WEBVTT
Kind: captions
Language: en
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See you know I am trying to bring in the context
of the context to explain why we need the
00:00:37.020 --> 00:00:45.480
Arzela-Ascoli Theorem, okay in our discussion,
okay so you know the as I was telling you
00:00:45.480 --> 00:00:52.080
in the last lecture the main thing we are
concerned about is compactness of families
00:00:52.080 --> 00:01:05.860
of functions, okay. So at the background is
we have the need to get a proof of the Picard
00:01:05.860 --> 00:01:10.149
theorems, okay and for that you have to study
compactness of families of Meromorphic functions,
00:01:10.149 --> 00:01:11.149
okay.
00:01:11.149 --> 00:01:20.240
And we have to understand that properly, alright
and therefore you must understand compactness
00:01:20.240 --> 00:01:28.170
from basic point of view first of all compactness
in the plain topological sense what it means
00:01:28.170 --> 00:01:33.280
for metric spaces and then you know what it
means for spaces of functions, okay and once
00:01:33.280 --> 00:01:42.660
you know all this then you your mind is now
then properly in tune to understand the you
00:01:42.660 --> 00:01:51.920
know the the proof of the so called Montel’s
theorem okay which is a key theorem that is
00:01:51.920 --> 00:01:58.040
used for going for getting a proof of the
Picard theorems, okay.
00:01:58.040 --> 00:02:05.869
So so again you know so let me tell you very
briefly let me again recall very briefly the
00:02:05.869 --> 00:02:12.019
topological background that I gave you last
class. So you know if you start with if you
00:02:12.019 --> 00:02:18.820
start with the topological space you know
there is a notion of compactness, okay namely
00:02:18.820 --> 00:02:25.740
every open cover has a finite sub cover and
that is a very that is a very general notion,
00:02:25.740 --> 00:02:30.780
alright and it can be defined for any topological
space because all you need for the definition
00:02:30.780 --> 00:02:35.920
is the idea of open set which is there in
any topological space which is fundamental
00:02:35.920 --> 00:02:38.060
to any topological space.
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Now but the point is that this this abstract
definition of compactness is not useful all
00:02:42.630 --> 00:02:48.050
the time because it is a very it is a condition
that you have to check for all covers for
00:02:48.050 --> 00:02:53.500
example if I want to check a topological space
is compact I have to take an arbitrary cover
00:02:53.500 --> 00:02:58.060
of open sets and then I have to from that
I have to show that I can pick out finitely
00:02:58.060 --> 00:03:05.670
many that is something that is not so easy
today because it is you know why it is not
00:03:05.670 --> 00:03:10.770
easy to do is because a cover can be very
abstract, it can be arbitrary and form something
00:03:10.770 --> 00:03:15.920
abstract it is very hard to extract something
that is very specific, okay.
00:03:15.920 --> 00:03:24.730
So so as usual as we do in mathematics normally
you have a definition which is very abstract
00:03:24.730 --> 00:03:27.850
and the reason why you like that definition
is because it has caught lot of power the
00:03:27.850 --> 00:03:33.100
abstraction gives it a lot of power, the abstract
definition of compactness gives you lot of
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power in the sense that you know it tells
you that whenever you have an open cover you
00:03:37.760 --> 00:03:39.410
can always get a finite sub cover.
00:03:39.410 --> 00:03:45.230
So it is a very powerful thing but practically
it is not so useful when you want to really
00:03:45.230 --> 00:03:51.579
use it in particular situations and just like
same situation mathematics most of the time
00:03:51.579 --> 00:03:58.829
you have some definition involving a property
which is very abstract and you make that definition
00:03:58.829 --> 00:04:02.680
because you know that property is very powerful
it is a very powerful strong property which
00:04:02.680 --> 00:04:03.959
you can use.
00:04:03.959 --> 00:04:12.040
But then how do you put it to use then the
theory tries to give equivalent conditions
00:04:12.040 --> 00:04:17.230
so that you can which are easier to verify
or which are more handier to verify, okay.
00:04:17.230 --> 00:04:25.350
So so in the same way if you look at compactness
okay then there are conditions which help
00:04:25.350 --> 00:04:33.700
you to verify compactness in a more easy way
by so what are equivalent conditions I was
00:04:33.700 --> 00:04:42.530
telling you yesterday that is you if you take
a metric space then for a metric space compactness
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is a same as compactness is a same as sequential
compactness and that is same as the Bolzano-Weierstrass
00:04:48.440 --> 00:04:50.430
property, okay.
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So so what is the sequential compactness it
is the property that every sequence has a
00:04:58.600 --> 00:05:04.820
convergent subsequence alright and and what
is a Bolzano-Weierstrass property? The Bolzano-Weierstrass
00:05:04.820 --> 00:05:13.180
property is that every infinite subset has
a limit point, okay and we are more used to
00:05:13.180 --> 00:05:22.860
looking at eucledian spaces, okay R n, n dimensional
real space and you can also take C n, n dimensional
00:05:22.860 --> 00:05:29.460
complex space the n C n can be thought of
as R 2n if you want, okay and these eucledian
00:05:29.460 --> 00:05:37.580
spaces they have the property that you know
compactness there is equivalent to closeness
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and boundedness, okay and that is what we
are whenever we are doing whenever we are
00:05:44.400 --> 00:05:48.370
working with eucledian spaces if you want
to check compactness all you will do is you
00:05:48.370 --> 00:05:53.931
will check it your set is bounded, you will
check it is closed, okay and that is it then
00:05:53.931 --> 00:05:55.630
you know it is compact, okay.
00:05:55.630 --> 00:06:03.880
So it is as easy it is that easy and mind
you just take for example the close close
00:06:03.880 --> 00:06:11.030
take a close disk on the complex plane take
a close disk on the complex plane it is easier
00:06:11.030 --> 00:06:17.980
to say that it is a close set and that it
that it is a bounded sets and therefore it
00:06:17.980 --> 00:06:24.910
is compact it is easier to say that then to
say then to take an arbitrary open cover and
00:06:24.910 --> 00:06:29.889
try to pick out a finite sub cover, okay it
is not easy if I give you an arbitrary open
00:06:29.889 --> 00:06:38.270
cover of a close disk, okay on the complex
plane it is not easy to pick out finite sub
00:06:38.270 --> 00:06:44.639
cover, alright. Whereas, I can assert that
this will be true because of compactness and
00:06:44.639 --> 00:06:48.180
why because I can check compactness by the
equivalent condition that it is both closed
00:06:48.180 --> 00:06:50.780
and bounded which is easy for me to check,
okay.
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So but ofcourse you know it is very important
that so this is see in all these issues we
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are in the context of metric spaces we are
not in the more general context of topological
00:07:03.610 --> 00:07:09.030
spaces because the metric is involved in all
these things for compact when you define compactness
00:07:09.030 --> 00:07:16.060
metric is not involved, okay but when you
define sequential compactness, okay you are
00:07:16.060 --> 00:07:22.440
worried about convergence of a sequence, okay
and that is easiest to define if you have
00:07:22.440 --> 00:07:25.460
a metric otherwise if you do not have a metric
you have to worry about nets and things like
00:07:25.460 --> 00:07:26.500
that, okay.
00:07:26.500 --> 00:07:35.730
And Bolzano-Weierstrass property, okay can
well in fact it can also be defined more generally
00:07:35.730 --> 00:07:40.850
but these properties are all very easily defined
when you have a distance function between
00:07:40.850 --> 00:07:48.260
two points which is a metric, okay. So you
see if you take ofcourse you know if you take
00:07:48.260 --> 00:07:54.520
a so what I was saying was that you know in
all these things we are working with a metric
00:07:54.520 --> 00:08:02.740
space, okay and and the point is that with
a metric space things are better because I
00:08:02.740 --> 00:08:07.430
mean you can visualize a lot of things because
there is a distance function, okay.
00:08:07.430 --> 00:08:13.419
So if limit if suppose you have if you say
limit x tends to x not in a metric space it
00:08:13.419 --> 00:08:17.990
makes sense because you are actually saying
the distance between x and x not is tending
00:08:17.990 --> 00:08:25.520
to 0, okay you are letting a function to tend
to 0 so it makes clear sense if you cannot
00:08:25.520 --> 00:08:31.772
make so easy sense of this if you did not
have a metric, okay. So so what I want I tell
00:08:31.772 --> 00:08:39.919
you is that now you know so the point is that
somehow if you are working with eucledian
00:08:39.919 --> 00:08:47.770
spaces then compactness is just the same as
closeness and boundedness put together, okay.
00:08:47.770 --> 00:08:54.090
But if you take an arbitrary topological space
one way is always true if something is compact,
00:08:54.090 --> 00:09:01.190
okay then it will always be if you take a
subspace of a topological space if it is compact,
00:09:01.190 --> 00:09:10.560
okay well I should say if you take an arbitrary
metric space not not not eucledian spaces,
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okay you take an arbitrary metric space if
you take a compact subset it will be closed
00:09:16.529 --> 00:09:24.880
and bounded, okay but the converse need not
be true, okay eucledian spaces are very special
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for which closeness and boundedness implies
compactness is equivalent to compactness.
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So the standard example is R infinity okay
you look at all the sequences of real numbers
00:09:37.680 --> 00:09:43.160
such that the sum of the if you write out
the series which consist of the sum of the
00:09:43.160 --> 00:09:52.480
squares of the modulus I the so called square
summable sequences, okay in in modulus, okay
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then so you know you take all the so you take
all sequences x1, x2, etc xn of real numbers
00:10:00.980 --> 00:10:09.880
such that sigma mod x i square okay you take
the modulus of the entries of the sequence
00:10:09.880 --> 00:10:14.610
and square it and you sum it mind you when
you sum it you are getting a series that series
00:10:14.610 --> 00:10:16.890
should converge, okay.
00:10:16.890 --> 00:10:21.870
The set of all these sequences they are called
square summable sequences the set of all these
00:10:21.870 --> 00:10:27.070
sequences is called R infinity is called the
infinite dimensional eucledian space, okay
00:10:27.070 --> 00:10:36.050
and in this R infinity it is a metric space,
okay because it is very easy to define a metric
00:10:36.050 --> 00:10:41.490
using just extending the same distance formula
only thing is that now you have infinitely
00:10:41.490 --> 00:10:46.520
many coordinates the usual distance formula
is what you take difference of coordinates
00:10:46.520 --> 00:10:52.110
square them, sum them up and then take square
root.
00:10:52.110 --> 00:10:56.110
Now what you do is you do it for all the coordinates
even though there are infinitely many coordinates
00:10:56.110 --> 00:11:01.630
this will work because I know that if you
sum all the if you take the sum of the squares
00:11:01.630 --> 00:11:06.950
of all the coordinates it converges, okay.
So what will happen is that you get this R
00:11:06.950 --> 00:11:12.830
infinity and in R infinity if you take the
if you take the open when you take the closed
00:11:12.830 --> 00:11:19.600
ball radius 1 centre the origin, okay you
are taking all those elements whose distance
00:11:19.600 --> 00:11:24.670
from the origin is less than or equal to 1,
okay then what happens is that this is ofcourse
00:11:24.670 --> 00:11:30.700
closed and bounded it is a closed ball so
it is close and its radius 1 so it is bounded
00:11:30.700 --> 00:11:36.030
but it is not compact, okay because if you
take the sequence which consist of only 1’s
00:11:36.030 --> 00:11:44.420
along the diagonal okay you take a sequence
of sequences, okay which if you write down
00:11:44.420 --> 00:11:49.120
one below the other so that you know you get
the only 1’s along the diagonal and 0’s
00:11:49.120 --> 00:11:58.529
elsewhere okay then that sequence you know
it will be in this in this closed ball but
00:11:58.529 --> 00:12:04.291
it will never have a convergence subsequence
and that is because the distance between any
00:12:04.291 --> 00:12:10.510
two members of sequence is always fixed at
positive quantity it is a constant value in
00:12:10.510 --> 00:12:13.890
fact it is it is root 2 actually, okay.
00:12:13.890 --> 00:12:20.399
So you can see that if the distance between
two members of the sequence is a constant
00:12:20.399 --> 00:12:23.940
such a sequence cannot have a convergent subsequence
because the convergent subsequence means it
00:12:23.940 --> 00:12:30.490
has to become Caushy and distance between
terms should come closer become very small,
00:12:30.490 --> 00:12:33.890
okay so it cannot have a Caushy subsequence.
So what it means that you are able to get
00:12:33.890 --> 00:12:38.370
a sequence which does not have a convergent
subsequence so it means it is not sequentially
00:12:38.370 --> 00:12:42.230
compact but if it is not sequentially compact
it cannot be compact because there is this
00:12:42.230 --> 00:12:46.320
theorem which says that for a metric space
compactness is same as sequential compactness,
00:12:46.320 --> 00:12:52.140
okay so R infinity is an example of metric
space where you closeness and boundedness
00:12:52.140 --> 00:12:54.529
does not imply compactness okay.
00:12:54.529 --> 00:13:00.540
So you are in this closed and bounded implies
compact works for us for most of the time
00:13:00.540 --> 00:13:05.360
because we are worried about only usually
we are worried about subsets of eucledian
00:13:05.360 --> 00:13:11.140
spaces, okay if you are working with R n or
C n, n dimensional real or complex space then
00:13:11.140 --> 00:13:19.170
you are in good shape okay. Now but then you
know the but you take an arbitrary metric
00:13:19.170 --> 00:13:26.470
space compactness will always imply closeness
and boundedness, okay a compact subset will
00:13:26.470 --> 00:13:31.970
always be closed and bounded, okay that will
always be true compactness is always stronger,
00:13:31.970 --> 00:13:37.680
compactness will give you closeness, compactness
will give you boundedness but if you want
00:13:37.680 --> 00:13:41.610
to get back compactness from closeness and
boundedness that will not work in an arbitrary
00:13:41.610 --> 00:13:46.980
metric space as we saw it does not work in
R infinity but it will work in R n or C n,
00:13:46.980 --> 00:13:48.029
okay.
00:13:48.029 --> 00:13:55.020
Now the truth is compactness is not only that
strong in fact it is far more stronger it
00:13:55.020 --> 00:13:59.149
gives you something called total boundedness
and what is this total boundedness? This total
00:13:59.149 --> 00:14:05.860
boundedness is as I was explaining to you
know last time by this diagram you can have
00:14:05.860 --> 00:14:14.730
a look at this diagram so you see I have this
space X which is a metric space mind you and
00:14:14.730 --> 00:14:20.230
the total boundedness is that you know you
give me any positive number epsilon, okay
00:14:20.230 --> 00:14:25.560
you think of it is a very small radius, okay
then what you can do is you can find finitely
00:14:25.560 --> 00:14:32.360
many points points of a subset A sub epsilon
called an epsilon net such that you know every
00:14:32.360 --> 00:14:37.390
point of X is within an epsilon distance from
one of these points atleast one of these points.
00:14:37.390 --> 00:14:44.019
So in other words if you take the if you take
the open balls centred at these points these
00:14:44.019 --> 00:14:53.450
finitely many points which are elements of
A epsilon so called epsilon net with radius
00:14:53.450 --> 00:14:59.410
epsilon, okay then the union of all these
balls will cover the whole space. So you know
00:14:59.410 --> 00:15:07.089
it is of mind you it is you are covering the
space by only finitely many open balls of
00:15:07.089 --> 00:15:15.160
given radius epsilon and this will and this
should work for every epsilon that is the
00:15:15.160 --> 00:15:21.060
that is even if you make epsilon smaller you
will still find another finite set of points
00:15:21.060 --> 00:15:25.120
at which centred at which if you take these
epsilon balls and you take the union it will
00:15:25.120 --> 00:15:29.670
still cover the whole space, okay it is a
very strong property.
00:15:29.670 --> 00:15:36.330
And this will this will ofcourse fall through
if it is compact why because you know the
00:15:36.330 --> 00:15:42.170
truth is that give me if the space is compact
give me any epsilon and you take the collection
00:15:42.170 --> 00:15:47.990
of all balls centred at various points all
possible points with radius epsilon you take
00:15:47.990 --> 00:15:51.300
all these open balls that is an open cover
and because it is compact it has to have a
00:15:51.300 --> 00:15:55.110
finite sub cover so I found an epsilon net
for every epsilon I can find an epsilon net
00:15:55.110 --> 00:16:00.390
just simply because of the very definition
of compactness so compactness will give you
00:16:00.390 --> 00:16:02.040
total boundedness.
00:16:02.040 --> 00:16:09.490
But total boundedness and ofcourse total boundedness
implies boundedness ofcourse because you can
00:16:09.490 --> 00:16:16.610
check as I was telling you last class that
the the diameter of the topological space
00:16:16.610 --> 00:16:21.910
can be compared is always less than or equal
to the diameter of the epsilon net plus 2
00:16:21.910 --> 00:16:26.690
times epsilon. So you will get this so I am
little short of space but let me write it
00:16:26.690 --> 00:16:38.060
here d of A is d of X not d of A d of X d
of X is less than or equal to d of A plus
00:16:38.060 --> 00:16:44.740
2 epsilon d of A epsilon plus 2 epsilon you
will get this for every epsilon you will get
00:16:44.740 --> 00:16:49.860
this and d of A epsilon mind you the finite
quantity because it is the diameter of a finite
00:16:49.860 --> 00:16:56.779
set diameter is supposed to be supremum of
the distances between two points on the set,
00:16:56.779 --> 00:17:03.770
okay it is trying to measure how large the
set is using the metric, okay.
00:17:03.770 --> 00:17:09.630
So and we put because sometimes the set may
be unbounded and in that case your diameter
00:17:09.630 --> 00:17:17.640
may be infinite, okay. So in any case what
this tells you is that if a space is compact
00:17:17.640 --> 00:17:22.370
if the metric space is compact and you know
this tells you that it is bounded because
00:17:22.370 --> 00:17:28.820
it has finite diameter alright and and ofcourse
I told you it will also be closed if a subspace
00:17:28.820 --> 00:17:34.890
of a metric space is compact it will be bounded
and closed ofcourse, okay and you also get
00:17:34.890 --> 00:17:40.049
an epsilon net for every epsilon which means
it is totally bounded alright and total boundedness
00:17:40.049 --> 00:17:43.620
a very strong condition it is a very strong
form of boundedness.
00:17:43.620 --> 00:17:47.810
Now the point is that from total boundedness
how do you get back compactness, can you get
00:17:47.810 --> 00:17:52.190
back compactness? And the theorem is that
you can get back compactness with this with
00:17:52.190 --> 00:18:01.009
this extra condition, okay. Suppose your space
is totally bounded and complete then it is
00:18:01.009 --> 00:18:13.190
compact, okay so it is a you know it is rather
powerful theorem so you know compactness so
00:18:13.190 --> 00:18:17.730
you know to check that something is compact
it is enough to check that your metric space
00:18:17.730 --> 00:18:21.480
if you have a subspace of a metric space suppose
you want to check its compact all you have
00:18:21.480 --> 00:18:26.720
to check is well now you have one more condition
you check it is complete that means you have
00:18:26.720 --> 00:18:33.279
checked that every Cauchy sequence converges
and you check that it is totally bounded,
00:18:33.279 --> 00:18:38.529
okay so this is this is another thing which
helps.
00:18:38.529 --> 00:18:43.269
And you know as far as general topological
spaces are concerned things are really bad,
00:18:43.269 --> 00:18:47.720
in fact you can have suppose you have a general
topological space which is not a metric space
00:18:47.720 --> 00:18:52.919
suppose it is not a metric space, okay then
you can have a horrible situation like this
00:18:52.919 --> 00:18:57.740
the whole space will compact you can have
a whole space which is compact not a metric
00:18:57.740 --> 00:19:02.299
space, topological space whole space is compact
you can have a (sub) you can have a subspace
00:19:02.299 --> 00:19:10.059
which is also compact but it is not closed
such a horrible thing can happen this can
00:19:10.059 --> 00:19:13.679
happen in an arbitrary topological space it
cannot happen in a metric space, in a metric
00:19:13.679 --> 00:19:19.020
space a compact subspace is always closed
and bounded, okay that is what you have to
00:19:19.020 --> 00:19:20.020
remember, okay.
00:19:20.020 --> 00:19:27.230
So so well now the point is that so what you
get is you know so let me continue with this
00:19:27.230 --> 00:19:35.720
discussion so you have compactness on the
one hand and then ofcourse we are working
00:19:35.720 --> 00:19:43.859
with metric spaces
you have compactness on one hand and that
00:19:43.859 --> 00:20:03.869
is you can get that from complete complete
and totally bounded
00:20:03.869 --> 00:20:08.659
and in fact this way also this is correct
because you see if a space is compact it is
00:20:08.659 --> 00:20:13.070
ofcourse totally bounded I have told you if
a metric space is compact it is totally bounded
00:20:13.070 --> 00:20:18.019
I told you it has finite diameter in fact
and finite diameter which is comparable to
00:20:18.019 --> 00:20:22.690
diameter of any epsilon net, okay which exist
because of total boundedness and compactness
00:20:22.690 --> 00:20:27.859
also implies completeness because you see
if it is compact it is sequentially compact,
00:20:27.859 --> 00:20:28.859
okay.
00:20:28.859 --> 00:20:34.380
So every every sequence has a convergence
subsequence. So in particular if a sequence
00:20:34.380 --> 00:20:39.991
is a Cauchy sequence if then it will have
a convergent subsequence but if a Cauchy sequence
00:20:39.991 --> 00:20:45.789
has a convergent subsequence then the Cauchy
sequence itself must be convergent, okay so
00:20:45.789 --> 00:20:49.860
so compactness is equivalent to complete and
totally bounded, okay. Now the problem is
00:20:49.860 --> 00:20:55.979
that you know this is this is what we have
in general topology if we are only worried
00:20:55.979 --> 00:21:03.720
about spaces okay not just topological spaces
but we are worried about more specific spaces
00:21:03.720 --> 00:21:07.629
namely metric spaces, okay but our context
is different our context is we are worried
00:21:07.629 --> 00:21:09.029
about functions, okay.
00:21:09.029 --> 00:21:14.190
If fact our application is we want to study
Meromorphic functions on a domain in the external
00:21:14.190 --> 00:21:19.229
complex maybe alright that is the context
where we have to go to. So somehow you know
00:21:19.229 --> 00:21:25.280
these results which are for spaces you have
to translate them they are not still good
00:21:25.280 --> 00:21:31.429
enough for our use you have to translate them
into results for spaces of functions, okay
00:21:31.429 --> 00:21:35.059
because we want to apply everything to spaces
of functions.
00:21:35.059 --> 00:21:40.019
So what spaces of functions will you think
of? So you know the you know if you take so
00:21:40.019 --> 00:21:48.649
let me recall again if X is a topological
space, okay then you know you can take this
00:21:48.649 --> 00:21:58.729
you can take the set of all maps from X to
R okay or you can take the set of all maps
00:21:58.729 --> 00:22:04.970
from X to C so you can take the set of all
real valued functions on X or you can take
00:22:04.970 --> 00:22:10.379
the set of all complex valued functions on
X this is these are algebras, okay.
00:22:10.379 --> 00:22:18.529
So you see you take the set of all maps from
the space to the real line these are just
00:22:18.529 --> 00:22:24.100
maps theoretic maps I am not assuming anything
continuity nothing, okay. So just maps from
00:22:24.100 --> 00:22:30.729
X to the real line or just you take the maps
on X to the complex plane, okay. So just real
00:22:30.729 --> 00:22:35.729
valued functions or complex valued functions,
okay now that is a these are algebras, algebras
00:22:35.729 --> 00:22:41.129
means that they are so for example we will
take the set of all real valued functions
00:22:41.129 --> 00:22:48.229
that is a vector space it is a real vector
space because you can add two functions and
00:22:48.229 --> 00:22:52.760
you can multiply a function by a constant
real constant it is a real vector space.
00:22:52.760 --> 00:22:59.690
And if if you think about it it is also an
algebra it is a commutative ring namely there
00:22:59.690 --> 00:23:05.019
is a multiplication in it which is commutative
and that is just multiplication of real valued
00:23:05.019 --> 00:23:09.260
functions if you multiply two real valued
functions point wise you get again another
00:23:09.260 --> 00:23:13.979
real valued function so this is a ring it
is a commutative ring which has a vector space
00:23:13.979 --> 00:23:17.999
structure and the vector space structure is
compatible with the ring structure multiplication
00:23:17.999 --> 00:23:22.639
distributes over addition in the right sense,
okay and so on and therefore this is a nice
00:23:22.639 --> 00:23:25.830
ring it is a ring plus a vector space such
a thing is called an algebra.
00:23:25.830 --> 00:23:31.450
So I have mentioned this before in one of
the earlier lectures, so this is set of all
00:23:31.450 --> 00:23:36.669
maps from X to R is a real algebra set of
all maps from X to C is a complex algebra,
00:23:36.669 --> 00:23:46.830
okay and the point is that if you are looking
at among these maps if you look at only bounded
00:23:46.830 --> 00:23:53.769
maps okay you look at maps whose images are
bounded, okay in the image I can talk about
00:23:53.769 --> 00:23:59.080
boundedness because the image is going to
lie either in R or in C if you take a if you
00:23:59.080 --> 00:24:03.529
take a real valued function the image is going
to be a subset of R, if you take a complex
00:24:03.529 --> 00:24:06.230
valued function the image is going to be a
subset of C, okay.
00:24:06.230 --> 00:24:13.309
So the images make sense and subsets of R
and C therefore boundedness of the image make
00:24:13.309 --> 00:24:19.960
sense therefore see what I can do is I can
look at so let me write this if I take the
00:24:19.960 --> 00:24:27.960
bounded maps from X to R or if I take the
bounded maps from X to C what will happen
00:24:27.960 --> 00:24:36.549
is that this is this is a subset of this,
okay in all these upto this point you know
00:24:36.549 --> 00:24:41.200
I do not even need X to be a topological space
X could have even be a non-empty set I am
00:24:41.200 --> 00:24:46.669
even the topology on X I am not going to use
I have not used okay because I will worry
00:24:46.669 --> 00:24:52.029
about the topology on X if for example I am
looking at something connected with maps which
00:24:52.029 --> 00:24:56.429
are connected with the topology namely continuous
maps, okay but I am just looking at maps.
00:24:56.429 --> 00:25:02.679
So X could need not even be a you know topological
space in fact I can just say let me I can
00:25:02.679 --> 00:25:12.039
just rub this and say X is just a non-empty
set. So I get these two algebras and I get
00:25:12.039 --> 00:25:18.049
the subsets of bounded maps, okay. Now these
bounded maps also if you if you check they
00:25:18.049 --> 00:25:22.710
also form sub algebras okay because the sum
of two bounded maps is bounded, product of
00:25:22.710 --> 00:25:25.570
two bounded maps is bounded and so on.
00:25:25.570 --> 00:25:31.279
So these will give you sub algebras and the
beautiful thing is that the boundedness allows
00:25:31.279 --> 00:25:37.289
you to define a norm the so called supremum
norm, okay. So what you can do is that you
00:25:37.289 --> 00:25:50.909
can define norm f to be supremum over x small
x in capital X mod f x for for f for f a bounded
00:25:50.909 --> 00:26:02.720
map, okay I can define this supremum norm
alright and the point is that this is a norm
00:26:02.720 --> 00:26:10.220
once it is a norm it is a norm on a vector
space so it becomes a norm linear space and
00:26:10.220 --> 00:26:16.320
once you have a norm linear space the norm
induces a distance function alright so you
00:26:16.320 --> 00:26:20.639
get a metric space, you get the metric induced
by a norm and once you have a once you have
00:26:20.639 --> 00:26:24.039
this metric you have a topology induced by
the metric.
00:26:24.039 --> 00:26:30.309
So these becomes very nice topological spaces
in fact they become Banach spaces, okay they
00:26:30.309 --> 00:26:35.359
become Banach algebras they become complete
norm linear spaces and the completeness is
00:26:35.359 --> 00:26:39.649
just because of the completeness of the target
it is because of completeness of real line
00:26:39.649 --> 00:26:44.340
completeness of complex numbers, okay. So
what you get is that you get these two even
00:26:44.340 --> 00:26:49.389
for a non-empty set you get this two beautiful
Banach spaces Banach algebras alright one
00:26:49.389 --> 00:26:53.149
is a real Banach algebra the other is a complex
Banach algebra depending on whether you are
00:26:53.149 --> 00:26:55.210
considering real valued functions or complex
valued functions.
00:26:55.210 --> 00:27:00.749
Now what you can do is, now you put one more
so so let me write that the distance between
00:27:00.749 --> 00:27:07.880
f and g is is just norm of f minus g I can
I can define this is the metric defined by
00:27:07.880 --> 00:27:14.360
the norm, alright and now we are in a nice
metric space norm okay and now what you can
00:27:14.360 --> 00:27:22.450
now what happens is that you know if X is
topological space then you can go one step
00:27:22.450 --> 00:27:28.119
down and then look at the set of all continuous
functions from continuous bounded functions
00:27:28.119 --> 00:27:32.820
from X to R or you can look at the set of
all continuous bounded functions from X to
00:27:32.820 --> 00:27:36.320
C, okay.
00:27:36.320 --> 00:27:42.690
Then what happens is that this you can check
that the set of all continuous functions that
00:27:42.690 --> 00:27:49.409
subset is a closed subset. The reason is because
if you take a sequence of if you take a sequence
00:27:49.409 --> 00:27:55.200
of continuous functions if it converges to
a limit function the convergence here will
00:27:55.200 --> 00:27:59.649
correspond to uniform convergence because
of the sup norm and you know a continuous
00:27:59.649 --> 00:28:03.510
a uniform limit of continuous functions is
continuous therefore what it means is that
00:28:03.510 --> 00:28:11.059
if you take a sequence of continuous bounded
continuous functions, okay and if it if it
00:28:11.059 --> 00:28:16.240
is a Cauchy sequence okay then you know it
will converge in the whole space because the
00:28:16.240 --> 00:28:21.429
whole space is ofcourse complete mind you
it is a Banach algebra the whole space is
00:28:21.429 --> 00:28:25.690
complete but the limit will also be continuous
because it is a uniform limit and why it is
00:28:25.690 --> 00:28:30.369
a uniform limit is because of the sup norm
if you check, okay.
00:28:30.369 --> 00:28:33.740
Therefore the space of all continuous real
valued functions bounded real valued functions
00:28:33.740 --> 00:28:41.070
C X R or the space of all continuous bounded
complex valued functions C X C that is a close
00:28:41.070 --> 00:28:46.679
it is a closed subset and you know a closed
subset of a complete space is again complete
00:28:46.679 --> 00:28:53.169
therefore this C X R and C X C are very beautiful
Banach algebras also they are Banach sub algebras
00:28:53.169 --> 00:28:54.700
closed sub algebras, okay.
00:28:54.700 --> 00:29:02.399
And now the point is suppose you are so now
you know you are coming to now slowly you
00:29:02.399 --> 00:29:11.090
know we have come into discussing spaces of
functions, okay now you know I would like
00:29:11.090 --> 00:29:22.169
to do topology on a subset of of this so I
would like to look at a space or a family
00:29:22.169 --> 00:29:28.619
or a subset or a collection of let us say
continuous real valued functions and on that
00:29:28.619 --> 00:29:36.479
set considering as a set A subset A of C X
R okay I wanted two topology on that and what
00:29:36.479 --> 00:29:38.539
kind of topology I am interested in compactness.
00:29:38.539 --> 00:29:46.749
So the question is suppose I start with an
A here or here the question is when is A compact
00:29:46.749 --> 00:29:53.549
this is my question you see because my aim
is what my aim is to study compactness of
00:29:53.549 --> 00:29:59.749
spaces of functions you know and mind you
let me again keep reminding you so that you
00:29:59.749 --> 00:30:05.259
do not get lost our final aim is to study
compactness of a family of Meromorphic functions,
00:30:05.259 --> 00:30:10.360
okay so I am trying to do it in the simplest
case in the case of just say topological the
00:30:10.360 --> 00:30:14.789
topological case of this continuous functions
continuous real valued bounded functions,
00:30:14.789 --> 00:30:15.789
okay.
00:30:15.789 --> 00:30:23.029
Now when is A compact? If you see by whatever
I told you because A is anyway a subset of
00:30:23.029 --> 00:30:29.659
a metric space, alright since A is subset
of a metric space to check A is compact I
00:30:29.659 --> 00:30:35.429
can check ofcourse I can check the usual definition
of compactness that every open cover has a
00:30:35.429 --> 00:30:41.080
finite sub cover that is highly in practical,
okay then I can check sequential compactness
00:30:41.080 --> 00:30:51.460
I can check that you know every every sequence
in A has a convergence subsequence, right
00:30:51.460 --> 00:30:55.009
that is another thing that I can check.
00:30:55.009 --> 00:31:00.299
The third thing is I can check that A has
a Bolzano-Weierstrass property because these
00:31:00.299 --> 00:31:07.849
are all the characterizations of compactness
on a metric space okay and you know of all
00:31:07.849 --> 00:31:16.399
the three one possible thing that I can check
for A is that it is sequentially compact,
00:31:16.399 --> 00:31:23.139
okay I can so I can check that if you give
me any sequence in A you give me a sequence
00:31:23.139 --> 00:31:28.809
of functions in A if I can check that there
is a convergent subsequence then also I will
00:31:28.809 --> 00:31:29.849
get compactness, okay.
00:31:29.849 --> 00:31:36.019
So that is something that is I have but mind
you if I have to check that I the sequence
00:31:36.019 --> 00:31:42.409
in A converges to it has a convergence subsequence
mind you the convergence is now uniform convergence
00:31:42.409 --> 00:31:46.779
because you are under the sup norm I have
to check uniform convergence, okay that is
00:31:46.779 --> 00:31:51.779
what it means, alright. So that is one thing
that I can do, what other things can I do?
00:31:51.779 --> 00:31:57.119
I just now told you that for metric spaces
as I have written above compactness is a same
00:31:57.119 --> 00:31:59.649
as completeness and total boundedness.
00:31:59.649 --> 00:32:09.519
So I can check that A is complete and totally
bounded but you know mind you I told you a
00:32:09.519 --> 00:32:15.649
compact subspace of a metric space is always
closed and bounded okay the converse may not
00:32:15.649 --> 00:32:22.549
be true okay as I have told you in the case
of R infinity okay. So if you expect A to
00:32:22.549 --> 00:32:28.539
be compact A should atleast be closed that
is a necessary condition okay so that is always
00:32:28.539 --> 00:32:36.059
that A has to be closed this this you cannot
avoid so you must have closeness alright.
00:32:36.059 --> 00:32:40.950
And what this means is that therefore if you
check that there is a if you so if you want
00:32:40.950 --> 00:32:46.269
to check sequential compactness in A it is
enough to check that every sequence as a Cauchy
00:32:46.269 --> 00:32:50.510
sequence that is enough because we will check
it has a Cauchy sequence then it means it
00:32:50.510 --> 00:32:56.559
has a convergent subsequence and the limit
will also be in A because it is closed, right.
00:32:56.559 --> 00:33:03.350
So A has to be closed you cannot avoid that
and what does this thing that I have written
00:33:03.350 --> 00:33:10.460
on top tell you to check that it is compact
I have to ofcourse it has to be closed I have
00:33:10.460 --> 00:33:13.779
to check that it is complete and totally bounded,
okay.
00:33:13.779 --> 00:33:19.629
And the point is completeness is automatic,
why because A is already a closed subspace
00:33:19.629 --> 00:33:24.509
of a complete metric space so it is automatically
complete. So the only thing I have to check
00:33:24.509 --> 00:33:28.539
is totally boundedness total boundedness.
00:33:28.539 --> 00:33:40.629
So so A is compact so let me write it here
A compact if and only if A is totally bounded
00:33:40.629 --> 00:33:48.789
okay this is what we get this is what we get
mind you ofcourse A is closed that is already
00:33:48.789 --> 00:33:56.119
already assume A is closed because if it is
not closed you cannot expect compactness okay.
00:33:56.119 --> 00:34:01.289
So we have ended up at this point we have
ended up at this point where you are saying
00:34:01.289 --> 00:34:07.710
so finally what does all this translate? Give
me a bunch of functions real valued continuous
00:34:07.710 --> 00:34:15.119
bounded functions on a space alright, how
do I check that as topologically it is compact
00:34:15.119 --> 00:34:17.409
how do I check it is compact?
00:34:17.409 --> 00:34:23.990
So this tells me check it is totally bounded,
okay. Now what does that mean? It means that
00:34:23.990 --> 00:34:29.010
you have to find an epsilon net for every
epsilon what does that mean it means given
00:34:29.010 --> 00:34:33.990
an epsilon positive I have to find finitely
many functions from this family such that
00:34:33.990 --> 00:34:39.770
the distance of any other function from atleast
one of these functions these finitely many
00:34:39.770 --> 00:34:44.210
functions is less than epsilon that is what
epsilon net means, okay.
00:34:44.210 --> 00:34:49.040
So I have to pick given an epsilon greater
than 0 I have to find finitely many functions
00:34:49.040 --> 00:34:57.740
from A such that the epsilon open open balls
at those centred at those finitely many functions
00:34:57.740 --> 00:35:03.750
with radius epsilon that covers A that is
what an epsilon net for A means okay and you
00:35:03.750 --> 00:35:09.120
see this is also very abstract you see it
is very very abstract it is if you if I start
00:35:09.120 --> 00:35:12.990
with an abstract family of functions usually
families of functions are abstract because
00:35:12.990 --> 00:35:18.770
you know they will depend on some property
I might have functions which have some differentiable
00:35:18.770 --> 00:35:24.740
property or some some they may be defined
by some abstract property and from an abstract
00:35:24.740 --> 00:35:32.020
collection it is not so easy to pick out finitely
many it may not be so easy to do it.
00:35:32.020 --> 00:35:35.920
So this total boundedness checking this total
boundedness for a family of function does
00:35:35.920 --> 00:35:43.370
not work, okay so what comes to help us here
is something that can really be checked for
00:35:43.370 --> 00:35:47.070
families of functions and that is the Arzela-Ascoli
Theorem.
00:35:47.070 --> 00:35:53.500
So let me write that down so so this is this
so let me write this this is not practical
00:35:53.500 --> 00:36:02.930
this is not practical by that I mean this
is not easily verifiable in practise okay
00:36:02.930 --> 00:36:09.330
you cannot demonstrate it in practise so easily,
okay so what comes to our help is so called
00:36:09.330 --> 00:36:14.390
Arzela-Ascoli Theorem. So what is that so
let me write that down.
00:36:14.390 --> 00:36:22.500
So here is the Arzela-Ascoli Theorem and what
is this theorem and what does this theorem
00:36:22.500 --> 00:36:35.410
say it says that suppose X is compact X is
a compact metric space you assume you are
00:36:35.410 --> 00:36:39.770
working on a compact metric space okay the
advantage of working on a compact metric space
00:36:39.770 --> 00:36:46.850
is that automatically all continuous functions
have bounded okay any continuous function
00:36:46.850 --> 00:36:56.050
on a compact any continuous real valued function
on a compact sets is you know it is bounded
00:36:56.050 --> 00:36:58.970
it attains it is bounded uniformly continuous
you know all these things.
00:36:58.970 --> 00:37:02.700
So if you are working with a compact space
then you do not have to restrict to bounded
00:37:02.700 --> 00:37:06.620
continuous functions every continuous every
continuous function is automatically bounded,
00:37:06.620 --> 00:37:21.270
okay so you work with a compact metric space
okay then a closed subset A of C X, R or C
00:37:21.270 --> 00:37:45.070
X, C is compact if and only if it is bounded
and equicontinous, okay. So this is this is
00:37:45.070 --> 00:37:51.400
this is a much more easier thing to verifying
principle.
00:37:51.400 --> 00:37:57.101
So what you do see what we had above is that
to check A is compact we have to check it
00:37:57.101 --> 00:38:04.280
is totally bounded and totally bounded is
not practical but checking its bounded that
00:38:04.280 --> 00:38:11.610
is more practical see checking the checking
a family of functions or a collection of functions
00:38:11.610 --> 00:38:17.730
is bounded is a very easy thing because you
have to check that there is a bound for all
00:38:17.730 --> 00:38:21.900
there is a common bound for all the functions
and mind you it is a uniform it is it will
00:38:21.900 --> 00:38:27.640
be a uniform or common bound because you are
the metric you are working with is induced
00:38:27.640 --> 00:38:29.590
by the norm and the norm is a sup norm.
00:38:29.590 --> 00:38:34.610
So when you say it is bounded you mean bounded
in the sup norm and that means it is uniformly
00:38:34.610 --> 00:38:41.820
bounded. So that means you must find a single
positive constant such that mod f x is always
00:38:41.820 --> 00:38:47.610
less than or equal to that positive constant
for all f in A and all X in X you must find
00:38:47.610 --> 00:38:51.840
uniform bound that is why this is sometimes
called a uniform boundedness principle, okay
00:38:51.840 --> 00:38:55.530
Arzela-Ascoli Theorem is sometimes called
the uniform boundedness principle.
00:38:55.530 --> 00:39:03.260
So what you do instead of checking set of
functions is totally bounded what you do you
00:39:03.260 --> 00:39:08.860
just check that there is uniform bound find
the bound for all the functions in your family
00:39:08.860 --> 00:39:13.110
which will work at once for all the functions
in a family that is one thing that you have
00:39:13.110 --> 00:39:19.130
to check, okay that is much more practically
easy easier and the other thing is you have
00:39:19.130 --> 00:39:23.680
to check this so called equicontinuity and
what is this equicontinuity? The equicontinuity
00:39:23.680 --> 00:39:29.460
is a very very simple thing what it says is
that no matter what points you chose okay
00:39:29.460 --> 00:39:37.870
the moment you decrease the distance between
points then the distance between the function
00:39:37.870 --> 00:39:42.691
values will decrease no matter what function
you chose in your family it will work for
00:39:42.691 --> 00:39:43.691
all.
00:39:43.691 --> 00:39:48.640
So you know usual definition of continuity
is given an epsilon you find delta, okay now
00:39:48.640 --> 00:39:54.870
that epsilon given an epsilon the delta will
depend on the point at which you are checking
00:39:54.870 --> 00:40:01.660
continuity and it will also depend on the
epsilon but what you want is you want given
00:40:01.660 --> 00:40:09.820
an epsilon, you want a delta which works for
any two points which works for any point and
00:40:09.820 --> 00:40:15.780
for any function at the same time in your
family that is equicontinuity, given an epsilon
00:40:15.780 --> 00:40:23.390
you find delta which works for every function
in your family and for all points in one go
00:40:23.390 --> 00:40:25.090
that is equicontinuity.
00:40:25.090 --> 00:40:29.390
And this is also something it is a property
of continuous functions so it can be checked
00:40:29.390 --> 00:40:35.220
unlike total boundedness where you have to
pick up some finitely many functions explicitly
00:40:35.220 --> 00:40:40.320
which is not so easy, okay that is why the
Arzela-Ascoli Theorem is a very useful a tool
00:40:40.320 --> 00:40:46.220
to check compactness of a family of functions,
okay and this is what you get from topology.
00:40:46.220 --> 00:40:50.960
Now what we will do in the next lecture is
that I will tell you we will try to understand
00:40:50.960 --> 00:40:55.320
how you can translate this to our situation
where we are working with analytic functions
00:40:55.320 --> 00:40:59.950
and Meromorphic functions you have to translate
this, okay so I will stop here.