WEBVTT
Kind: captions
Language: en
00:00:29.310 --> 00:00:41.739
Let us continue with our discussion of this
Arzela-Ascoli Theorem, okay. So you know the
00:00:41.739 --> 00:00:48.370
situation is that we are trying to understand
what compactness means for functions, okay
00:00:48.370 --> 00:00:52.570
basically we want to understand compactness
for families of Meromorphic functions defined
00:00:52.570 --> 00:01:00.660
on a domain in the complex plane, okay or
external complex plane okay but then we want
00:01:00.660 --> 00:01:07.869
to get some idea about what compactness means
from basic topology, okay.
00:01:07.869 --> 00:01:18.360
So as I was telling you suppose you are having
a topological space, you have the notion of
00:01:18.360 --> 00:01:24.740
compactness, okay and if the topological space
is is a metric space then compactness is the
00:01:24.740 --> 00:01:33.570
same as it is equivalent to sequential compactness
and it is also equivalent to the Bolzano-Weierstrass
00:01:33.570 --> 00:01:37.759
property, the sequential compactness is the
property that every sequence has a convergent
00:01:37.759 --> 00:01:41.990
subsequence, okay the Bolzano-Weierstrass
property is a property that every infinite
00:01:41.990 --> 00:01:46.929
subset has a limit point, okay and these three
are equivalent, right.
00:01:46.929 --> 00:01:56.590
And then there is also this this this implication
that if you have a compact subsets it is always
00:01:56.590 --> 00:02:00.630
closed and bounded if you are in a metric
space, okay but the converse is not true.
00:02:00.630 --> 00:02:08.179
So if you take R infinity you can easily look
at the unit ball in R infinity the closed
00:02:08.179 --> 00:02:12.910
unit ball it will be closed and bounded but
it will not be compact because you can write
00:02:12.910 --> 00:02:19.020
out your sequence which does not admit any
convergent subsequence namely the sequence
00:02:19.020 --> 00:02:27.730
consisting of 1 at the ith place and 0 elsewhere,
okay any two terms of this sequence differ
00:02:27.730 --> 00:02:34.120
by a distance of root 2, okay are separated
by a distance of root 2 and therefore this
00:02:34.120 --> 00:02:39.590
sequence cannot have a convergent subsequence,
okay because terms of the sequence do not
00:02:39.590 --> 00:02:49.650
come you cannot find a subsequence whose terms
whose elements come closer and closer together,
00:02:49.650 --> 00:02:50.650
okay.
00:02:50.650 --> 00:02:58.569
So but ofcourse in any in any metric space
compactness implies closeness and boundedness
00:02:58.569 --> 00:03:04.099
the converse is not true the converse is true
for eucledian spaces and in fact I even told
00:03:04.099 --> 00:03:09.680
you that if you take a Banach space the condition
that it is finite dimension is equivalent,
00:03:09.680 --> 00:03:15.230
every closed subspace every closed bounded
subspace is compact if you want to put that
00:03:15.230 --> 00:03:19.709
condition then the Banach space has to be
finite dimensional it cannot happen if it
00:03:19.709 --> 00:03:20.709
is a infinite dimension Banach space, okay.
00:03:20.709 --> 00:03:29.690
So now you see alright so the so there is
a related notion that is this notion of total
00:03:29.690 --> 00:03:38.060
boundedness, okay which is implied by compactness,
alright total boundedness is that is basically
00:03:38.060 --> 00:03:46.390
the property that you know you can cover the
whole space by open balls of a fixed radius
00:03:46.390 --> 00:03:53.000
no matter how small the radius is the point
is ofcourse by finitely many such open balls
00:03:53.000 --> 00:03:54.000
okay.
00:03:54.000 --> 00:04:00.650
So given any positive radius epsilon I should
be able to find just finitely many open balls
00:04:00.650 --> 00:04:05.450
of radius epsilon whose union is the whole
space and I should I should be able to do
00:04:05.450 --> 00:04:10.049
this for every epsilon greater than 0 this
is total boundedness is very strong form of
00:04:10.049 --> 00:04:17.030
boundedness this implies boundedness, okay
and in fact space that is totally bounded
00:04:17.030 --> 00:04:22.720
is actually even it even has finite diameter,
okay because the diameter of the space can
00:04:22.720 --> 00:04:29.780
be compared to the diameter of this finite
collection of centres of these open disks
00:04:29.780 --> 00:04:36.900
which is called a net an epsilon net if the
radius you are talking about is epsilon, okay.
00:04:36.900 --> 00:04:47.940
And the point is that total boundedness itself
though it is a strong form of boundedness
00:04:47.940 --> 00:04:55.060
total boundedness is not enough to give compactness,
okay what you have to add to it is completeness.
00:04:55.060 --> 00:05:00.610
So if you have something that is complete
and totally bounded a metric space that is
00:05:00.610 --> 00:05:08.759
complete and totally bounded then it is compact,
okay and so that is the so that is another
00:05:08.759 --> 00:05:10.190
equivalent version of compactness.
00:05:10.190 --> 00:05:18.190
So you know so we check to check compactness
either you check the the abstract definition
00:05:18.190 --> 00:05:22.740
of compactness which is very hard you have
to take an arbitrary open cover and you have
00:05:22.740 --> 00:05:30.940
to show that there is a finite sub cover that
is very difficult in practise, okay and probably
00:05:30.940 --> 00:05:37.510
if you are given a particular open cover specific
open cover may be you might be able to because
00:05:37.510 --> 00:05:41.259
of your knowledge of geometry and topology
you might be able to extract a finite sub
00:05:41.259 --> 00:05:50.550
cover but you cannot do this in an arbitrary
way so it is abstract, okay trying to get
00:05:50.550 --> 00:05:54.960
hold of a finite sub cover from an any given
open cover is a very abstract thing it is
00:05:54.960 --> 00:05:56.610
not easy to check, okay.
00:05:56.610 --> 00:06:01.819
So we end up checking sequential compactness,
okay namely you take a sequence and show that
00:06:01.819 --> 00:06:06.090
there is a convergent subsequence if you check
that then that is equivalent to compactness.
00:06:06.090 --> 00:06:12.580
The other thing you can check is that the
space has the metric space has the Bolzano-Weierstrass
00:06:12.580 --> 00:06:20.849
property show that every infinite subset has
a limit point, okay
00:06:20.849 --> 00:06:25.889
and ofcourse if you are in if you are working
with subsets eucledian space you know what
00:06:25.889 --> 00:06:31.199
to do you will just check that the subset
is that you want to say is compact is both
00:06:31.199 --> 00:06:34.300
closed and bounded so there is not much to
do, alright.
00:06:34.300 --> 00:06:40.450
But point is that and ofcourse now there is
also this new there is also this extra condition
00:06:40.450 --> 00:06:48.509
that you know if you want to check a spaces
compact you check that it is complete and
00:06:48.509 --> 00:06:56.379
it is totally bounded, okay and ofcourse completeness
is the condition that every Cauchy sequence
00:06:56.379 --> 00:07:08.699
converges, okay that is you know whether that
can be easily checked or not depends on the
00:07:08.699 --> 00:07:13.490
particular case you are looking at okay it
is also not so easy, okay.
00:07:13.490 --> 00:07:19.840
And the other thing is total boundedness and
that is also very abstract, okay total boundedness
00:07:19.840 --> 00:07:24.139
is very abstract thing you have to say that
you know you can find an epsilon net for every
00:07:24.139 --> 00:07:29.220
epsilon greater than 0 and that is so you
have to produce an epsilon net for every epsilon
00:07:29.220 --> 00:07:35.220
greater than 0 and epsilon net is those finitely
many points in the space centred at which
00:07:35.220 --> 00:07:39.400
if you take open balls of various epsilon
these balls will cover the whole space, okay
00:07:39.400 --> 00:07:44.629
you have to you have to show the existence
of those finitely many points that epsilon
00:07:44.629 --> 00:07:48.500
net and you should do this for every epsilon
it is not an easy thing to do, alright.
00:07:48.500 --> 00:07:51.810
So these are the various versions of compactness
that we have but you know we are interested
00:07:51.810 --> 00:07:58.310
in compactness of functions okay. Now for
functions what is it that what is that is
00:07:58.310 --> 00:08:05.560
easy to you know verify normally what you
normally easily verify about functions you
00:08:05.560 --> 00:08:13.310
verify continuity okay you verify differentiability,
you verify boundedness of a function, okay
00:08:13.310 --> 00:08:19.250
try to find a constant which bounds the modulus
of the function values, okay.
00:08:19.250 --> 00:08:26.659
So these are the things that you verify, okay
so you need something in the situation of
00:08:26.659 --> 00:08:32.510
functions if you are working with a space
of functions then you need better conditions
00:08:32.510 --> 00:08:39.350
for compactness and that is where the Arzela-Ascoli
Theorem steps in. So basically you know we
00:08:39.350 --> 00:08:46.500
are looking at a compact metric space X, okay
so X is a compact metric space and so let
00:08:46.500 --> 00:08:54.480
me go back to what I wrote last lecture X
is a compact metric space and let me let me
00:08:54.480 --> 00:09:06.920
use a different color now X is a compact metric
space and you are looking at a I am looking
00:09:06.920 --> 00:09:15.700
at a I am looking at this C X, R which is
the you know it is a Banach algebra it is
00:09:15.700 --> 00:09:22.220
a complete norm linear space of continuous
functions on X with values in real numbers,
00:09:22.220 --> 00:09:23.220
okay.
00:09:23.220 --> 00:09:28.790
And you can also take with values in the complex
numbers you will get the other Banach algebra
00:09:28.790 --> 00:09:39.120
C X, C okay and the question is that you want
to and ofcourse mind you since X is compact
00:09:39.120 --> 00:09:43.790
and a continuous function on a compact set
A is uniformly continuous and always attains
00:09:43.790 --> 00:09:50.649
its bounds every function that we are worried
about is already uniformly continuous and
00:09:50.649 --> 00:10:01.180
it is bounded, okay so normally when you write
C X, R or C X, C you is you mean also not
00:10:01.180 --> 00:10:05.160
only just continuous functions but also continuous
bounded functions, okay but here boundedness
00:10:05.160 --> 00:10:08.190
is automatic because X is compact alright.
00:10:08.190 --> 00:10:16.190
So what I want I want I want to take a close
subset of I want to look at a subset of functions
00:10:16.190 --> 00:10:19.570
I want to look at a family of functions or
a collection of functions, okay which you
00:10:19.570 --> 00:10:25.200
think of as a subset of the space of functions
C X, R or C X, C and I want to say that it
00:10:25.200 --> 00:10:30.180
is compact that is my aim I want to get a
nice condition to say that a collection of
00:10:30.180 --> 00:10:36.440
functions is compact, okay. Now you know since
we are in the metric space something that
00:10:36.440 --> 00:10:40.700
has to be compact a subset has to be compact
means that it has to be both closed and bounded.
00:10:40.700 --> 00:10:46.510
So certainly the subset to begin with should
be closed, okay so we put this pre condition
00:10:46.510 --> 00:10:50.240
that it is already a closed set, if it is
not a closed set it is not going to be compact
00:10:50.240 --> 00:10:55.700
because compact implies its closeness, okay
and ofcourse it has to be bounded because
00:10:55.700 --> 00:11:00.380
I told you that in any metric space compact
subset is always closed and bounded, okay.
00:11:00.380 --> 00:11:08.540
So if you start with the closed subset A the
condition that it is compact is equivalent
00:11:08.540 --> 00:11:13.680
to being bounded and the here is the extra
condition the extra condition is equicontinuity,
00:11:13.680 --> 00:11:23.680
okay so equicontinuity is something that that
helps us to get hold of compactness, right.
00:11:23.680 --> 00:11:29.839
So you know I want to see at this point I
want at this point say the following thing
00:11:29.839 --> 00:11:37.240
I want to say that if you look at it from
the view point of total boundedness, okay
00:11:37.240 --> 00:11:48.160
then you see to say that A is compact it is
enough to say that A is totally bounded, okay
00:11:48.160 --> 00:11:59.040
because if A is totally bounded, alright then
A is also complete not then I mean A is already
00:11:59.040 --> 00:12:05.399
complete because A is a close subset of complete
metric space so it is complete.
00:12:05.399 --> 00:12:09.040
And if you have total boundedness together
with completeness you will get compactness
00:12:09.040 --> 00:12:13.829
because that is another characterization of
compactness I told you okay. So if A is totally
00:12:13.829 --> 00:12:22.050
bounded then I know it is compact, okay so
the only issue is with replacing this total
00:12:22.050 --> 00:12:26.800
boundedness with something else and I told
you this total boundedness is not so easy
00:12:26.800 --> 00:12:30.590
to verify because you have to show the existence
of an epsilon net for every epsilon that is
00:12:30.590 --> 00:12:32.470
not practical, okay.
00:12:32.470 --> 00:12:43.030
So whereas total boundedness being difficult
to verify we have boundedness is easier to
00:12:43.030 --> 00:12:49.899
verify, okay so you from the total boundedness
you remove the total and you just put boundedness
00:12:49.899 --> 00:12:56.790
which is an easy thing for verify for functions,
okay and you because you made the condition
00:12:56.790 --> 00:13:02.790
weaker from total boundedness to boundedness
you have to put something else to make it
00:13:02.790 --> 00:13:08.680
strong enough to get compactness to give compactness
and that strong enough thing is the equicontinuity,
00:13:08.680 --> 00:13:09.680
okay.
00:13:09.680 --> 00:13:16.790
So I will tell you what this what this equicontinuity
is, okay so so basically the idea is very
00:13:16.790 --> 00:13:22.899
very simple the idea is see if you want function
a function to be continuous at a point then
00:13:22.899 --> 00:13:27.760
what you do is you say that that is usual
you know you have the usual you know you have
00:13:27.760 --> 00:13:33.250
the usual definition of the epsilon delta
definition of continuity at a point and what
00:13:33.250 --> 00:13:40.899
is that give me epsilon greater than 0, I
can find a delta such that whenever the whenever
00:13:40.899 --> 00:13:48.860
the point is within a delta of the given particular
point then the function value is within an
00:13:48.860 --> 00:13:54.380
epsilon of the image of the function value
at that point, okay.
00:13:54.380 --> 00:14:09.170
Now so the pole issue here is that if you
change the point, okay so you are looking
00:14:09.170 --> 00:14:15.570
at continuity at a point for a given epsilon
you have to verify it, okay now the delta
00:14:15.570 --> 00:14:20.570
that you get will depend on the epsilon ofcourse
because if you make epsilon smaller you might
00:14:20.570 --> 00:14:26.240
expect that the delta must become smaller,
okay and so the delta depends on the epsilon
00:14:26.240 --> 00:14:32.970
the delta also depends on the point and the
delta is particular to that function, okay.
00:14:32.970 --> 00:14:38.029
So the delta depends on three things, okay
delta depends on three things if you are so
00:14:38.029 --> 00:14:48.959
you know so let me write this down so what
is this equicontinuity? So you see what is
00:14:48.959 --> 00:15:02.050
usual continuity usual continuity at let us
say x not for f what is this? Given epsilon
00:15:02.050 --> 00:15:11.820
greater than 0 there exist a delta greater
than 0 such that whenever the distance between
00:15:11.820 --> 00:15:28.660
x and x not is less than delta then we have
that the distance between f x and f x not
00:15:28.660 --> 00:15:34.600
is less than epsilon this is the usual good
old definition of continuity epsilon delta
00:15:34.600 --> 00:15:36.200
definition of continuity.
00:15:36.200 --> 00:15:45.760
And you see the point here is that ofcourse
this this the first d that I used in d of
00:15:45.760 --> 00:15:53.610
x, comma x not is the metric on the space
on which the function is defined whose points
00:15:53.610 --> 00:15:58.209
are x and x not, x not is a fixed point, x
is a variable point, okay. So basically this
00:15:58.209 --> 00:16:03.980
d x, comma x not less than delta refers to
you know basically it refers to all the points
00:16:03.980 --> 00:16:11.150
in a delta open ball centred at x not in your
space, okay and what you are saying is that
00:16:11.150 --> 00:16:17.300
whenever a point whenever you take a point
delta open ball centred at x not then its
00:16:17.300 --> 00:16:22.899
image is in an epsilon open ball centred at
f x not that is what you are saying, okay.
00:16:22.899 --> 00:16:31.190
So the image of this delta open ball under
f goes into this epsilon open ball centred
00:16:31.190 --> 00:16:37.300
at f x centred at f x not, okay that is what
usual continuity is but the point is that
00:16:37.300 --> 00:16:49.390
this epsilon you see given an epsilon okay
delta depends on what? See depends on ofcourse
00:16:49.390 --> 00:16:56.779
epsilon it depends on x not and you know it
depends on f also ofcourse we are looking
00:16:56.779 --> 00:17:02.279
at a single function so we forget the function
because if you are dealing with only one function
00:17:02.279 --> 00:17:06.560
there is no confusion but if you are looking
at a family of functions then you will have
00:17:06.560 --> 00:17:14.920
different functions f okay and as you change
the function for the same epsilon and even
00:17:14.920 --> 00:17:20.170
for the same point okay if you have a family
of functions which are continuous at a point
00:17:20.170 --> 00:17:26.959
x not, okay then for that family of functions
even if you take a single epsilon the same
00:17:26.959 --> 00:17:32.060
delta will not work the delta will change
with the function, okay and ofcourse delta
00:17:32.060 --> 00:17:33.240
will change with the point.
00:17:33.240 --> 00:17:37.890
Now equicontinuity is the fact that you are
is the condition that you know you are able
00:17:37.890 --> 00:17:43.550
to get a delta that works for all functions
in one stroke that is equicontinuity, okay.
00:17:43.550 --> 00:18:01.530
So collection collection of functions of functions
so well A so let me call it as A is called
00:18:01.530 --> 00:18:23.030
equicontinuous at x not you know if given
epsilon greater than 0 we can get a delta
00:18:23.030 --> 00:18:35.580
that works for all functions in that collection
that is equicontinuity you get a common delta
00:18:35.580 --> 00:18:37.810
that is equicontinuity, alright.
00:18:37.810 --> 00:18:50.040
And well and ofcourse you say that a family
is equicontinuous on a subset of the space
00:18:50.040 --> 00:18:57.290
if it is equicontinuous at every point of
the space, okay. So you make a point wise
00:18:57.290 --> 00:19:02.850
differentiation and then you say that this
definition holds if it holds on a holds for
00:19:02.850 --> 00:19:08.910
a subset of points if it holds for every point
in that subset, okay. So this is just like
00:19:08.910 --> 00:19:13.170
saying that function is continuous on a set
if it is continuous at every point of the
00:19:13.170 --> 00:19:14.570
set it is like that.
00:19:14.570 --> 00:19:36.370
So A is equicontinuous at or let me say on
on let me say subset Y in X if it is let me
00:19:36.370 --> 00:19:52.580
just explain if it is equicontinuous at each
point of X prime, okay. So so this equicontinuity
00:19:52.580 --> 00:19:57.280
is basically if you want to you know think
about it in a very simple way equicontinuity
00:19:57.280 --> 00:20:01.890
is given an epsilon you find it delta that
works for all functions that is the point,
00:20:01.890 --> 00:20:06.600
okay a single delta will work an epsilon for
every function in your collection then that
00:20:06.600 --> 00:20:12.030
collection is called equicontinuity an equicontinuous
collection or equicontinuous family of functions,
00:20:12.030 --> 00:20:13.030
okay.
00:20:13.030 --> 00:20:18.320
And what the Arzela-Ascoli Theorem says is
that well the Arzela-Ascoli Theorem says that
00:20:18.320 --> 00:20:27.470
if you put equicontinuity together with boundedness
then that is good enough to say that your
00:20:27.470 --> 00:20:34.580
collection is compact, okay so compactness
of a family of functions is equivalent to
00:20:34.580 --> 00:20:39.180
that family be equicontinuous and boundedness
and and you this is a much much nicer condition
00:20:39.180 --> 00:20:45.960
then total boundedness is okay equicontinuity
is it is a kind of continuity that you have
00:20:45.960 --> 00:20:49.210
to check it is just continuity you have to
check but make sure that you get one delta
00:20:49.210 --> 00:20:55.030
for all the functions for a given epsilon,
okay that is like checking continuity and
00:20:55.030 --> 00:21:03.100
that should be far more easier checking boundedness
is not a it should not be a big problem, okay.
00:21:03.100 --> 00:21:06.670
So these are all things that we normally check
for functions but given a family of functions
00:21:06.670 --> 00:21:12.340
you never it is not common that you check
total boundedness total boundedness is you
00:21:12.340 --> 00:21:17.380
have to find finitely given an epsilon you
have to find finitely many functions so that
00:21:17.380 --> 00:21:22.380
the distance of any function is within an
epsilon to one of these finitely many functions
00:21:22.380 --> 00:21:26.560
you have to find this epsilon net that is
not easy that is not common, okay that is
00:21:26.560 --> 00:21:28.060
not practical.
00:21:28.060 --> 00:21:32.890
So Arzela-Ascoli Theorem helps you by saying
that well you want to check family of functions
00:21:32.890 --> 00:21:38.880
is compact do not do much you just check that
it is bounded, check that it is equicontinuous
00:21:38.880 --> 00:21:48.680
and you are done, okay. So I just wanted to
you know
00:21:48.680 --> 00:21:53.950
indicate something about the about this this
equivalence of compactness with boundedness
00:21:53.950 --> 00:21:58.280
in equicontinuity because it is a because
it is a very very fundamental thing and you
00:21:58.280 --> 00:22:01.200
need to know how things work, okay.
00:22:01.200 --> 00:22:09.090
So so let me so let me do the following thing
I will I will again change colour and use
00:22:09.090 --> 00:22:26.400
a different colour so I will say proof of
Arzela-Ascoli Theorem. So so let me go through
00:22:26.400 --> 00:22:32.340
the proof because I wanted to understand the
ideas involved, okay you will also get familiar
00:22:32.340 --> 00:22:40.400
with all these notions of total boundedness,
epsilon net and how equicontinuity is actually
00:22:40.400 --> 00:23:00.590
used, okay. So so start with a closed start
with a closed A inside C X, R start with the
00:23:00.590 --> 00:23:06.400
closed set what do you have to show? You have
to show that A is compact if and only if it
00:23:06.400 --> 00:23:08.480
is bounded and equicontinuous, okay.
00:23:08.480 --> 00:23:20.960
So assume A is compact assume A is compact
alright now what do you have to show so this
00:23:20.960 --> 00:23:25.710
is one way of the theorem you have to show
that if it is compact it is bounded and is
00:23:25.710 --> 00:23:33.430
equicontinuous, okay. I told you that in a
metric space subset is compact implies it
00:23:33.430 --> 00:23:39.730
is both closed and bounded, okay so it is
bounded, okay compactness always implies closeness
00:23:39.730 --> 00:23:41.870
and boundedness in a metric space, right.
00:23:41.870 --> 00:23:47.830
So and mind you that is true only for metric
spaces, if you go to arbitrary topological
00:23:47.830 --> 00:23:53.970
spaces things can become very bad you can
have a compact subspace of a compact topological
00:23:53.970 --> 00:23:59.590
space which is not closed, okay it can behave
terribly you can have a whole space which
00:23:59.590 --> 00:24:03.410
is compact, you can have a subspace which
is compact but the subspace is not closed
00:24:03.410 --> 00:24:07.140
in the whole space that this can happen for
an arbitrary topological space, it cannot
00:24:07.140 --> 00:24:11.970
happen for a metric space. For a metric space
compactness always forces closeness and boundedness
00:24:11.970 --> 00:24:20.310
so house of spaces also house of spaces if
you want ya ya.
00:24:20.310 --> 00:24:43.460
So you see so assume A is compact then A is
A is bounded bounded, okay because compactness
00:24:43.460 --> 00:24:47.580
will imply closeness and boundedness it is
all A is already closed, okay but if you want
00:24:47.580 --> 00:24:53.850
boundedness that that is implied by compactness.
We only have to show that A is equicontinuous,
00:24:53.850 --> 00:25:12.830
okay we only need to show A is equicontinuous,
okay that is what you have to show. So you
00:25:12.830 --> 00:25:19.930
see so incidentally let me make a remark here
to in this case we are anyway going to the
00:25:19.930 --> 00:25:28.290
way we prove the Arzela-Ascoli Theorem is
by using the other version of compact characterization
00:25:28.290 --> 00:25:33.001
of compactness which is compactness is equivalent
to A being totally bounded, okay this total
00:25:33.001 --> 00:25:38.910
boundedness will come into the picture, okay
atleast in the proof, okay that is the key
00:25:38.910 --> 00:25:43.210
tool that helps in the proof of Arzela-Ascoli
Theorem.
00:25:43.210 --> 00:25:49.350
So if you think of since if you think of that
given that A is compact I know it is totally
00:25:49.350 --> 00:25:54.100
bounded and I told you totally bounded also
is a strong form of boundedness so it implies
00:25:54.100 --> 00:25:59.670
boundedness totally bounded is the condition
that there is an epsilon net for every epsilon,
00:25:59.670 --> 00:26:07.950
okay and you know that diameter of the space
can be compared to the diameter of any epsilon
00:26:07.950 --> 00:26:14.740
net, okay to within an epsilon net to within
an epsilon, okay or two epsilon, okay.
00:26:14.740 --> 00:26:20.290
So therefore A is always bounded in fact it
has finite diameter alright. We only need
00:26:20.290 --> 00:26:24.851
to show that A is equicontinuous and for the
equicontinuity also you use the fact that
00:26:24.851 --> 00:26:26.580
it is totally bounded, okay.
00:26:26.580 --> 00:26:52.360
So since A is totally bounded well A is A
admits an epsilon net for every epsilon greater
00:26:52.360 --> 00:26:59.930
than 0, okay so here I am using the fact that
compactness implies total boundedness, right
00:26:59.930 --> 00:27:03.970
and compactness implies total boundedness
is a very very it is a very simple thing there
00:27:03.970 --> 00:27:11.560
is nothing complicated about it, okay basically
what you are saying is that you should cover
00:27:11.560 --> 00:27:20.980
the space by finitely many open balls of the
fixed radius, okay that radius being epsilon
00:27:20.980 --> 00:27:25.500
and that is very that is very easy to see
because if you give if you take all open balls
00:27:25.500 --> 00:27:30.580
of radius epsilon that is the cover for the
space you allow every point of the space to
00:27:30.580 --> 00:27:36.390
be a centre and you take the collection of
all open balls of radius epsilon that is obviously
00:27:36.390 --> 00:27:40.500
an open cover for the space and if you know
it is compact it has to admit a finite cover
00:27:40.500 --> 00:27:45.730
and that finite cover if you take the centres
that will give you the epsilon net that you
00:27:45.730 --> 00:27:51.840
want, okay. So it is very that compactness
implies total boundedness is absolutely easy
00:27:51.840 --> 00:27:55.100
to see, okay. So there is a there is an epsilon
net, okay.
00:27:55.100 --> 00:28:12.480
So so let epsilon greater than 0 be given
and let us take an so here is a you know this
00:28:12.480 --> 00:28:18.160
is the this is the kind of thing you fiddle
with to get epsilon finding your answer so
00:28:18.160 --> 00:28:23.540
you know usually the idea is that you use
a triangle inequality you know finally all
00:28:23.540 --> 00:28:29.170
these epsilon arguments they finally end up
with a triangle inequality if you are breaking
00:28:29.170 --> 00:28:35.220
a distance into two pieces then you try to
and you want that distance to be less than
00:28:35.220 --> 00:28:39.590
epsilon you know you try to make each piece
less than epsilon by 2, if you are breaking
00:28:39.590 --> 00:28:44.860
it into 3 pieces then you know you make each
piece less than epsilon by 3. So what I am
00:28:44.860 --> 00:28:49.200
going to do is I am not going to take an epsilon
net I am going to take an epsilon by 3 net,
00:28:49.200 --> 00:28:50.200
okay.
00:28:50.200 --> 00:28:58.240
So so let epsilon greater than 0 be given
and let us take an epsilon by 3 net for A,
00:28:58.240 --> 00:29:07.450
okay. So what does this mean? This means that
so there exist functions f 1, etc upto f m
00:29:07.450 --> 00:29:27.860
which are in A such that well the any function
in A such that for any function in A there
00:29:27.860 --> 00:29:43.070
exist an i such that i with the distance between
f i and f less than epsilon by 3, okay so
00:29:43.070 --> 00:29:49.980
this is what epsilon net means basically you
cover all points of the space by looking at
00:29:49.980 --> 00:29:53.700
open balls of radius epsilon by 3, centred
at the points which correspond to the epsilon
00:29:53.700 --> 00:29:59.560
by 3 net, right this is this is this is what
you get.
00:29:59.560 --> 00:30:05.900
Now I wanted to you know at this point you
know I want you to remember what is this distance
00:30:05.900 --> 00:30:12.650
here? See this distance is the distance in
this distance is the distance in the in the
00:30:12.650 --> 00:30:21.540
Banach algebra C X, R okay it is mind you
all these what is A? A is a subset of C X,
00:30:21.540 --> 00:30:28.930
R and C X, R are you know functions which
are continuous functions on X real valued
00:30:28.930 --> 00:30:38.060
functions, okay and so what is the and ofcourse
X is compact X is compact so these all these
00:30:38.060 --> 00:30:43.350
functions are bounded, okay continuous functions
on a compact set is bounded and in fact it
00:30:43.350 --> 00:30:46.390
is uniformly continuous it bounds, right.
00:30:46.390 --> 00:30:55.200
So so what is the distance the distance is
the supremum now, so what is this this quantity
00:30:55.200 --> 00:31:07.010
is supremum as X belongs to capital X as X
varies over capital X of you know mod of f
00:31:07.010 --> 00:31:15.880
i of x minus f of x this is what it is, okay.
In fact it is actually it is precisely norm
00:31:15.880 --> 00:31:22.260
of f i minus f that is what the distance is
the distance between f i and f is just norm
00:31:22.260 --> 00:31:27.920
of f i minus f and the norm is the sup norm
so you calculate f i minus f at each point
00:31:27.920 --> 00:31:34.330
and then you take the mod of that and then
you take the supremum, okay and this should
00:31:34.330 --> 00:31:40.260
be less than epsilon by 3 this is what you
have, okay this is what you have.
00:31:40.260 --> 00:31:47.220
Now you see now we are kind of in more or
less in a very good shape because see now
00:31:47.220 --> 00:31:51.580
you can get equicontinuity very easily. So
let us check equicontinuity at a point, okay.
00:31:51.580 --> 00:32:06.350
So let let us check let us check equicontinuity
at x not in X let us check equicontinuity
00:32:06.350 --> 00:32:13.520
there okay then what is what is it that is
going to happen well what do you have to check
00:32:13.520 --> 00:32:22.170
for equicontinuity you have to find a delta
such that whenever mod x minus whenever the
00:32:22.170 --> 00:32:28.770
distance between x minus x and x not is less
than delta, okay the the distance between
00:32:28.770 --> 00:32:36.530
f x and f x not is less than epsilon and this
should work for all f in your family A in
00:32:36.530 --> 00:32:38.090
your collection A, okay.
00:32:38.090 --> 00:32:50.380
So given for the given for the given epsilon
greater than 0, we seek a delta greater than
00:32:50.380 --> 00:33:01.270
0 such that mod I keep saying mod because
you know at the back of at the back of once
00:33:01.270 --> 00:33:06.580
when one keeps thinks of eucledian space but
it is not mod you should replace it with d
00:33:06.580 --> 00:33:21.160
so the distance between x and x not less than
delta implies that mod f x the the distance
00:33:21.160 --> 00:33:28.460
between f x and f x not which is mod f x minus
f x not is less than epsilon and this should
00:33:28.460 --> 00:33:36.040
work for all f in A this is what you want,
alright.
00:33:36.040 --> 00:33:40.500
So you see how do you do this the idea is
very very simple that is this one the one
00:33:40.500 --> 00:33:49.270
hand you have an X which is a variable point
you have this point x not and you have a arbitrary
00:33:49.270 --> 00:33:58.090
f alright and you will have to now connect
that arbitrary f in A with these particular
00:33:58.090 --> 00:34:05.600
f i’s which are the elements of the epsilon
net, okay. So you do that by a you know basically
00:34:05.600 --> 00:34:11.190
by using a triangle inequality to break down
some distance into three pieces, okay.
00:34:11.190 --> 00:34:20.060
So what you do is that well you write this
mod f x minus f x not you write this as you
00:34:20.060 --> 00:34:29.220
know I have already told you that given this
f there is an f i with the property that the
00:34:29.220 --> 00:34:34.339
norm of f i minus f is less than epsilon by
3 that is already there so use that f i. So
00:34:34.339 --> 00:34:48.169
you write this as modulus of you know well
f of x minus f i of x plus mod f i of x minus
00:34:48.169 --> 00:34:56.110
f i of x not plus modulus of so I should put
less than or equal to modulus of f i of x
00:34:56.110 --> 00:35:05.480
not minus (f i) f of x not this is what I
will have to do, okay. So I introduced this
00:35:05.480 --> 00:35:12.060
this f i cleverly this f i values at x and
f i values at x not to break this.
00:35:12.060 --> 00:35:20.970
And then what happens is that you see the
you know f for any x in x okay for any x in
00:35:20.970 --> 00:35:28.610
x the modulus of f i x f x is always less
than epsilon by 3. So the point is that this
00:35:28.610 --> 00:35:35.800
fellow here this is less than or this is less
than epsilon by 3, okay and so is this fellow
00:35:35.800 --> 00:35:41.960
here this is also epsilon by 3, okay I have
to be worried only about the central term
00:35:41.960 --> 00:35:47.620
this is the only term I have to worry about
with that this is I cannot apply that epsilon
00:35:47.620 --> 00:35:53.520
by 3, bound to that because the points are
different the function is the same it is the
00:35:53.520 --> 00:35:59.750
same f i okay the same f i but the points
are x and x not that is in the first term
00:35:59.750 --> 00:36:06.210
the point is x, the third term the point is
x not, the term in the middle has two points
00:36:06.210 --> 00:36:10.260
with the same function, okay.
00:36:10.260 --> 00:36:16.340
What do I do that what do I do here? So here
what is do is basically I I use the fact that
00:36:16.340 --> 00:36:20.650
all these f i's are all uniformly continuous
in fact they are all continuous they are continuous
00:36:20.650 --> 00:36:24.460
on a compact set so they are uniformly continuous.
So because you are uniformly continuous I
00:36:24.460 --> 00:36:30.840
can make sure that you know if I if I chose
a delta sufficiently small whenever the distance
00:36:30.840 --> 00:36:36.160
between x and x not is less than delta I can
make the central term less than epsilon by
00:36:36.160 --> 00:36:37.660
3 I can do this.
00:36:37.660 --> 00:36:44.010
In fact I can simply do it because of continuity
of f i at x not but the point is that I can
00:36:44.010 --> 00:36:51.730
do this for all f i at the same time because
there are only finitely many f i's, okay there
00:36:51.730 --> 00:36:57.260
are only finitely many f i's. See if I want
f i x minus f i x not to be less than epsilon
00:36:57.260 --> 00:37:04.860
by 3 I can ofcourse chose delta such that
I can find a delta for which whenever the
00:37:04.860 --> 00:37:12.140
distance between x and x not is not less than
delta then mod f i x minus f i x not is less
00:37:12.140 --> 00:37:16.340
than epsilon by 3 I can do this because of
continuity of f i at x not, okay.
00:37:16.340 --> 00:37:25.200
But then this delta may depend on the f i,
okay and if I change the i to other f i's
00:37:25.200 --> 00:37:28.950
okay then the delta will change but there
are only finitely many of these f i's so I
00:37:28.950 --> 00:37:34.810
could have taken the minimum of them and that
would work for all f i's all the elements
00:37:34.810 --> 00:37:42.460
of the epsilon net in one go alright and mind
you you could even forget the point x not
00:37:42.460 --> 00:37:48.080
that is because all these functions that we
are studying they are uniformly continuous,
00:37:48.080 --> 00:37:55.230
okay so there is also you can you can not
only do this you can not only do this for
00:37:55.230 --> 00:37:59.672
all the f i's because they are finitely many,
you can also do it irrespective of the point
00:37:59.672 --> 00:38:06.250
x not because of uniform continuity that is
because all the f i's all the functions we
00:38:06.250 --> 00:38:11.840
are considering are continuous functions on
a compact space and when you have continuous
00:38:11.840 --> 00:38:15.660
functions on a compact space you get uniform
continuity.
00:38:15.660 --> 00:38:20.930
Uniform continuity is that whenever the distance
between source points is less than a delta
00:38:20.930 --> 00:38:25.110
then the distance between the image points
is less than epsilon it does not matter what
00:38:25.110 --> 00:38:31.380
the source points are the only condition is
they have to be within a delta you can find
00:38:31.380 --> 00:38:34.470
such an delta that is uniform continuity,
okay.
00:38:34.470 --> 00:38:49.140
So so let me so let me write here can find
a delta greater than 0 such that mod f i of
00:38:49.140 --> 00:38:58.960
x minus so let me write f j of x minus f j
of x not is less than epsilon by 3 if d of
00:38:58.960 --> 00:39:13.820
x, comma x not is less than delta (because)
for all j so let me write for all j because
00:39:13.820 --> 00:39:36.690
of continuity of f j at x not and and the
fact that there are only finitely many
00:39:36.690 --> 00:39:49.100
only finitely many f j, okay I can do this,
right. Not only that I can do more in fact
00:39:49.100 --> 00:39:54.580
because all of these are uniformly continuous
I can do this irrespective of x not okay I
00:39:54.580 --> 00:39:56.220
can even forget x not, okay.
00:39:56.220 --> 00:40:16.630
In fact since X is compact and f j are uniformly
continuous so I am abbreviating it to ufly
00:40:16.630 --> 00:40:34.850
for uniformly uniformly continuous we can
do this for any x not, okay. So you can find
00:40:34.850 --> 00:40:41.890
given an epsilon you can find a delta which
neither depends on the f nor does it depend
00:40:41.890 --> 00:40:49.070
on the x x not, okay so you have verified
equicontinuity in a very in uniform way, okay
00:40:49.070 --> 00:40:52.970
and that is how so you have got a equicontinuity
that is it, you have found a delta that works
00:40:52.970 --> 00:41:05.330
for every f, okay. So thus A is equicontinuous,
okay.
00:41:05.330 --> 00:41:13.470
So so the moral of the story is we are able
to see one way, why if you have a compact
00:41:13.470 --> 00:41:23.790
space of functions then these functions must
form an equicontinuous family, okay. So equicontinuity
00:41:23.790 --> 00:41:33.880
is something that is is very very important
and mind you I will have to prove the other
00:41:33.880 --> 00:41:40.570
way of the theorem but then before I do that
which I will do in the next lecture what I
00:41:40.570 --> 00:41:45.900
will now say is I will tell you why this is
called why Arzela-Ascoli Theorem is sometimes
00:41:45.900 --> 00:41:51.700
called uniform boundedness principle, it is
because of the following reason you see the
00:41:51.700 --> 00:42:00.340
condition is that a close subset of functions
is compact if and only if it is bounded and
00:42:00.340 --> 00:42:01.340
equicontinuous, okay.
00:42:01.340 --> 00:42:08.190
Now this bounded is actually bounded with
respect to the metric on the space of functions
00:42:08.190 --> 00:42:14.810
and that is that is given by the sup norm,
okay and boundedness under the sup norm, okay
00:42:14.810 --> 00:42:20.790
is actually uniform boundedness it means see
what does boundedness normally mean? It means
00:42:20.790 --> 00:42:25.810
that you are able to find positive constant
such that the modulus of the function values
00:42:25.810 --> 00:42:31.270
at all points is bounded above by this positive
constant, okay.
00:42:31.270 --> 00:42:35.630
But ofcourse this positive constant would
change if you change the function, so if it
00:42:35.630 --> 00:42:40.750
take different functions in a family each
function may individually be bounded but you
00:42:40.750 --> 00:42:46.390
may not be able to find the common bound for
all the functions in the family, okay and
00:42:46.390 --> 00:42:50.960
that is what uniform boundedness does. It
gives you a common bound for all functions
00:42:50.960 --> 00:42:54.450
in your family or collection or subset, okay.
00:42:54.450 --> 00:43:01.530
So boundedness in C X, R actually means uniform
boundedness and Arzela-Ascoli Theorem is just
00:43:01.530 --> 00:43:06.130
that if you have a uniformly bounded family
which is equicontinuous then it is compact
00:43:06.130 --> 00:43:11.901
and conversely that is what Arzela-Ascoli
Theorem is and that is why it is called the
00:43:11.901 --> 00:43:14.940
uniform boundedness principle, okay. So I
will continue in the next lecture and try
00:43:14.940 --> 00:43:19.020
to tell you the other way of the the other
way of the proof, okay.