See you know I am trying to bring in the context of the context to explain why we need the
Arzela-Ascoli Theorem, okay in our discussion, okay so you know the as I was telling you
in the last lecture the main thing we are concerned about is compactness of families
of functions, okay. So at the background is we have the need to get a proof of the Picard
theorems, okay and for that you have to study compactness of families of Meromorphic functions,
okay.
And we have to understand that properly, alright and therefore you must understand compactness
from basic point of view first of all compactness in the plain topological sense what it means
for metric spaces and then you know what it means for spaces of functions, okay and once
you know all this then you your mind is now then properly in tune to understand the you
know the the proof of the so called Montel’s theorem okay which is a key theorem that is
used for going for getting a proof of the Picard theorems, okay.
So so again you know so let me tell you very briefly let me again recall very briefly the
topological background that I gave you last class. So you know if you start with if you
start with the topological space you know there is a notion of compactness, okay namely
every open cover has a finite sub cover and that is a very that is a very general notion,
alright and it can be defined for any topological space because all you need for the definition
is the idea of open set which is there in any topological space which is fundamental
to any topological space.
Now but the point is that this this abstract definition of compactness is not useful all
the time because it is a very it is a condition that you have to check for all covers for
example if I want to check a topological space is compact I have to take an arbitrary cover
of open sets and then I have to from that I have to show that I can pick out finitely
many that is something that is not so easy today because it is you know why it is not
easy to do is because a cover can be very abstract, it can be arbitrary and form something
abstract it is very hard to extract something that is very specific, okay.
So so as usual as we do in mathematics normally you have a definition which is very abstract
and the reason why you like that definition is because it has caught lot of power the
abstraction gives it a lot of power, the abstract definition of compactness gives you lot of
power in the sense that you know it tells you that whenever you have an open cover you
can always get a finite sub cover.
So it is a very powerful thing but practically it is not so useful when you want to really
use it in particular situations and just like same situation mathematics most of the time
you have some definition involving a property which is very abstract and you make that definition
because you know that property is very powerful it is a very powerful strong property which
you can use.
But then how do you put it to use then the theory tries to give equivalent conditions
so that you can which are easier to verify or which are more handier to verify, okay.
So so in the same way if you look at compactness okay then there are conditions which help
you to verify compactness in a more easy way by so what are equivalent conditions I was
telling you yesterday that is you if you take a metric space then for a metric space compactness
is a same as compactness is a same as sequential compactness and that is same as the Bolzano-Weierstrass
property, okay.
So so what is the sequential compactness it is the property that every sequence has a
convergent subsequence alright and and what is a Bolzano-Weierstrass property? The Bolzano-Weierstrass
property is that every infinite subset has a limit point, okay and we are more used to
looking at eucledian spaces, okay R n, n dimensional real space and you can also take C n, n dimensional
complex space the n C n can be thought of as R 2n if you want, okay and these eucledian
spaces they have the property that you know compactness there is equivalent to closeness
and boundedness, okay and that is what we are whenever we are doing whenever we are
working with eucledian spaces if you want to check compactness all you will do is you
will check it your set is bounded, you will check it is closed, okay and that is it then
you know it is compact, okay.
So it is as easy it is that easy and mind you just take for example the close close
take a close disk on the complex plane take a close disk on the complex plane it is easier
to say that it is a close set and that it that it is a bounded sets and therefore it
is compact it is easier to say that then to say then to take an arbitrary open cover and
try to pick out a finite sub cover, okay it is not easy if I give you an arbitrary open
cover of a close disk, okay on the complex plane it is not easy to pick out finite sub
cover, alright. Whereas, I can assert that this will be true because of compactness and
why because I can check compactness by the equivalent condition that it is both closed
and bounded which is easy for me to check, okay.
So but ofcourse you know it is very important that so this is see in all these issues we
are in the context of metric spaces we are not in the more general context of topological
spaces because the metric is involved in all these things for compact when you define compactness
metric is not involved, okay but when you define sequential compactness, okay you are
worried about convergence of a sequence, okay and that is easiest to define if you have
a metric otherwise if you do not have a metric you have to worry about nets and things like
that, okay.
And Bolzano-Weierstrass property, okay can well in fact it can also be defined more generally
but these properties are all very easily defined when you have a distance function between
two points which is a metric, okay. So you see if you take ofcourse you know if you take
a so what I was saying was that you know in all these things we are working with a metric
space, okay and and the point is that with a metric space things are better because I
mean you can visualize a lot of things because there is a distance function, okay.
So if limit if suppose you have if you say limit x tends to x not in a metric space it
makes sense because you are actually saying the distance between x and x not is tending
to 0, okay you are letting a function to tend to 0 so it makes clear sense if you cannot
make so easy sense of this if you did not have a metric, okay. So so what I want I tell
you is that now you know so the point is that somehow if you are working with eucledian
spaces then compactness is just the same as closeness and boundedness put together, okay.
But if you take an arbitrary topological space one way is always true if something is compact,
okay then it will always be if you take a subspace of a topological space if it is compact,
okay well I should say if you take an arbitrary metric space not not not eucledian spaces,
okay you take an arbitrary metric space if you take a compact subset it will be closed
and bounded, okay but the converse need not be true, okay eucledian spaces are very special
for which closeness and boundedness implies compactness is equivalent to compactness.
So the standard example is R infinity okay you look at all the sequences of real numbers
such that the sum of the if you write out the series which consist of the sum of the
squares of the modulus I the so called square summable sequences, okay in in modulus, okay
then so you know you take all the so you take all sequences x1, x2, etc xn of real numbers
such that sigma mod x i square okay you take the modulus of the entries of the sequence
and square it and you sum it mind you when you sum it you are getting a series that series
should converge, okay.
The set of all these sequences they are called square summable sequences the set of all these
sequences is called R infinity is called the infinite dimensional eucledian space, okay
and in this R infinity it is a metric space, okay because it is very easy to define a metric
using just extending the same distance formula only thing is that now you have infinitely
many coordinates the usual distance formula is what you take difference of coordinates
square them, sum them up and then take square root.
Now what you do is you do it for all the coordinates even though there are infinitely many coordinates
this will work because I know that if you sum all the if you take the sum of the squares
of all the coordinates it converges, okay. So what will happen is that you get this R
infinity and in R infinity if you take the if you take the open when you take the closed
ball radius 1 centre the origin, okay you are taking all those elements whose distance
from the origin is less than or equal to 1, okay then what happens is that this is ofcourse
closed and bounded it is a closed ball so it is close and its radius 1 so it is bounded
but it is not compact, okay because if you take the sequence which consist of only 1’s
along the diagonal okay you take a sequence of sequences, okay which if you write down
one below the other so that you know you get the only 1’s along the diagonal and 0’s
elsewhere okay then that sequence you know it will be in this in this closed ball but
it will never have a convergence subsequence and that is because the distance between any
two members of sequence is always fixed at positive quantity it is a constant value in
fact it is it is root 2 actually, okay.
So you can see that if the distance between two members of the sequence is a constant
such a sequence cannot have a convergent subsequence because the convergent subsequence means it
has to become Caushy and distance between terms should come closer become very small,
okay so it cannot have a Caushy subsequence. So what it means that you are able to get
a sequence which does not have a convergent subsequence so it means it is not sequentially
compact but if it is not sequentially compact it cannot be compact because there is this
theorem which says that for a metric space compactness is same as sequential compactness,
okay so R infinity is an example of metric space where you closeness and boundedness
does not imply compactness okay.
So you are in this closed and bounded implies compact works for us for most of the time
because we are worried about only usually we are worried about subsets of eucledian
spaces, okay if you are working with R n or C n, n dimensional real or complex space then
you are in good shape okay. Now but then you know the but you take an arbitrary metric
space compactness will always imply closeness and boundedness, okay a compact subset will
always be closed and bounded, okay that will always be true compactness is always stronger,
compactness will give you closeness, compactness will give you boundedness but if you want
to get back compactness from closeness and boundedness that will not work in an arbitrary
metric space as we saw it does not work in R infinity but it will work in R n or C n,
okay.
Now the truth is compactness is not only that strong in fact it is far more stronger it
gives you something called total boundedness and what is this total boundedness? This total
boundedness is as I was explaining to you know last time by this diagram you can have
a look at this diagram so you see I have this space X which is a metric space mind you and
the total boundedness is that you know you give me any positive number epsilon, okay
you think of it is a very small radius, okay then what you can do is you can find finitely
many points points of a subset A sub epsilon called an epsilon net such that you know every
point of X is within an epsilon distance from one of these points atleast one of these points.
So in other words if you take the if you take the open balls centred at these points these
finitely many points which are elements of A epsilon so called epsilon net with radius
epsilon, okay then the union of all these balls will cover the whole space. So you know
it is of mind you it is you are covering the space by only finitely many open balls of
given radius epsilon and this will and this should work for every epsilon that is the
that is even if you make epsilon smaller you will still find another finite set of points
at which centred at which if you take these epsilon balls and you take the union it will
still cover the whole space, okay it is a very strong property.
And this will this will ofcourse fall through if it is compact why because you know the
truth is that give me if the space is compact give me any epsilon and you take the collection
of all balls centred at various points all possible points with radius epsilon you take
all these open balls that is an open cover and because it is compact it has to have a
finite sub cover so I found an epsilon net for every epsilon I can find an epsilon net
just simply because of the very definition of compactness so compactness will give you
total boundedness.
But total boundedness and ofcourse total boundedness implies boundedness ofcourse because you can
check as I was telling you last class that the the diameter of the topological space
can be compared is always less than or equal to the diameter of the epsilon net plus 2
times epsilon. So you will get this so I am little short of space but let me write it
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
2 epsilon d of A epsilon plus 2 epsilon you will get this for every epsilon you will get
this and d of A epsilon mind you the finite quantity because it is the diameter of a finite
set diameter is supposed to be supremum of the distances between two points on the set,
okay it is trying to measure how large the set is using the metric, okay.
So and we put because sometimes the set may be unbounded and in that case your diameter
may be infinite, okay. So in any case what this tells you is that if a space is compact
if the metric space is compact and you know this tells you that it is bounded because
it has finite diameter alright and and ofcourse I told you it will also be closed if a subspace
of a metric space is compact it will be bounded and closed ofcourse, okay and you also get
an epsilon net for every epsilon which means it is totally bounded alright and total boundedness
a very strong condition it is a very strong form of boundedness.
Now the point is that from total boundedness how do you get back compactness, can you get
back compactness? And the theorem is that you can get back compactness with this with
this extra condition, okay. Suppose your space is totally bounded and complete then it is
compact, okay so it is a you know it is rather powerful theorem so you know compactness so
you know to check that something is compact it is enough to check that your metric space
if you have a subspace of a metric space suppose you want to check its compact all you have
to check is well now you have one more condition you check it is complete that means you have
checked that every Cauchy sequence converges and you check that it is totally bounded,
okay so this is this is another thing which helps.
And you know as far as general topological spaces are concerned things are really bad,
in fact you can have suppose you have a general topological space which is not a metric space
suppose it is not a metric space, okay then you can have a horrible situation like this
the whole space will compact you can have a whole space which is compact not a metric
space, topological space whole space is compact you can have a (sub) you can have a subspace
which is also compact but it is not closed such a horrible thing can happen this can
happen in an arbitrary topological space it cannot happen in a metric space, in a metric
space a compact subspace is always closed and bounded, okay that is what you have to
remember, okay.
So so well now the point is that so what you get is you know so let me continue with this
discussion so you have compactness on the one hand and then ofcourse we are working
with metric spaces you have compactness on one hand and that
is you can get that from complete complete and totally bounded
and in fact this way also this is correct because you see if a space is compact it is
ofcourse totally bounded I have told you if a metric space is compact it is totally bounded
I told you it has finite diameter in fact and finite diameter which is comparable to
diameter of any epsilon net, okay which exist because of total boundedness and compactness
also implies completeness because you see if it is compact it is sequentially compact,
okay.
So every every sequence has a convergence subsequence. So in particular if a sequence
is a Cauchy sequence if then it will have a convergent subsequence but if a Cauchy sequence
has a convergent subsequence then the Cauchy sequence itself must be convergent, okay so
so compactness is equivalent to complete and totally bounded, okay. Now the problem is
that you know this is this is what we have in general topology if we are only worried
about spaces okay not just topological spaces but we are worried about more specific spaces
namely metric spaces, okay but our context is different our context is we are worried
about functions, okay.
If fact our application is we want to study Meromorphic functions on a domain in the external
complex maybe alright that is the context where we have to go to. So somehow you know
these results which are for spaces you have to translate them they are not still good
enough for our use you have to translate them into results for spaces of functions, okay
because we want to apply everything to spaces of functions.
So what spaces of functions will you think of? So you know the you know if you take so
let me recall again if X is a topological space, okay then you know you can take this
you can take the set of all maps from X to R okay or you can take the set of all maps
from X to C so you can take the set of all real valued functions on X or you can take
the set of all complex valued functions on X this is these are algebras, okay.
So you see you take the set of all maps from the space to the real line these are just
maps theoretic maps I am not assuming anything continuity nothing, okay. So just maps from
X to the real line or just you take the maps on X to the complex plane, okay. So just real
valued functions or complex valued functions, okay now that is a these are algebras, algebras
means that they are so for example we will take the set of all real valued functions
that is a vector space it is a real vector space because you can add two functions and
you can multiply a function by a constant real constant it is a real vector space.
And if if you think about it it is also an algebra it is a commutative ring namely there
is a multiplication in it which is commutative and that is just multiplication of real valued
functions if you multiply two real valued functions point wise you get again another
real valued function so this is a ring it is a commutative ring which has a vector space
structure and the vector space structure is compatible with the ring structure multiplication
distributes over addition in the right sense, okay and so on and therefore this is a nice
ring it is a ring plus a vector space such a thing is called an algebra.
So I have mentioned this before in one of the earlier lectures, so this is set of all
maps from X to R is a real algebra set of all maps from X to C is a complex algebra,
okay and the point is that if you are looking at among these maps if you look at only bounded
maps okay you look at maps whose images are bounded, okay in the image I can talk about
boundedness because the image is going to lie either in R or in C if you take a if you
take a real valued function the image is going to be a subset of R, if you take a complex
valued function the image is going to be a subset of C, okay.
So the images make sense and subsets of R and C therefore boundedness of the image make
sense therefore see what I can do is I can look at so let me write this if I take the
bounded maps from X to R or if I take the bounded maps from X to C what will happen
is that this is this is a subset of this, okay in all these upto this point you know
I do not even need X to be a topological space X could have even be a non-empty set I am
even the topology on X I am not going to use I have not used okay because I will worry
about the topology on X if for example I am looking at something connected with maps which
are connected with the topology namely continuous maps, okay but I am just looking at maps.
So X could need not even be a you know topological space in fact I can just say let me I can
just rub this and say X is just a non-empty set. So I get these two algebras and I get
the subsets of bounded maps, okay. Now these bounded maps also if you if you check they
also form sub algebras okay because the sum of two bounded maps is bounded, product of
two bounded maps is bounded and so on.
So these will give you sub algebras and the beautiful thing is that the boundedness allows
you to define a norm the so called supremum norm, okay. So what you can do is that you
can define norm f to be supremum over x small x in capital X mod f x for for f for f a bounded
map, okay I can define this supremum norm alright and the point is that this is a norm
once it is a norm it is a norm on a vector space so it becomes a norm linear space and
once you have a norm linear space the norm induces a distance function alright so you
get a metric space, you get the metric induced by a norm and once you have a once you have
this metric you have a topology induced by the metric.
So these becomes very nice topological spaces in fact they become Banach spaces, okay they
become Banach algebras they become complete norm linear spaces and the completeness is
just because of the completeness of the target it is because of completeness of real line
completeness of complex numbers, okay. So what you get is that you get these two even
for a non-empty set you get this two beautiful Banach spaces Banach algebras alright one
is a real Banach algebra the other is a complex Banach algebra depending on whether you are
considering real valued functions or complex valued functions.
Now what you can do is, now you put one more so so let me write that the distance between
f and g is is just norm of f minus g I can I can define this is the metric defined by
the norm, alright and now we are in a nice metric space norm okay and now what you can
now what happens is that you know if X is topological space then you can go one step
down and then look at the set of all continuous functions from continuous bounded functions
from X to R or you can look at the set of all continuous bounded functions from X to
C, okay.
Then what happens is that this you can check that the set of all continuous functions that
subset is a closed subset. The reason is because if you take a sequence of if you take a sequence
of continuous functions if it converges to a limit function the convergence here will
correspond to uniform convergence because of the sup norm and you know a continuous
a uniform limit of continuous functions is continuous therefore what it means is that
if you take a sequence of continuous bounded continuous functions, okay and if it if it
is a Cauchy sequence okay then you know it will converge in the whole space because the
whole space is ofcourse complete mind you it is a Banach algebra the whole space is
complete but the limit will also be continuous because it is a uniform limit and why it is
a uniform limit is because of the sup norm if you check, okay.
Therefore the space of all continuous real valued functions bounded real valued functions
C X R or the space of all continuous bounded complex valued functions C X C that is a close
it is a closed subset and you know a closed subset of a complete space is again complete
therefore this C X R and C X C are very beautiful Banach algebras also they are Banach sub algebras
closed sub algebras, okay.
And now the point is suppose you are so now you know you are coming to now slowly you
know we have come into discussing spaces of functions, okay now you know I would like
to do topology on a subset of of this so I would like to look at a space or a family
or a subset or a collection of let us say continuous real valued functions and on that
set considering as a set A subset A of C X R okay I wanted two topology on that and what
kind of topology I am interested in compactness.
So the question is suppose I start with an A here or here the question is when is A compact
this is my question you see because my aim is what my aim is to study compactness of
spaces of functions you know and mind you let me again keep reminding you so that you
do not get lost our final aim is to study compactness of a family of Meromorphic functions,
okay so I am trying to do it in the simplest case in the case of just say topological the
topological case of this continuous functions continuous real valued bounded functions,
okay.
Now when is A compact? If you see by whatever I told you because A is anyway a subset of
a metric space, alright since A is subset of a metric space to check A is compact I
can check ofcourse I can check the usual definition of compactness that every open cover has a
finite sub cover that is highly in practical, okay then I can check sequential compactness
I can check that you know every every sequence in A has a convergence subsequence, right
that is another thing that I can check.
The third thing is I can check that A has a Bolzano-Weierstrass property because these
are all the characterizations of compactness on a metric space okay and you know of all
the three one possible thing that I can check for A is that it is sequentially compact,
okay I can so I can check that if you give me any sequence in A you give me a sequence
of functions in A if I can check that there is a convergent subsequence then also I will
get compactness, okay.
So that is something that is I have but mind you if I have to check that I the sequence
in A converges to it has a convergence subsequence mind you the convergence is now uniform convergence
because you are under the sup norm I have to check uniform convergence, okay that is
what it means, alright. So that is one thing that I can do, what other things can I do?
I just now told you that for metric spaces as I have written above compactness is a same
as completeness and total boundedness.
So I can check that A is complete and totally bounded but you know mind you I told you a
compact subspace of a metric space is always closed and bounded okay the converse may not
be true okay as I have told you in the case of R infinity okay. So if you expect A to
be compact A should atleast be closed that is a necessary condition okay so that is always
that A has to be closed this this you cannot avoid so you must have closeness alright.
And what this means is that therefore if you check that there is a if you so if you want
to check sequential compactness in A it is enough to check that every sequence as a Cauchy
sequence that is enough because we will check it has a Cauchy sequence then it means it
has a convergent subsequence and the limit will also be in A because it is closed, right.
So A has to be closed you cannot avoid that and what does this thing that I have written
on top tell you to check that it is compact I have to ofcourse it has to be closed I have
to check that it is complete and totally bounded, okay.
And the point is completeness is automatic, why because A is already a closed subspace
of a complete metric space so it is automatically complete. So the only thing I have to check
is totally boundedness total boundedness.
So so A is compact so let me write it here A compact if and only if A is totally bounded
okay this is what we get this is what we get mind you ofcourse A is closed that is already
already assume A is closed because if it is not closed you cannot expect compactness okay.
So we have ended up at this point we have ended up at this point where you are saying
so finally what does all this translate? Give me a bunch of functions real valued continuous
bounded functions on a space alright, how do I check that as topologically it is compact
how do I check it is compact?
So this tells me check it is totally bounded, okay. Now what does that mean? It means that
you have to find an epsilon net for every epsilon what does that mean it means given
an epsilon positive I have to find finitely many functions from this family such that
the distance of any other function from atleast one of these functions these finitely many
functions is less than epsilon that is what epsilon net means, okay.
So I have to pick given an epsilon greater than 0 I have to find finitely many functions
from A such that the epsilon open open balls at those centred at those finitely many functions
with radius epsilon that covers A that is what an epsilon net for A means okay and you
see this is also very abstract you see it is very very abstract it is if you if I start
with an abstract family of functions usually families of functions are abstract because
you know they will depend on some property I might have functions which have some differentiable
property or some some they may be defined by some abstract property and from an abstract
collection it is not so easy to pick out finitely many it may not be so easy to do it.
So this total boundedness checking this total boundedness for a family of function does
not work, okay so what comes to help us here is something that can really be checked for
families of functions and that is the Arzela-Ascoli Theorem.
So let me write that down so so this is this so let me write this this is not practical
this is not practical by that I mean this is not easily verifiable in practise okay
you cannot demonstrate it in practise so easily, okay so what comes to our help is so called
Arzela-Ascoli Theorem. So what is that so let me write that down.
So here is the Arzela-Ascoli Theorem and what is this theorem and what does this theorem
say it says that suppose X is compact X is a compact metric space you assume you are
working on a compact metric space okay the advantage of working on a compact metric space
is that automatically all continuous functions have bounded okay any continuous function
on a compact any continuous real valued function on a compact sets is you know it is bounded
it attains it is bounded uniformly continuous you know all these things.
So if you are working with a compact space then you do not have to restrict to bounded
continuous functions every continuous every continuous function is automatically bounded,
okay so you work with a compact metric space okay then a closed subset A of C X, R or C
X, C is compact if and only if it is bounded and equicontinous, okay. So this is this is
this is a much more easier thing to verifying principle.
So what you do see what we had above is that to check A is compact we have to check it
is totally bounded and totally bounded is not practical but checking its bounded that
is more practical see checking the checking a family of functions or a collection of functions
is bounded is a very easy thing because you have to check that there is a bound for all
there is a common bound for all the functions and mind you it is a uniform it is it will
be a uniform or common bound because you are the metric you are working with is induced
by the norm and the norm is a sup norm.
So when you say it is bounded you mean bounded in the sup norm and that means it is uniformly
bounded. So that means you must find a single positive constant such that mod f x is always
less than or equal to that positive constant for all f in A and all X in X you must find
uniform bound that is why this is sometimes called a uniform boundedness principle, okay
Arzela-Ascoli Theorem is sometimes called the uniform boundedness principle.
So what you do instead of checking set of functions is totally bounded what you do you
just check that there is uniform bound find the bound for all the functions in your family
which will work at once for all the functions in a family that is one thing that you have
to check, okay that is much more practically easy easier and the other thing is you have
to check this so called equicontinuity and what is this equicontinuity? The equicontinuity
is a very very simple thing what it says is that no matter what points you chose okay
the moment you decrease the distance between points then the distance between the function
values will decrease no matter what function you chose in your family it will work for
all.
So you know usual definition of continuity is given an epsilon you find delta, okay now
that epsilon given an epsilon the delta will depend on the point at which you are checking
continuity and it will also depend on the epsilon but what you want is you want given
an epsilon, you want a delta which works for any two points which works for any point and
for any function at the same time in your family that is equicontinuity, given an epsilon
you find delta which works for every function in your family and for all points in one go
that is equicontinuity.
And this is also something it is a property of continuous functions so it can be checked
unlike total boundedness where you have to pick up some finitely many functions explicitly
which is not so easy, okay that is why the Arzela-Ascoli Theorem is a very useful a tool
to check compactness of a family of functions, okay and this is what you get from topology.
Now what we will do in the next lecture is that I will tell you we will try to understand
how you can translate this to our situation where we are working with analytic functions
and Meromorphic functions you have to translate this, okay so I will stop here.