WEBVTT
Kind: captions
Language: en
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So what we are supposed to worry about is
you know see we are worried about compactness
00:00:39.140 --> 00:00:45.540
of families of meromorphic functions, okay.
So so basically you are trying to to topology
00:00:45.540 --> 00:00:51.950
on collection on meromorphic functions and
see this is the technical background that
00:00:51.950 --> 00:00:57.910
is required to prove the Picard theorems and
many others theorems in fact because the root
00:00:57.910 --> 00:01:01.800
is through so called Montelâ€™s theorem, okay.
00:01:01.800 --> 00:01:06.140
So you know what I wanted to do is I want
you I want to go back to some topology and
00:01:06.140 --> 00:01:14.970
tell you about compactness, okay so that you
realize how whatever we are going to do is
00:01:14.970 --> 00:01:21.780
connected with all this we will have to bring
into the discussion Arzela-Ascoli theorem
00:01:21.780 --> 00:01:33.829
and then Montel's theorem, okay and then we
will you see so let me say the following thing
00:01:33.829 --> 00:01:36.220
you know what we have done so far is the following.
00:01:36.220 --> 00:01:43.130
We have defined a spherical derivative, alright.
So first of all so let me sum up what we have
00:01:43.130 --> 00:01:50.680
done so far, we have first we have tried to
think of a Meromorphic function as a continuous
00:01:50.680 --> 00:01:58.159
function even at a pole, okay that is because
we allowed the value infinity and so we are
00:01:58.159 --> 00:02:02.440
not only look at looking at complex valued
functions we are looking at functions with
00:02:02.440 --> 00:02:04.560
values in the extended complex plane.
00:02:04.560 --> 00:02:08.840
So we allow the value infinity the advantage
of allowing the value infinity is that a Meromorphic
00:02:08.840 --> 00:02:15.580
function at a pole can be given the value
infinity and it becomes a continuous map it
00:02:15.580 --> 00:02:19.879
becomes a continuous map when you consider
it as a map into the extended complex plane
00:02:19.879 --> 00:02:25.200
which is identified with the Riemann sphere,
okay you know it is a complete compact matrix
00:02:25.200 --> 00:02:27.530
space, alright.
00:02:27.530 --> 00:02:35.019
Now so first we have to deal with the point
at infinity, okay so we try to think of infinity
00:02:35.019 --> 00:02:40.049
as a isolated singularity when is infinity
an essential singularity, when is infinity
00:02:40.049 --> 00:02:45.879
a removable singularity, when is infinity
a pole, okay all these things we discussed
00:02:45.879 --> 00:02:51.450
behaviour at infinity, okay and then value
of the function at infinity that also we have
00:02:51.450 --> 00:03:05.849
we worried about, okay. So you allow in principle
you allow functions not only to take the value
00:03:05.849 --> 00:03:12.739
infinity but you also want to study functions
at infinity, okay so the you see these are
00:03:12.739 --> 00:03:21.920
two different concepts in the in the in the
in the co-domain of the function usually we
00:03:21.920 --> 00:03:27.260
are interested only complex functions but
now you allow the value infinity the advantage
00:03:27.260 --> 00:03:34.989
is that you can make a Meromorphic function
in a continuous map even at a pole, okay.
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Then not only that in the domain normally
the domain of the function is usually a domain
00:03:40.760 --> 00:03:45.889
in the complex plane but then you also want
to study the function at infinity itself.
00:03:45.889 --> 00:03:52.109
So you want to put infinity also in the domain,
okay so you have to define you have to understand
00:03:52.109 --> 00:03:58.359
the behaviour of the function at infinity,
okay. So a function may have a pole at infinity,
00:03:58.359 --> 00:04:02.569
it may go to infinity at infinity which is
the case for example if you take polynomials
00:04:02.569 --> 00:04:06.249
one constant polynomials they all have poles
at infinity.
00:04:06.249 --> 00:04:10.180
So you want to be able to work in this kind
of generality that is the reason why we have
00:04:10.180 --> 00:04:16.500
to study the function behaviour at infinity
thing of infinity as a isolated singularity
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and classify that kind of singularity and
we also want infinity to be a value taken
00:04:21.900 --> 00:04:28.970
by the function. For example the value of
a Meromorphic function at a pole, okay so
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we had to deal with infinity that was the
first thing.
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Then the second thing is is we were worried
about this defining spherical derivative,
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okay we were concerned about defining spherical
derivative and see the important thing about
00:04:47.100 --> 00:04:51.972
the spherical derivative is that the spherical
derivative will not change you can first of
00:04:51.972 --> 00:04:58.700
all you can define it for a Meromorphic function,
okay. So it is a derivative that will work
00:04:58.700 --> 00:05:03.440
even at a pole. See if you take a Meromorphic
function by which by definition is a function
00:05:03.440 --> 00:05:09.110
which is which has only pole singularities,
okay of course it may be completely holomorphic,
00:05:09.110 --> 00:05:14.540
may be completely analytic but we are interested
in in the situations we are going to really
00:05:14.540 --> 00:05:17.680
encounter those in which they are actually
poles, okay.
00:05:17.680 --> 00:05:23.690
So if you look at Meromorphic functions Meromorphic
functions you take a pole at the pole it is
00:05:23.690 --> 00:05:28.970
not differentiable because after all you know
at the pole the function goes to infinity
00:05:28.970 --> 00:05:34.390
and it is not differentiable because it is
a singular point it is not a removable singularity
00:05:34.390 --> 00:05:40.180
it is a pole, the function is not differentiable
in the usual sense of the term, okay and the
00:05:40.180 --> 00:05:43.750
function value is also not defined in the
usual sense of the term but what we do is
00:05:43.750 --> 00:05:48.210
we define the function value at the pole to
be infinity that is an extra definition we
00:05:48.210 --> 00:05:54.120
make and then since you cannot (def) you cannot
differentiate the function at a pole.
00:05:54.120 --> 00:06:03.560
So what you do is you do this of differentiating
not with respect to the usual metric which
00:06:03.560 --> 00:06:08.460
is Euclidean metric but you try to differentiate
with respect to the spherical metric so you
00:06:08.460 --> 00:06:15.460
introduce what is called the spherical derivative,
okay so that gives you a derivative of a function
00:06:15.460 --> 00:06:20.220
which will work even at a pole you see that
is the advantage, okay.
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If I take a Meromorphic function at a pole
I cannot differentiate it but if I take the
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spherical derivative the spherical derivative
will exist and I have told you that spherical
00:06:28.880 --> 00:06:35.340
derivative we calculated it last time I think
it was 2 2 divided by the modulus of the residue
00:06:35.340 --> 00:06:41.220
at the simple pole if it is a simple pole
and it is 0 if it is if it is not a simple
00:06:41.220 --> 00:06:43.280
pole if it is a pole of higher order.
00:06:43.280 --> 00:06:50.000
So even the spherical derivative make sense
and on top of all this one more beautiful
00:06:50.000 --> 00:06:52.820
thing about the spherical derivative is that
the spherical derivative will not change if
00:06:52.820 --> 00:06:57.650
you change the function by its reciprocal
that is if you take the Meromorphic function
00:06:57.650 --> 00:07:03.360
f and calculate the spherical derivative we
will get the same thing if you took 1 by f,
00:07:03.360 --> 00:07:09.500
okay mind you which is also Meromorphic with
only the only thing is that the poles and
00:07:09.500 --> 00:07:14.610
zeros will get interchanged when you move
from f to 1 by f but for 1 by f also if we
00:07:14.610 --> 00:07:17.140
calculate the spherical derivative you will
again get the same thing as a spherical derivative
00:07:17.140 --> 00:07:18.280
of f.
00:07:18.280 --> 00:07:24.020
So what it tells you is it if you are studying
the spherical derivative you can actually
00:07:24.020 --> 00:07:30.210
apply the thrift analytic functions and stop
worrying about even poles because at a pole
00:07:30.210 --> 00:07:35.550
of f I can simply if I am working with the
spherical derivative in the neighbourhood
00:07:35.550 --> 00:07:42.780
of a pole of f it is a same as a spherical
derivative in a neighbourhood of that point
00:07:42.780 --> 00:07:49.510
for 1 by f but for 1 by f that point is 0,
okay and therefore it is analytic 1 by f becomes
00:07:49.510 --> 00:07:53.230
analytic at that point that is the advantage.
00:07:53.230 --> 00:07:58.520
So working with a spherical derivative allows
you to reduce to analytic functions, okay
00:07:58.520 --> 00:08:01.510
you do not have to even worry about poles
that is an advantage and the other thing is
00:08:01.510 --> 00:08:07.180
it gives you a derivative of that works even
at poles, okay. Now so this is the this is
00:08:07.180 --> 00:08:11.831
this is what we have done so far. Now why
did we do all this we did do the idea is that
00:08:11.831 --> 00:08:18.740
there are two concepts on the one hand we
are worried about compactness of a family
00:08:18.740 --> 00:08:23.930
of Meromorphic functions that is our main
aim we want to do topology on a collection
00:08:23.930 --> 00:08:29.520
of Meromorphic functions on the space of Meromorphic
functions, we want to what kind of topology
00:08:29.520 --> 00:08:35.019
of course topology means there are many things
right there is connectedness connectedness,
00:08:35.019 --> 00:08:39.890
compactness so on and so forth but we are
interested in compactness, okay.
00:08:39.890 --> 00:08:46.529
And so that is on that is on the one end on
the other end what we have is this spherical
00:08:46.529 --> 00:08:54.370
derivative that is that is what we have which
is close to a derivative in the case of a
00:08:54.370 --> 00:09:03.009
in the case of a Meromorphic function, okay.
So now I need to I need to tell you people
00:09:03.009 --> 00:09:08.370
how I need to tell you people how to connect
these two things, okay so we need to do some
00:09:08.370 --> 00:09:10.870
topology. So I will give you some topological
background.
00:09:10.870 --> 00:09:23.360
So topological background so this is very
very important because only then you will
00:09:23.360 --> 00:09:30.509
understand what is going on, okay in the broad
sense what are we trying to do, okay. So if
00:09:30.509 --> 00:09:35.970
you want to get an idea of that this is very
very important. So so what we will do is we
00:09:35.970 --> 00:09:46.649
start with let us say let us say you are working
with a metric space suppose you are working
00:09:46.649 --> 00:09:52.249
with a metric space, okay mind you the topology
I am worried about the topological property
00:09:52.249 --> 00:09:54.790
that I am worried about is compactness, okay.
00:09:54.790 --> 00:10:01.249
So we will try to do try to understand everything
connected with compactness, right so start
00:10:01.249 --> 00:10:04.980
with a metric space which is a simplest kind
of topological space that you can think off
00:10:04.980 --> 00:10:22.019
which naturally occurs, okay then what do
you have? The following are equivalent is
00:10:22.019 --> 00:10:44.160
a compactness, number 2 is sequential compactness
and number 3 is the so called Bolzano-Weierstrass
00:10:44.160 --> 00:10:57.839
property, okay.
00:10:57.839 --> 00:11:04.600
So so we have these three these three properties
are equivalent, okay so I am just trying to
00:11:04.600 --> 00:11:12.610
recall what is equivalent to compactness,
okay just it helps to translate a property
00:11:12.610 --> 00:11:18.670
in different ways to find out equivalent properties
so that you can work with them, okay. So compactness
00:11:18.670 --> 00:11:23.910
is so this is a this is something that you
should have done in the first course in topology,
00:11:23.910 --> 00:11:28.600
what is compactness? Compactness is that every
open cover admits a finite sub cover, okay
00:11:28.600 --> 00:11:34.129
that is when you are are given a collection
of open sets whose union is a full space then
00:11:34.129 --> 00:11:41.110
it is enough to pick only finitely many among
those collections among in that collection
00:11:41.110 --> 00:11:45.809
whose union is also the whole space you can
extract a finite sub cover from every open
00:11:45.809 --> 00:11:47.379
cover that is compactness, okay.
00:11:47.379 --> 00:11:52.730
It is a very it is you see it is defined only
in terms of open sets and it is a very general
00:11:52.730 --> 00:11:57.329
thing so it works for any topological space
compactness space can be defined for any topological
00:11:57.329 --> 00:12:04.490
space because for any topological space open
sets make sense, okay defining the collection
00:12:04.490 --> 00:12:11.110
of open sets is exactly the what giving a
topology is, okay so compactness make sense
00:12:11.110 --> 00:12:16.829
for any topological space but it is a very
abstract notion at least for metric spaces
00:12:16.829 --> 00:12:21.810
where the topology is induced by a metric,
okay that means that you know your open sets
00:12:21.810 --> 00:12:30.939
are defined to be unions of open balls and
open balls are they are the analog of open
00:12:30.939 --> 00:12:35.730
balls in euclidean space you take points of
the space and then you take all points which
00:12:35.730 --> 00:12:41.369
whose distance from the given fixed end point
is less than some positive number which you
00:12:41.369 --> 00:12:43.529
call the radius of the open ball, okay.
00:12:43.529 --> 00:12:47.899
And of course you say strictly less than because
if you put less than or equal to then you
00:12:47.899 --> 00:12:51.589
are also include the boundary and it will
not remain an open set it will become a close
00:12:51.589 --> 00:12:55.760
set, okay so you put strictly less than the
distance should be strictly less than some
00:12:55.760 --> 00:13:02.420
positivities, okay and if I said is called
open if it is union of such open balls and
00:13:02.420 --> 00:13:07.649
this is how you and you know this involves
the notion of distance that is why the metric
00:13:07.649 --> 00:13:09.399
in the space is used.
00:13:09.399 --> 00:13:14.690
So the metric space the metric induces a topology
so when we say metric space and you think
00:13:14.690 --> 00:13:20.119
of it as a topological space you always mean
the topology induced by the metric, okay the
00:13:20.119 --> 00:13:24.149
open sets are precisely those which are given
by union of open balls and open balls are
00:13:24.149 --> 00:13:30.610
defined by the metric alright. Now for such
a metric space compactness which is a very
00:13:30.610 --> 00:13:35.820
abstract thing is connected with what is is
equivalent to sequential compactness, what
00:13:35.820 --> 00:13:40.329
is sequential compactness it has to do with
sequences what it says is that you give me
00:13:40.329 --> 00:13:46.240
any sequence of points in the space I can
always find a convergent subsequence that
00:13:46.240 --> 00:13:49.050
is what sequential compactness is, okay.
00:13:49.050 --> 00:13:55.110
If you give me a sequence in the space the
sequence itself may not converge but at the
00:13:55.110 --> 00:14:01.800
worst you can pick out a subsequence which
converges, okay that is sequential compactness
00:14:01.800 --> 00:14:05.920
and that is equivalent to compactness is what
this basic result says. And then there is
00:14:05.920 --> 00:14:11.459
a third property which is called the Bolzano-Weierstrass
property what is Bolzano-Weierstrass property?
00:14:11.459 --> 00:14:17.959
It is just a property which is satisfied by
the euclidean spaces which you namely which
00:14:17.959 --> 00:14:23.480
you would have come across namely the fact
that you take any infinite subsets it has
00:14:23.480 --> 00:14:29.879
an accumulation point or a limit point, okay
given any infinite subset all there is a cluster
00:14:29.879 --> 00:14:38.899
point there is a point where there is a point
at the space such that if you take any open
00:14:38.899 --> 00:14:42.809
neighbourhood of that point and delete that
point there is a point of your infinite subset
00:14:42.809 --> 00:14:43.809
there, okay.
00:14:43.809 --> 00:14:48.910
So points of your infinite subset come closer
and closer and closer to atleast one point
00:14:48.910 --> 00:14:54.589
of the space and that point is a limit point
of that set, okay. Now that every infinite
00:14:54.589 --> 00:14:59.550
subset has a limit point is Bolzano-Weierstrass
property and that is also equivalent the space
00:14:59.550 --> 00:15:07.089
having this property is also is also compact
okay so all these three are three different
00:15:07.089 --> 00:15:16.170
avatars of compactness, okay alright sequential
compactness and then Bolzano-Weierstrass property,
00:15:16.170 --> 00:15:17.660
okay.
00:15:17.660 --> 00:15:26.790
And well if you are looking at euclidean spaces
okay that is R n n dimensional real spaces
00:15:26.790 --> 00:15:34.220
finite dimensional real spaces then what happens
is that this is also equivalent to if you
00:15:34.220 --> 00:15:38.540
look at a subset of euclidean space, compactness
is equivalent to closeness and boundedness
00:15:38.540 --> 00:15:44.610
put together, okay and that is what we most
of the time when you are working in R n n
00:15:44.610 --> 00:15:51.050
dimensional real space we keep using that
all the time. Whenever you want to say something
00:15:51.050 --> 00:15:55.040
is compact you say it is you just verify that
it is closed end bounded.
00:15:55.040 --> 00:15:59.240
For example if you take the close disk in
the complex plane that is close disk in the
00:15:59.240 --> 00:16:07.829
complex plane is compact because it is disk
of finite radius so it is bounded and it is
00:16:07.829 --> 00:16:12.489
closed so it is both closed and bounded so
it is compact so we keep using this all the
00:16:12.489 --> 00:16:13.489
time, okay.
00:16:13.489 --> 00:16:23.910
So let me write that down for for euclidean
spaces
00:16:23.910 --> 00:16:47.509
R to the n we also we also have equivalence
of the above with with 4 so this is for if
00:16:47.509 --> 00:16:54.889
fact I should say for subsets of for subsets
of euclidean spaces. So the subset should
00:16:54.889 --> 00:17:10.420
be closed and boundedness, okay. So if something
is closed and bounded is compact and conversely,
00:17:10.420 --> 00:17:20.670
okay. So mind you you know you know my bag
what is the background of our trying to understand
00:17:20.670 --> 00:17:25.240
all this the background of our trying to understand
all this is you want to do this for functions
00:17:25.240 --> 00:17:28.990
for space of functions you want to do this
for space of functions.
00:17:28.990 --> 00:17:36.590
For a space of functions if you take a space
of functions it will be a subset of all functions
00:17:36.590 --> 00:17:40.000
of the given type. So for example if you take
a space of continuous functions, real valued
00:17:40.000 --> 00:17:45.820
functions it will be a subset of space of
all continuous if you want continuous bounded
00:17:45.820 --> 00:17:51.910
real valued functions, okay or you might be
looking at a space of analytic functions or
00:17:51.910 --> 00:17:57.240
you might be looking at a space of Meromorphic
functions that is the that is the background
00:17:57.240 --> 00:18:00.850
in which that is the generality in which you
want to do all this and you want to make sense
00:18:00.850 --> 00:18:04.140
of compactness for such a set of functions.
00:18:04.140 --> 00:18:09.370
So usually we use various sometimes we say
family of functions if you want to specify
00:18:09.370 --> 00:18:15.330
an index set, or sometimes we say sequence
of functions if you want to think of sequence
00:18:15.330 --> 00:18:23.960
of elements which each is a function or you
take a subset of the space of all functions
00:18:23.960 --> 00:18:28.920
okay so you refer to it in different ways
but then basically you are looking at a subset
00:18:28.920 --> 00:18:33.810
of functions and you want to study compactness
for that, okay.
00:18:33.810 --> 00:18:44.520
Now now you see the question is ofcourse that
you know how do you how do you go from this
00:18:44.520 --> 00:18:51.880
to something else. So there is a very important
there is a very very important property and
00:18:51.880 --> 00:19:03.930
that is called total boundedness, okay there
is something called total boundedness, okay.
00:19:03.930 --> 00:19:08.020
Now what is this total boundedness? It is
a very very strong form of boundedness it
00:19:08.020 --> 00:19:10.530
is a very very strong form of boundedness.
00:19:10.530 --> 00:19:14.940
So what is this total boundedness so I will
try to explain to you so basically what happens
00:19:14.940 --> 00:19:24.410
is that you know you have some space x okay
and let us assume that x is x is a say metric
00:19:24.410 --> 00:19:34.890
space. Suppose x is a metric space there is
something so the idea of total boundedness
00:19:34.890 --> 00:19:44.930
is like is to you know fill out the whole
space by finitely many open disks of a fixed
00:19:44.930 --> 00:19:51.640
radius, okay no matter how small that radius
may be that is the idea.
00:19:51.640 --> 00:20:07.000
So total boundedness
so here is my space x it is a metric space,
00:20:07.000 --> 00:20:15.410
okay and then for every epsilon positive no
matter how small it is there exist a subset
00:20:15.410 --> 00:20:21.301
A epsilon subset of x A epsilon and this is
the point is this is a finite set so it is
00:20:21.301 --> 00:20:31.950
only a finite collection of points A epsilon
finite, okay such that you see the union if
00:20:31.950 --> 00:20:49.300
you take the union of all the if I take the
union of all the open balls centred at points
00:20:49.300 --> 00:21:00.171
x i of A epsilon and take radius epsilon and
I do this for i equal to i so you know in
00:21:00.171 --> 00:21:06.010
fact let me not put a subscript let me get
rid of the subscript and just put x belongs
00:21:06.010 --> 00:21:10.780
to A epsilon. So when I say x belongs to A
epsilon there are only finitely many such
00:21:10.780 --> 00:21:13.230
x because A epsilon is finite.
00:21:13.230 --> 00:21:20.560
And for each such x i so you know so here
is one x here and then I have this this ball
00:21:20.560 --> 00:21:30.880
centred at x this open ball centred at x and
radius epsilon, okay and I do this for all
00:21:30.880 --> 00:21:36.310
the points of A epsilon I take the open ball
centred at each of the points of A epsilon
00:21:36.310 --> 00:21:42.890
with radius epsilon, okay and if I take the
union that should be equal to x that is the
00:21:42.890 --> 00:21:53.580
that is the requirement. So I can cover x
by finitely many such open balls and the beautiful
00:21:53.580 --> 00:22:01.130
thing is that the all these balls have the
same radius epsilon, okay and there are only
00:22:01.130 --> 00:22:05.020
finitely many of them they cover all of x,
okay.
00:22:05.020 --> 00:22:12.250
And this must happen for every positive epsilon
this should happen for every epsilon if it
00:22:12.250 --> 00:22:18.530
happens for a particular epsilon such a collection
of points finitely many points see epsilon
00:22:18.530 --> 00:22:23.840
is called an epsilon net, okay so this is
called an epsilon net so this is called an
00:22:23.840 --> 00:22:36.320
epsilon net and this is the net condition,
okay. Now this is this this see you are saying
00:22:36.320 --> 00:22:44.230
that no matter how small an epsilon you take
I can make sure I can find I can make sure
00:22:44.230 --> 00:22:50.280
to find only finitely many points in x such
that every other point of x is at a distance
00:22:50.280 --> 00:22:54.740
less than epsilon from atleast one of these
balls that is what you are saying, right so
00:22:54.740 --> 00:22:59.470
let me repeat it what is this epsilon net
condition? Given an epsilon no matter how
00:22:59.470 --> 00:23:06.080
small, okay you are able to find finitely
many points that they will constitute their
00:23:06.080 --> 00:23:10.970
elements of the set A epsilon such that given
any point of x it is distance from atleast
00:23:10.970 --> 00:23:15.180
one of these points is less than epsilon that
way you cover every point of x, okay.
00:23:15.180 --> 00:23:21.620
It is a very very strong point and you know
the point is that this is this is a very strong
00:23:21.620 --> 00:23:27.650
form of boundedness because this implies boundedness
because you see why does this imply boundedness
00:23:27.650 --> 00:23:39.680
if you see you know so so let me say it inverts
so let me put this here this implies rather
00:23:39.680 --> 00:23:53.240
let me write it above I will put it here this
implies boundedness and why is that true?
00:23:53.240 --> 00:23:59.840
See in fact what it will tell you is that
you know it will tell you that diameter of
00:23:59.840 --> 00:24:05.560
x is comparable to the diameter of any of
these A epsilons that is what it will tell
00:24:05.560 --> 00:24:12.130
you. See what is the diameter of a space a
metric space the diameter is supremum of the
00:24:12.130 --> 00:24:18.440
lens between two of its points and you allow
those two points to just vary so it is like
00:24:18.440 --> 00:24:24.850
trying to draw the longest line segment through
that space if you want to think of it and
00:24:24.850 --> 00:24:30.970
measure the length of that ofcourse this longest
may not exist so it might become infinite
00:24:30.970 --> 00:24:37.160
so your space may have infinite diameter.
So that is the reason instead of taking maximum
00:24:37.160 --> 00:24:38.530
you take supremum.
00:24:38.530 --> 00:24:43.710
So basically what you do is you take supremum
of the distances between two points of your
00:24:43.710 --> 00:24:49.900
space and you allow the points to vary, okay.
If that has a finite value that is called
00:24:49.900 --> 00:24:56.600
the diameter of your space and the point is
if your space is totally bounded then its
00:24:56.600 --> 00:25:01.370
diameter can be compared to any epsilon net.
So for example you know if you take an epsilon
00:25:01.370 --> 00:25:07.380
net such as A epsilon, okay and you measure
the distance between two points of the space,
00:25:07.380 --> 00:25:12.740
what you can do is that each of these points
is within an epsilon from one of the points
00:25:12.740 --> 00:25:17.500
in the net and the distance between two points
and the net cannot exceed the diameter of
00:25:17.500 --> 00:25:23.370
A epsilon mind you A epsilon is only a finite
set so it has a finite diameter the finite
00:25:23.370 --> 00:25:27.510
subset always has a finite diameter because
you are just going to take supremum of the
00:25:27.510 --> 00:25:33.160
finitely many distances between pairs of points
in that set and that is only finitely many
00:25:33.160 --> 00:25:34.710
pairs, okay.
00:25:34.710 --> 00:25:44.410
So the diameter of any finite subset is of
course finite, alright and and you know if
00:25:44.410 --> 00:25:49.870
you look at the diameter of A epsilon okay
that will be an upper bond for the distance
00:25:49.870 --> 00:25:55.220
between any two points of A epsilon, okay.
Now if you take any two points of the space
00:25:55.220 --> 00:26:00.380
for each point I can find a point of A epsilon
which is to within an epsilon so what this
00:26:00.380 --> 00:26:04.900
comparison will tell you by triangle inequality
is that the diameter of the space cannot exceed
00:26:04.900 --> 00:26:11.070
the diameter of A epsilon plus 2 times epsilon
if you write it out, alright if you use a
00:26:11.070 --> 00:26:15.750
triangle inequality the diameter of the space
cannot exceed the diameter of A epsilon plus
00:26:15.750 --> 00:26:19.600
two epsilon for every epsilon greater than
0 that will tell you that the space has finite
00:26:19.600 --> 00:26:20.760
diameter, okay.
00:26:20.760 --> 00:26:27.310
So you can see it is a very very strong condition
and ofcourse if the diameter is finite it
00:26:27.310 --> 00:26:32.950
means the space is bounded if the diameter
of the space is finite that means if the diameter
00:26:32.950 --> 00:26:38.540
of the space is say lambda positive number
lambda then the space is ofcourse bounded
00:26:38.540 --> 00:26:45.020
because you take any point in that space and
take disk take an open ball of radius greater
00:26:45.020 --> 00:26:51.010
than lambda the whole space will be contained
in that so it becomes bounded. So totally
00:26:51.010 --> 00:26:54.360
bound is very strong it implies boundedness,
alright.
00:26:54.360 --> 00:27:02.130
And in fact actually for euclidean spaces
you see boundedness is same as total boundedness,
00:27:02.130 --> 00:27:09.450
okay and in fact more more generally if you
take a Banach space you take a complete norm
00:27:09.450 --> 00:27:16.210
linear space you take a Banach space even
for a Banach space you see the fact that every
00:27:16.210 --> 00:27:20.340
subset that is bounded is also totally bounded
is a very strong condition it will happen
00:27:20.340 --> 00:27:24.570
if and only if the Banach space is finite
dimensional, okay it cannot happen in infinite
00:27:24.570 --> 00:27:26.850
dimensional Banach space, okay.
00:27:26.850 --> 00:27:31.810
So if you go to infinite dimensional spaces
okay which is like non-euclidean kind of spaces
00:27:31.810 --> 00:27:38.160
then you are in trouble, okay there is a difference
between total boundedness and boundedness,
00:27:38.160 --> 00:27:42.580
okay but total boundedness A priory is a very
very strong condition, right. So for example
00:27:42.580 --> 00:27:50.029
you know if you take R infinity infinite sequences
of you know infinite sequences of real numbers.
00:27:50.029 --> 00:27:56.000
Then what will happen is that if you take
the unit ball there that is ofcourse you know
00:27:56.000 --> 00:28:01.130
bounded but it is not totally bounded because
if you take the diagonal sequence which consist
00:28:01.130 --> 00:28:08.440
of 0 everywhere 1 in the ith place for i equal
to 1, 2, 3, 4 is called the diagonal sequence,
00:28:08.440 --> 00:28:14.240
okay then that sequence will never have a
convergence subsequence because distance between
00:28:14.240 --> 00:28:19.000
any two points of that sequence is finite
quantity.
00:28:19.000 --> 00:28:26.000
So it is a finite positive quantity it is
a constant, okay and therefore you cannot
00:28:26.000 --> 00:28:29.080
have a convergence subsequence because if
there is a convergence subsequence then distance
00:28:29.080 --> 00:28:33.870
between points should come closer and closer
but this does not happen all distance between
00:28:33.870 --> 00:28:41.360
any two points in that sequence is equal to
some fixed positive quantity, okay. So if
00:28:41.360 --> 00:28:50.290
you take R infinity the unit ball is bounded
but this is certainly not totally bounded,
00:28:50.290 --> 00:28:51.290
okay.
00:28:51.290 --> 00:28:55.550
And what I am trying to say here is basically
a theorem in fact what I am trying to say
00:28:55.550 --> 00:29:01.570
is that you know if you have compactness which
I have written on the left side in its various
00:29:01.570 --> 00:29:06.330
avatars in its various avatars I have written
compactness sequential compactness. See all
00:29:06.330 --> 00:29:16.200
these things they all implied total boundedness,
okay compactness or sequential compactness
00:29:16.200 --> 00:29:20.970
or Bolzano-Weierstrass property they are imply
total boundedness ofcourse what I wrote below
00:29:20.970 --> 00:29:30.550
is that you know they they all imply for euclidean
spaces they all imply closeness and boundedness,
00:29:30.550 --> 00:29:36.910
okay but it is not just compactness in general
gives you very strong thing it gives you total
00:29:36.910 --> 00:29:37.910
boundedness.
00:29:37.910 --> 00:29:44.110
Now the question is how do you come back from
total boundedness how to you come back to
00:29:44.110 --> 00:29:52.880
compactness, okay and the answer to that is
theorem if you want to come back this side
00:29:52.880 --> 00:30:00.310
what you need to do is you will have to put
the condition that your space is complete,
00:30:00.310 --> 00:30:07.200
okay. So with completeness so let me let me
try to use a different color so that you understand
00:30:07.200 --> 00:30:19.620
the implication that is involved with completeness.
So if I go like this plus completeness. If
00:30:19.620 --> 00:30:28.400
I take a metric space that is totally bounded
and I add completeness to it, okay completeness
00:30:28.400 --> 00:30:33.340
is the condition that every Cauchy sequence
convergences, okay I will put this completeness
00:30:33.340 --> 00:30:40.700
condition then you will get compactness, okay
this is a so you know in so what I am trying
00:30:40.700 --> 00:30:46.200
to tell you is see we are trying to move from
compactness which is a very abstract thing
00:30:46.200 --> 00:30:49.620
to something that is related to boundedness,
okay.
00:30:49.620 --> 00:30:55.370
And why we are doing this is because when
you are studying functions or spaces of function
00:30:55.370 --> 00:31:00.710
it is easier to verify something is bounded
if you want to say a function is bounded that
00:31:00.710 --> 00:31:08.650
is easy to verify, okay where is if I want
to say that a collection of functions is compact
00:31:08.650 --> 00:31:14.770
it is very very abstract, okay. So boundedness
is something that for functions it is easy
00:31:14.770 --> 00:31:20.440
to verify under under many situations.
00:31:20.440 --> 00:31:24.820
So that is why we are trying to move from
compactness to boundedness and this is the
00:31:24.820 --> 00:31:32.200
route compactness implies boundedness for
example in euclidean space, okay and if fact
00:31:32.200 --> 00:31:38.080
it is equivalent to closeness and boundedness
but if you forget euclidean spaces, compactness
00:31:38.080 --> 00:31:42.750
gives you total boundedness which is a very
strong form of boundedness but from total
00:31:42.750 --> 00:31:46.630
boundedness if you want to come back to get
compactness you need to complete this.
00:31:46.630 --> 00:31:55.340
So the translation so far is we so basic topology
teaches us that you can translate from compactness
00:31:55.340 --> 00:32:02.520
to completeness plus total boundedness, okay.
Now what I need to do is that I will have
00:32:02.520 --> 00:32:12.090
to now translate all this to functions spaces
of functions, okay and that is where what
00:32:12.090 --> 00:32:19.230
we will come across is the so called Arzela
Ascoli theorem and then so what we will do
00:32:19.230 --> 00:32:26.000
is there we will try to see how to decide
a certain collection of functions is compact,
00:32:26.000 --> 00:32:27.000
okay.
00:32:27.000 --> 00:32:32.070
So you will you can expect that you know you
will say the condition that will be total
00:32:32.070 --> 00:32:36.059
boundedness and completeness but completeness
you will get if the collection is already
00:32:36.059 --> 00:32:39.350
a close subset because a close subset of a
complete space is complete. So if you are
00:32:39.350 --> 00:32:44.750
working for example with the Banach space
of real valued functions or complex valued
00:32:44.750 --> 00:32:52.840
bounded continuous functions, okay then any
subset any close subset that any close subset
00:32:52.840 --> 00:32:54.590
will automatically be complete.
00:32:54.590 --> 00:32:58.670
So the only thing that is required for it
to be compact by what I just said is that
00:32:58.670 --> 00:33:04.080
it should be totally bounded, okay but then
from total boundedness you want to even remove
00:33:04.080 --> 00:33:09.430
the totalness and come down to boundedness
that is where you have to bring in the Arzela
00:33:09.430 --> 00:33:11.680
Ascoli theorem, okay. So I will explain that
in the next lecture.