diagram no more no more drawing squares all the way down but um I just wanted to show all these ideas are deeply related and each one of them is interesting to study was in its own respect now granted all the center squares Norm spaces spawning spaces and inner product spaces these are mostly talked about in functional analysis which roughly makes sense because you need Vector spaces in order to move forward um but yeah because again functional analysis is done on arm spaces and more specifically usually product spaces okay I want to know one small application to 18 and 101 and then I'm going to talk about the application to differential equations in 18101 analysis and manifolds in your studying of course manifolds now what is a manifold a manifold is just some smooth enough blob and you can picture it in euclidean space it's a to be even more specific a picture you have like an orange the main properties that we like about an orange or sphere is that locally it looks that right just like it is in calculus a function locally looks like a line in higher generality a manifold is just a space that locally looks flat um the reason I bring this up is because in the definition of a manifold you assume or at least you define a manifold you assume it is metrizable it's a space where it's a metric on it or at least a metric that induces the norm so why is this so important right why do we study my child's ability in our manifold well we want there to be a notion of distance on our manifold which makes sense the reason why metrics ability I bring up here is because in reality when you're studying matricable spaces you don't often care what the metric is sometimes you do sometimes you want to say oh once you know they're exist symmetric then x y and z but most of the time you just use the properties of the open sets so most of the time you just want to use the fact that balls of radius Epsilon are open in doing so lets you define smooth functions it lets you define integration it lets you define I had one more thing I mentioned it you can Define integration and you can define a vector fields as you might have seen in 1802 all this is just letting us do calculus on weirder shapes right doing calculus on a sphere doing calculus on a smooth enough tree things like that but really all we really need is that the balls of radius Epsilon are open to start doing so so sometimes in 18101 you won't see how metrics come up in particular but the intuition is still there right you still want to have a notion of distance on your manifold so yeah that's all I'll say about 18101 because of course it takes like a month to actually get into manifold Theory because you're redefining multivariable calculus so I'll leave that there for now yeah the last example I want to talk about today is specifically the application to differential equations and we already briefly talked about this right last time we talked about um integral operators specifically as an application of the Bonnet fixed Point theorem so let me just write this down last time we considered Odes of the form uh what was it G of x plus the integral from A to B k x y f of x sorry F of y d y so last time we considered pde that leads to this nice integral operator if this is unfamiliar that's totally fair but I'm just bringing up the fact that we've already seen one small application of demonic fixed Point theorem to differential equations but in fact you can use differential equations to motivate the notion of compact sets oh I should have written eighteen one five two intro to partial differential equations so we can use the notion or we can use ode or pde to motivate compact sets and let me just briefly explain why this is the case so picture um picture some subset Omega some of set subset Omega of R2 and picture this as a metal sheet so I just have some subset out here Omega I'm going to assume it's nice and connected um and I want to consider it as a metal sheet why do I do that because from here let's say I just heat up one tiny portion of it let's say I take a little blow torch and I heat it up right here we want to know how the temperature is affected by this in particular the temperature let's say that you of x y is the temperature at x y in Omega what you can derive using physics or using um just differential uh looking at it from a differential viewpoint you can Define um you can show that the heat equation that you'll get from heating up this tiny spot is of the form derivative of U with respect to X twice plus derivative of U with respect to y twice this is known as the heat equation and in fact if it reaches um if we let this metal sheet reach equilibrium if we let the blowtorch heat dissipate to all over the metal sheet then you'll get that this equation is equal to zero which is the differential equation that we're particularly interested in if I have a solution of this form what what do I know this is in fact so important this this operator that we just call it the laplacian of you if you've heard of that before where the laplacian is just the derivatives squared applied each time and summing over them okay so the question is what if I only know the temperature at the boundary so here the boundary is just we could just draw it out pictorially it's the point it's just along the edge suppose just along the edge I knew the temperature and that's all I knew so f equals U on Boundary and we denote the boundary as partial Omega this is just how we Define that's just the notation that's used the question is is given this equation is true and given that I know the temperature values along the boundary does their exist or I should just write out exist a u satisfying this so specifically does there exist to you such that the laplacian of U is zero and such that U is the function when restricted to the boundary so this is this will be what I call q1 the question is quite a bit difficult to answer at least immediately in fact that's how most pde questions go like it's particularly difficult to show existence of a solution but we can perhaps think about it more physically What If instead of viewing it in terms of its temperature which can be a little bit weird we try to minimize the energy right if we minimize the energy then maybe then we'll have reached thermal equilibrium and this is exactly what mathematicians did at the time sorry one second oh I should note this this question is known as the dear slay problem you would have saw or you've likely seen a discrete version of this if you've done 18701 on the first problem set sometimes professors include it sometimes they don't it's totally fair if not um but yeah the first question is does there exist to you satisfying this and mathematicians at the time just try to study the energy so defined the energy of a function U to be one half times the integral over Omega of the gradient of U squared d a where a is the area differential on a on Omega this is known as the energy of the function U you can think about it as like the heat energy um but now the question is is if I minimize this is it a solution delivered to the differential equation and that's what mathematicians in particular did they said question question two um does there exist a function U specifically U in D in C2 because we want to be able to differentiate it twice such that energy of U is equal to the infamous of the energy so it's equal to e if where e is defined as Z Infamous of the energy so just take the infamous this assume that I know that in the end the energy should be insert value here does there exist to you that minimizes that energy and I'll come back to these questions in a moment but the thing I want to know is that what mathematicians did is they said oh if there exists a function in U that satisfies this then it'll solve the dirichlet problem um but the issue here is that sorry let me just read through this one more time so what I'm trying to say the issue here is that we still have to find this function right the question of existence is finding one that works and what was done is they were taking sequences of functions so what was done is you would study functions u n converging to U so you would construct u n such that u n would convert to the function U that you wanted and once this was done you would get that the energy of u n converges to the energy of U the proof of this fact just follows from taking the limit right here and applying uniform continuity so um but yeah this was the idea and this idea did not work weirdly Why didn't it work because even if there exists a sequence of these functions in C2 that converges to you how do we know that U is C2 question is you in C2 the answer isn't is not directly yes because the C2 functions are not complete are there not a bionic space or yeah it's an autobotic space the way that you can see this so not necessarily not compact the way that you see that it's non-compact is you just consider the sequence xn x to the n as your un each of these is differentiable twice um but U.N will converge Point wise to 1 or sorry to U of x equal to one when X is not equal when X is in zero to one and zero otherwise I.E if it's equal to zero and this is not twice differentiable it's not differentiable at the point zero it's discontinuous so the issue is that this function wasn't in C2 necessarily um as the way that you see that this isn't compact recall is by the fact that this would imply that it's not sequentially compact now what else could we do well mathematicians then also showed that if the sequence of u n is instead in C1 does that work and what they were able to show is that if the sequence is in C1 and such and the submit point Sorry if U1 was in C1 and this limit existed then you can show that U was in C2 so retroactively we'd be done but the issue is again sorry not that U was in C2 but that each of the UNS would be in C2 the issue is that again C1 is not compact so we're still not done now in the end we were able to solve this problem um with techniques that are quite beyond the scope of this class it's sometimes talked about in 18102 so see 18 and 102 sometimes it's talked about there it's talked about in the lecture notes that are on ocw so if you want to read more about that there but that's like the very very last lecture so it's difficult yes but interesting nonetheless but it's an interesting question how should we be approaching these problems should we be approaching them physically or should we be approaching them super rigorously and the answer is unclear right like it makes sense there's no direct answer to this question this gave us the framework to think about compactness even though the solution didn't work it motivated the development of compact sets for metric spaces right we have the C1 and C2 are metric spaces we want to understand if limit points are in your set and even though it wasn't the case for C1 and C2 it still developed all this terminology that as you've already seen is very very important so um yeah they were able to solve this problem eventually just using much different techniques okay so that's all I've prepared for today I mean this was mostly just a broad overview of where the material goes from here this the intuition that comes from metric spaces shows up all the time like for instance in foia analysis you want to understand a foyer series converts sheer function and in which sense and in that way it's related to metric spaces we've talked about how it applies to manifold Theory we've talked about functional analysis topology differential equations these are like five major applications of the material but the intuition that you gain for metric spaces will continue to work throughout your time at MIT so for instance um in general proving things axiomatically are functions that satisfy certain properties like Norms inner products and metrics is a useful skill but even more in particular I think a class like this and the class like real analysis is interesting because of all these different subsets and bigger subsets or bigger subsets is the terminology right we started off with studying euclidean space which was just a tiny little Dot in the set of metric spaces but there's obviously much more to be done here we talked about metric spaces but it makes sense to start right in the middle right and then work your way out work your way out towards topological spaces and then work your or potentially also work your way towards Norm spaces and bionic spaces in whichever order you choose but the main important thing the main reason I've been teaching this class for two years is to highlight you know the fact that um is is to give you the tools to be able to move forward into studying topological spaces in our spaces with some amount of intuition as much as that's possible okay so unless you have any questions well n 30 or 25 minutes early