l infinity norm
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okay so welcome to this next video in the playlist on functional analysis so were continuing with our study of metric spaces and were going to look at an even more abstract metric space now were going to look at the L infinity space L infinity space our infinity space okay and this is a really really important space for us in functional analysis so the N Infinity space consists of a set so the underlying set which well call X is going to be equal to or you could what is the L infinity for L infinity the L infinity space is the set along with the metric so well call it L infinity but remember that the L infinity space is not just this set but also the metrical structure well it has more than just metrical structure but after now were just viewing it with its metrical structure and its the set of all sequences of real numbers and is equal to 1 to infinity of real numbers so xn is an element of the real numbers such that for all for all thats cool lets call this sequence this overall sequence were going to call an element X so for all for all for all for all X is an element of this space if all element for all X of well and hmm no no Im going to write this differently for all n for all n is an element of the natural numbers X n is less than or equal to some M which is dependent on the sequence so let me explain what I mean its basically saying oh I want the set of all sequences which are bounded so for instance if I take the function if I take the sequence which goes lets say 0 1 0 1 0 1 0 1 and just keeps switching between 0 & 1 then that would be a sequence in this space the reason being that if we call this sequence the overall sequence is going to be called axe even though the individual terms are you know X 1 X 2 X 3 it is overall the overall sequence has a name and thats going to be called X and basically all the elements X n are less than or equal to some M which depends on X so for each sequence there is a different M but or for which all of the terms are less than or equal to that M so in this case I could just that M be equal to 1 and indeed all the elements of this sequence would be there in fact I think you want strict inequality there you want m to be great so lets just take one point 1 obviously if you can find an M for which equality holds then then you obviously can find an M for which which is greater than all of the terms but I think usually you do put the greater than and you want your m to be greater than all other terms so at one point one ha is greater than all these terms oh and Ive made a terrible mistake we want the modular sign there you want to take the modulus of it for instance I could take a sequence that goes off to negative infinity different since if I took the sequence zero negative one negative two negative three negative four prior to my correction of my definition this would have been an element of this set I do not want that to be an element of the set I want the set of all bounded sequences so this is the set of all bounded sequences meaning that if I take some NX some NX and I draw a bool one interval of size R of radius MX I take the interval negative n x2 plus M X then all of the terms of the sequence are within there and obviously you will have to vary this M for each sequence its going to be different for each sequence but providing there is such an EM that there exists such an M for which all the terms of the sequence are within this interval negative M x2 plus MX then your sequence is said to be bounded and your sequence will in the set so this is the set of all bounded sequences of real numbers of real numbers so this is not an element of this set because this is not bounded below it goes off to negative infinity so it should be bounded are both above and below by the same value M X so we can say we can encapsulate that by saying that the modulus of X M must be less than or less than this M X the must exist so I should put for all and is an element with natural numbers there exists an M X such that the modulus of xn is less than M X okay so thats good so how so thats our set so thats the first complicated thing about this silly metric space that the actual set on which it stood as our ion which is defined is this convoluted complicated thing that we cant well we
can do our best to visualize it but how big is this thing we have no idea we cant write it down enough set up paper so its its quite an abstract set to be defining a metric space drip-drop and now were going to define the metric space structure so the were going to take a function D which is going to map an ordered pair x and y or onto the non-negative real numbers and remember x is a sequence so x is some x1 x2 all the way off to a countably infinite ly many terms and y is also a sequence y1 y2 all the way off – cant be infinitely many terms will we define the distance between x and y to be the supremum over N is an element of the natural numbers of the modulus of y ym minus XM so basically what we do is we go along every single every single term term by term so basically what we do is we construct y minus x which is going to be equal to a new sequence which is y1 minus x1 y2 minus x2 you.why 3 minus X 3 so all Im doing is subtracting all the components of X from Y and we get this brand new sequence and basically because these were both bounded sequences this is also going to be a bounded sequence because any term in here is going to be have modulus less than or equal to M and any term in here is going to have much oh well lets lets do this rigorously and so Y minus X is a bounded sequence the reason being that for all n is an element of the natural numbers what the modulus of yn is lets not eat less than some my and the modulus of xn is the less than some M X so the modulus of yn plus xn is going to be less than my plus MX so for the same reason as we did in the previous video was it where we showed that if you have two inequalities like this it implied this inequality here what you do is you add on M X to both sides here you then say what would you do then you then say lets add all so lets just go through this again so to this one what we do is we add on M X onto both sides so we get that this inequality holds and then to this one what we do is we add on the modulus of yn onto both sides we get XM plus mod of Y n is less than M X plus the mod of Y M and then basically we use we use transitivity to say therefore that XM + ym which is less than this bit here here is less therefore less than this because this is less than this so basically it happens because this and this are the same thing so we overall get that this here which is the same thing as here is less than this thing here which is the same thing here okay so thats just basic properties of the order fielder for your numbers so we get that and we also know of course that the modulus of yn oh dear Ive made this tight mistake that because we dont actually want well we want the modulus of y and minus XM but that is going to be less than or equal to the modulus of yn plus the modulus of negative x n Seaview were using this inequality again the modulus of a plus B is less than the modulus of a plus the more just B so Im just swallowing this negative into here and get saying that its plus negative x cents right this is equal to well its less than or equal to the modulus of y and plus the modulus of negative xn but thats obviously is equal to the modulus of XM so then I can apply this and say that the modulus of yn minus xn is less than my plus M X so thats how I know that this sequence this sequence yn minus XM is bounded and now it is bounded it the set of all values in this sequence is a bounded set of real numbers so by the least upper bound property it has a supremum so this this distance function is indeed well-defined so by at least upper bound property property supremum exists so thats excellent so we have to find a welder whoa so weve got a well-defined function now what we want to make sure is that this actually obeys the properties of a metric space okay so lets rewrite out what our function was the distance between x and y is equal to the supremum over n is an element of the natural numbers of the modulus of yn minus xn so just beware that obviously this were taking the modulus here rather than just having the value yn minus xn so were not taking the supremum over were not actually taking the supremum over this entire set of values of the sequence were taking the supremum over all of these values
in the sequence the modulus of them but if they are all bounded then obviously the modulus of them is also bounded so the supremum would exist again okay and so this is our definition of the distance function for our and infinity space so lets make sure it Abbes the X into a metric space so the first one is that the distance between X and y is an element of the norm negative real numbers so how are we going to make sure of that well Y and minus xn is going to be a banded real number but it might be positive or negative but when I take the absolute value of it its obviously going to be positive so Im taking the soup premium of a bunch shop well Im sorry its going to be non-negative it could be zero of course so were taking the supremum of a bunch of non-negative real numbers so it is indeed going to be greater than or equal to zero and so is indeed going to be a non-negative real number excellent so second property that the distance between X and X is equal to zero so if we have X and X then were taking the supremum the distance between X and X is the supremum over N is an element of the natural numbers of the modulus of xn minus XM well the absolute value of xn minus xn is going to be equal to zero because xn minus xn is 0 and the absolute value of zero is the so were taking the supremum over n is an element of the natural numbers of zero effectively and for every single natural number n its going to stay zero so this is a set basically containing just zero so this is a set containing zero and nothing else because for every single natural number it is zero so this is equal to the supremum of the set containing zero and obviously that is equal to zero so the least of the bands are set containing zero is zero so thats one way now the other way is if the distance between x and y is equal to zero I want you to show me the x and y are in fact equal to one another okay so if distance between x and y is equal to zero that implies that the supremum over all natural numbers of the of ym minus xn is equal to zero so that implies because were taking the supremum over a great big sets and remember what this means this means the supremum of the set containing the mortise of y1 minus x1 the month the modulus of y2 minus x2 the modulus of y 3 minus x3 and you go on and on and on and on and on okay so you have these huge know you have a huge number of of non-negative real numbers now if the supremum of this set is equal to zero these are all greater than equals there it implies every single one of these must equal zero because if one if even one was greater than zero remember none of these can be negative theyve all got to be either zero or a refer or positive if one of these was positive then that would imply that the supremum was going to be greater than zero so if the supreme is equal to 0 it implies that for all n is an element of the natural numbers are yn minus xn is equal to zero which implies since if the absolute value is equal to zero that implies that yn minus xn is equal to there the only number which the app for which the absolute value is zero is zero so it implies that yn is equal to X n for all N is an element of the natural numbers which implies that the entire two sequences are exactly the same because y1 is equal to X 1 Y 2 is equal to X 2 y 3 is equal to X 3 every single component is the same which implies that the whole sequence X is equal to the whole sequence Y so thats excellent okay so the third property is symmetry that the distance between x and y is equal to the distance between y and x so the distance between x and y is the supremum of n its an element of the natural numbers of the mod yn minus xn now clearly the absolute value doesnt care which or do we do these to it so this is equal to the supremum and its an element of the natural numbers of xn minus yn which is then equal to the distance between x and y so symmetry is usually a very easy properties to show and now weve got the triangle inequality so how do we show the triangle inequality so want to show the triangle inequality so for so we take a third sequence which is Z which is Z 1 Z 2 all the way on and we want to show that to show that the XY is less than or equal to D XZ plus d zy ok so now the distance between x
and y we know is equal to the supremum over N is Homam is an element of the natural numbers of the modulus of y and minus xn well we apply our same trick as always its equal to the supremum of n is an element of the natural numbers of yn minus Zn plus Zn minus xn so theres nothing I have not done something I have done I have I have played a trick here but this is perfectly true you cannot stop me doing that theres nothing incorrect about what I have done its a bit odd initially but you cannot stop me from doing that so what I can then say is that this number yn minus ZM plus Z minus XM is less than or equal to yn minus ZM plus Z then mod of Zn minus xn and that just follows because mod a plus B is less than or equal to mod a plus mod B weve used this inequality a lot now so a were viewing here has been wire miners there then and be reviewing as being Z then minus XM okay so that implies that if we take the supremum over all natural numbers of this it implies that the supremum over all natural numbers of this of this of this object here why a minus there then plus Z then minus xn that that if we take that its going to be greater than or equal to the supremum over N is an element of the natural numbers of yn minus Zn plus there then minus XM so basically if I take the supremum over all natural numbers of this these values its going to be less than or equal to the supremum over all natural numbers of these values because all of these values are going to be either greater than or equal to each of the corresponding values in here so therefore when I take the least upper bound of this set it has to be greater than or equal to the least upper bound at this set okay so more paper now the supremum has a beautiful property of adding like this so this here is equal to the supremum of m is an element of the natural numbers of ym minus ZM plus the supremum and is an element of the natural numbers of ZN minus xn that works because these two are both are going both going to be non-negative real numbers so basically what you have is you are saying if were taking the supreme over this overall set every single element in here will be made up of two other so if you think about this being an element of this set up here so that call that at the Sun that its going to be made up of two elements from each of these sets so lets call this this West warm and the 2s2 and basically if we take the supremum of all these sets pause the Supreme of this set here and then that overall has to add up to the supremum of this set here and the intuition is that youre asking when youre taking the supremum over set what youre basically saying is youve got loads of numbers and youre asking what is its least upper bound what is it almost converging on what is the least value that is greater than or equal to every single value here and basically if you have if you split each of these elements up each of these elements in this set up into two components two non- components then and then take the supremum over both of those sets then the the absolute maximum value this is plus the absolute maximum this is has to be equal to this because if it was less than then you wouldnt possibly be able to get this value here and if it was greater than then you get a bigger value here basically so the supremum here is indeed equal to that so what we get is that the supremum over N is an element of the natural numbers of yn minus Zn plus Zn minus XM is the less than or equal to the supremum and is an element of the natural numbers YN minus seven plus the supremum n is an element of the natural numbers Zn minus XM okay so this thing here is the distance between y and X but this thing here is the distance between y and set distance between y sorry we wrote this originally as the distance between x and y and this thing here is the distance between Z and Y because Y appears first in here of Y appears secondary is trivial but just to make the notation consistent and this is the distance between X and z so just by the commutative property of the real numbers this is the distances X or between x and y is less than or equal to the distance between X and said first the distance between Z and Y so the triangle inequality does indeed hold true and so what we have done is we constructed a set of bounded sequences and we have defined a successful metric on that set
Functional Analysis (Field Of Study)
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