Now, there are a few things that are remaining
in the discussion of the series and those
we shall complete today. We discussed for
a long time the series of non negative terms.
Only in the last class we discussed the series
of the terms which may have positive and negative
signs and we discussed what is known as a
bells test.
A series which is convergent, but which is
not absolutely convergent it has a special
name it is called conditionally convergent.
That is to give of more formal definition,
suppose if this is a series sigma a n, n going
from 1 to infinity. This is said to be conditionally
convergent or we say it converges conditionally.
If the series converges that is if sigma a
n converges, but it is not absolutely convergent.
That means and a sigma mod a n diverges. We
have seen an example of a series of this kind.
For example, we saw this series yesterday
1 minus half plus 1 by 3 minus 1 by 4 etcetera.
We have seen that this series converges, but
it is not an absolutely convergent series.
So, this is an example of a conditionally
convergent series. Now, in case of this conditional
and absolute convergence, there appears one
very important question and that is what we
shall also discuss today.
That is what is called rearrangement. Now,
to just motivate this after all series is
something like an infinite sum. Now, if we
take a let us say a finite sum, suppose we
take say something like a 1 plus, a 2 plus,
a 3 plus, a 4 plus, a 5 etcetera. Since the
addition is commutative as well as associative
it does not matter.
In what way I take the sum, whether I add
a 1 to a 2, then add to a 3 or I take a 1,
a 5 here or a 3. There it does not matter
whichever way you write this the sum is going
to be the same right, but that may or may
not happen in case of the infinite series.
So, now here actually what is meant by rearrangement,
it is that you write these terms of the series
in some different order that is called a rearrangement.
To make it more precise, suppose we take a
map tau from n to n, suppose it is a bisection.
It is 1 1 on to as you know such maps are
also called permutation. If it is a finite
set, you call 1 1 on to map is permutation.
So, if you consider a series sigma a suffix
tau n that is, suppose your given series is
sigma a n instead of that you consider sigma
a suffix tau n. Then this is called rearrangement
of sigma a n. That means basically what the
term says that rearrangement, that you just
rearrange the terms of the series write the
terms in some different order. Now, what is
the obvious question here, if the original
series converges does this rearrangement also
converge and does it converge to the same
sum if it converges does it converge to the
same sum
Now, it so turns out that that is true if
the series is absolutely convergent and the
things are very bad if the series is conditionally
convergent. We can just see an example here,
let us take this same example we have seen
that this series is 1 minus etcetera. We have
seen that this is a convergent series suppose
its sum is s suppose its sum is s. Now, let
us just rewrite the series in some different
order what I will do now is that, I will write
the series as follows. So, 1 minus half minus
1 by 4, that instead of taking 1 by 3 I take
the next term as 1 by 4 then I will take plus
1 by 3, then I will take the next 2 immediate
term 1 minus 6 minus what is a.
So, 1 minus 4 the next negative term will
be minus 1 by 8 right, then take the next
positive term that is 1 by 5 and then minus
1 by 10, minus 1 by 12 etcetera. I suppose
you are not the original series was there
was 1 positive term 1 negative term etcetera,
followed by that. Now, what I am doing is
that I am taking that same series I am taking
the first positive term, then the next tqo
negative terms then the next positive term.
Again followed by next two negative terms
and do like that all right. So, it is a, this
is a rearrangement of this same series. So,
all right now let us see a few things here
for example, this 1 minus half is half 1 minus
half is half. So, this is 1 by 2 minus, 1
by 4 plus, again this 1 by 3 minus 1 by 6.
You can say that is same as 1 by 6 that is
same as. So, 1 minus. So, the next will be
1 by 6 minus 1 by 8.
Then similarly, you can say that 1 by 5 minus
1 by 10 again 1 by ten. So, that is. So, the
next 2 terms will be 1 by 10 minus 1 by 12
etcetera this will be the series after simplification
right. You can see that I can this half is
a common factor from all of them ok. So, suppose
we take that common factor out what remains
half into 1 minus this will be 1 by 2, that
will be plus 1 by 3 then minus 1 by 4 etcetera
plus 1 by 5, minus 1 by 6 etcetera all right.
Do you see that it is the same series here
ok.
So, now what is its sum it is half s it 1
by 2 s right. So, it is the rearrangement
of the same series, but it converges to a
different number right, it converges to different
number. So, this idea of this example is to
show that, if the conditional convergence
behaves very badly with respect to rearrangements.
In fact what is known is something much worse
not only that rearrangements will converge
different number, in fact given any real number
we can find some rearrangement of the series,
such that that rearrangement converges to
that given real number. Not only that we can
also find a rearrangement. So, that that real,
the new series with that that new rearranges
that new rearrangement diverges. On the other
hand, if the series converges absolutely then
all its rearrangements converge ok.
All those rearrangements converge to the same
sum. So, let us just write this as a theorem.
So, if sigma a n converges absolutely, then
all it rearrangements converge 
and converge through the same sum. On the
other hand, if sigma a n converges conditionally
then given any real number, we can find a
rearrangement such that that rearrangement
converge to that re real number. So, what
I was saying that for every s in R there exist
a rearrangement sigma, I will call it rearrangement
sigma a suffix tau n converging to s.
Also we can find a rearrangement. So, that
that rearrangement diverges. So, we can say
that also 
there exist rearrangement sigma a tau n that
diverges. So, that is the importance of absolute
convergence. If you know that a series is
absolutely convergent or the series is of
non- negative terms, remember this if the
series is of non- negative terms, there is
no difference between convergence and absolute
convergence.
So, in that case you can rearrange the terms
of the series in any manner you like and that
will not change the convergence or divergence
or it will not also change the sum. Whereas
in case of conditional convergence things
are quite bad all right. Now, there are a
few elementary properties of the series which
perhaps we should not discuss immediately
after discussing the series and those are
as follows.
Suppose we take two series sigma a n, n going
from 1 to infinity and let us say sigma b
n, n going from 1 to infinity. Suppose both
of them converge then we can say that if you
take the series sigma a n plus b n that should
also converge. It is sum should be same as
the sum of sigma a n and sigma b n. So, let
us let us just see that, if sigma a n and
sigma b n converge then, sigma a n plus b
n also converges and sigma a n plus b n n
going from 1 to infinity, this is same as
sigma a n plus sigma b n equal to. By the
way, you may wonder we are not going to discuss
the proof of this theorem here because it
is proof is somewhat lengthy.
Those a few who are interested in proof you
can see the proof in Roudin's book. This theorem
is given in Ruded coming back to this. So,
what it says is that if the two series converge
their sum also converge. Similarly, if you
multiply the series by some real number lambda
then the new series will also converge. That
is also we can say that also sigma lambda
a n this is also convergent series and it
sum will be same as lambda times sigma a n
n going from 1 to infinity all right.
This will follow simply by taking the partial
sums. Suppose s n is a partials of the series,
suppose s n is a 1 plus a 2 plus a n and t
n is say b 1 plus b 2 plus b n, then saying
that sigma a n converges is same as saying
that s n converges, s n converges to s and
t n converges let us say t. Then use the corresponding
theorem about the sequences then s n plus
t n converges to s plus t.
Similarly, lambda s n converges to lambda
times s that is all in there is a proof. I
said already that, whatever we want to prove
say or prove about the series, everything
can be done using the sequence of partial
sums. Using the corresponding theorem about
a sequences, but what you will also notice
further is that, we cannot give a similar
characterization if we take the product. For
example, if we take this series.
Let me again recall, that is suppose we take
this s n as the sum sigma. Let us say sigma
a j, j going from 1 to n and say t n as sigma
b j, j going from 1 to n. Then partial sum
of the series sigma a j a n plus b n that
is same as s n plus t n that is fine, but
suppose I take these series, sigma a n b n,
n going from 1 to infinity. Then the partial
sum of this will be a 1 b 1 plus a 2 b 2 plus
a n b n and that is not the product of s n
and t n right that is not the product of s
n and t n. So, we cannot say that if s n converges
and t n converges, this series also converges.
So, in general we cannot say that if the two
series converges that is, if the sigma a n
and sigma b n, if both of them converge we
cannot infer from that that sigma a n b n
is also a convergent series. So, to consider
the products there is totally different notion,
what is called cauchy products. Cauchy product
is something like this for example, the first
term suppose I denote the term, say product
as sigma c n I denote that terms that are
wrote as sigma c n. I think going from 1 to
infinity, then this first term let, I think
for this for considering this, it is convenient
to start with 0 to infinity, take both the
series, starting from 0 to infinity.
That is sigma a n also going from 0 to infinity
and sigma b n also going from 3 to infinity
just a minor convenience here. So, sigma c
n will also I will take from o to infinity.
So, the first number here is c naught, the
first number here is c naught. So, that is
taken as a naught b naught, that is just the
product of the corresponding terms. Then the
next number c 1, that is taken as a 1 b naught
plus a naught b 1, a 1 b naught plus a naught
b 1 right. Now, you can understand how we
will proceed for example, next number c 2
that will be taken as a 2 b naught, plus a
1 b 1, plus a naught b 2 that is what we are
doing here.
We are taking all those indices such that
the sum becomes 2, here. Now, we can understand
that how the general term will be. So, in
general the term c n that will be sigma a
k b n minus k, k going from o to n ok. So,
suppose you form a series like this, then
that series is called the cauchy product of
these 2 series, sigma a n and sigma b n. We
can say something about the cauchy product,
if sigma a n and sigma b n converges then
whether cauchy product also converges are
not, there are some conditions for that, but
since that is not very important right.
Now, for us we shall not go into theorems
of those kind for the time being. All that
you should remember is that convergence of
these two series sigma a n and sigma b n certainly
does not imply that sigma a n b n converges
right. Of course, directly it also does not
imply that sigma c n converges you need some
additional conditions for that ok all right.
Now I think for the time being we shall close
the discussion of the series and let us move
on to next topic, but to move out to the next
topic, let us start something with which depends
on the series. I shall define a new set.
Let us say I will call that set l 1, l super
script 1 this is also let us say this is a
set of real sequences and a real sequence.
As you know a sequence is nothing, but a real
sequence is nothing but a function from n
to r. So, suppose I denote any such function
as x, x from n to r. Then image of any number
n here by using this notation I should denote
by x of n right, but in a sequence it is customary
to denote it is as x suffix n all right. We
can continue to use this notation. So, I will
just say that x means this sequence x n, x
means the function from n going to R and that
is nothing but same as the sequence x n.
Now, this notation is some times more convenient,
when you also want to talk about sequence
of sequences. Then for example, suppose I
want to talk of sequence of this sequences,
then you will need some more either super
script or sub script. So, instead of that
for that this notation is more convenient
all right. So, what I want to do is that I
shall take the set of all sequences x. So,
x is a sequence, such that sigma x n is absolutely
convergent, that is take the series sigma
x n, either write x n or this way whichever
way you want like sigma x n is absolutely
convergent all right.
That means what sigma mod x n is convergent,
that is what sigma mod x n is convergent for
example, if you look at these theorem. Here
can I say if sig a sigma a n and sigma b n
if both are absolutely convergent will it
follow that sigma a n plus b n is also absolutely
convergent right because you take mod a n
plus b n, that is less nor equal to mod a
n plus mod b n. Since sigma mod a n and sigma
mod b n those are convergence. So, what you
can say that sigma mod a n plus mod b n that
is a convergent series.
Then use comparison test right. So, that is
trivial. So, if there are two series which
are absolutely convergent, then their sum
is also absolutely convergent. In other words
if I use this notation I can say that if x
and y x and y belong to l 1, then x plus y
also belongs to l 1 all right all right. Next
is I will say that if x belongs to l 1 and
let us say some alpha belongs to r, then alpha
x also belongs to l 1 right because the series
alpha x n will be nothing, but mod alpha times
mod x n, that is also convergent series right.
So, what it means is that if you take these
set l 1, then if you take any two elements
in this set their sum is also in this set.
Product of a real number and any element in
this is also in this set, in other words this
set these two operations, addition of two
elements and multiplication of a scalar. A
element in this right scalar real scalar is
real number. So, what is the obvious thing
to do next right, you are also learning linear
algebra simultaneously right. So, you have
all heard of vector spaces right.
So, vector spaces is structure with has which
has these two operations. So, what is the
next obvious question that we should ask that,
whether these is a vector space and what is
the answer right because see after all what
is the operation here. If you take two any
way it is a sum of two functions operation.
So, we can say operation is co-ordinate wise
x plus y nth entry of x plus y is x n plus
y n right. So, what is the zero element, constant
sequence 0 right. So, it is this operation
of x plus y, it is associative commutative
and simply also you can also verify all other
axioms that alpha times x plus y is alpha
x plus alpha etcetera all right. So, I will
simply say that l 1 is a vector space.
Well that is the thing right, but there is
something more about this. Now, because this
I could have said even if I did not use this
for absolutely convergent. Suppose I taken
the sequences only of convergent sequences
still I could have done this. I could have
converted that into vector space, that is
also done. Now I shall make use of this fact
because of this fact what I can now is that
the series sigma mod x n is a convergent series.
So, what I will do is that i shall call that,
I shall give some notation for that sum I
shall call it norm of x this is used it is
called norm of x it is nothing but sigma n
going from 1 to infinity, mod x n. We know
that once x is in l 1, once x is in l 1 this
is a real number that a it is absolutely convergent.
So, this is defined. So, norm of x is defined.
So, norm now it means it is a function from
l 1 to R it is a function from l 1 to R right.
Now, let ask some very obvious questions,
what properties does this function have.
So, we have function norm, which goes from
l 1 to R right. Of course, we are using the
notation in a slightly different manner since
if norm is a function from l 1 to R I should
have denoted norm of x by this right. Something
like f of x right, but any way this is customary
to denote bring this x here all right. What
are the properties let me just say first property.
For example, can you say that this norm of
x is bigger naught equal to 0 for all x in
l 1, norm of x is bigger naught equals to
0 for all x in l 1 all right.
What is the norm of the 0 element 0 all right.
Suppose norm of some element is 0, then what
can you say if sigma mod x n is 0. Then all
of this x n 's must be 0. So, can we say this
that norm of x is equal to 0 if and only if
x is equal to 0 norm of x is equal to 0 if
and only if x is equal to 0 all right. Second
property I want to say something about norm
of x plus y norm of x, suppose you take 2
elements x and y in l 1. I want to know how
is norm of x plus y related to norm x and
norm y right. That is something we saw just
now right. We can say that for example, what
we are asking is this, what is the relationship
between norm of x y is nothing but sigma mod
x n plus y n right.
That is norm of x plus y and what is no R
m x it is sigma mod x n and what is mod y
it is sigma mod y n. Now, are these 3 numbers
related, it is clear because mod of x n plus
y n is less not equals to mod x n. So, this
is less not equal to this right. So, we can
say that this is less not equal to norm of
x plus norm of y, for all x and y in l 1 ok.
Lastly I want to say that norm of alpha times
x norm of alpha that is alpha. Suppose, alpha
is an real number, now alpha times x means
sigma mod alpha x n. Is it clear that this
should be same as mod alpha times norm x mod
alpha times norm x.
This should be true for every x in l 1 and
alpha in R right. Now, what can you say about
these three properties, have you seen something
similar earlier. It is these are properties
very similar to the absolute value function
on the real line, on the real line we have
defined the function modulus of real number.
When we listed a properties of that function
those properties were very similar right.
All these things for example, say suppose
x were a real number all these things are
true, if x and y are real number.
So, this is a function which basically has
a properties which very similar to the properties
what is called absolute value of real number.
Now, such functions can be defined on several
vector spaces. When it can be done that function
is called a norm. The corresponding vector
space is called a normed vector space or norm
linear space ok.
So, let us just make a formal definition as
follows that is suppose v is a real vector
space 
and a function, let us say norm going from
v to R is called a norm 
is called a norm on v, if it satisfies these
three properties all right, if you want we
will write once again. If first is norm of
x is bigger not equals to 0, for every x in
v and norm of x is equal to 0 if and only
if x is equal to 0 all right. Second property
is norm of x plus y is less nor equal to norm
of x plus norm of y, for every x y in v. Third
property is norm of alpha times x is same
as mod alpha times norm of x for every x in
v and alpha in R right.
What I said last is that a norm linear space,
a norm 
linear space is a pair v norm is a pair v
norm. So, this is a term that where called
norm linear space or we can also call norm
vector space, where v is a vector space and
this is a norm on v. So, is a pair where v
is vector space and norm is a norm on v all
right. Why we say that why we talk in terms
of this pairs. So, it is possible that on
the same vector space there may exist several
functions satisfying this right. On the same
vector space you may be able to define different
we will see examples of these kind of things
little later.
So, as a vector space those two objects will
be same, but as norm linear spaces those two
objects will be different. So, we with sum,
let us say norm 1 and we with norm 2 as vector
space is no other those are the same, but
as norm linear spaces is those are different.
So, that is why we usually talk about a pair
of course, again as is the practice where
it is clear from the discussion what is the
norm that we are talking about. Then we will
simply say that v is a norm linear space also
there is one.
Obvious question here why we are taking real
vector space. What can we not take vector
space on some other field of course, we can
also take vector space on complex numbers,
you can take complex vector space. Then see
as far as these first two are concerned there
is it has no reference to the scalar. Only
this last axiom, that refers to the scalars
and that will change this will become for
every x in v and alpha in c every x in v and
alpha in c these two. That will be call complex
vectors space and.
So, corresponding it will be a complex norm
linear space and this is what we can call
real norm linear space right. Since there
is not much difference as far as the definition
is concerned we shall not bother too much
about this. Now, this is an example of a real
vector space. Now, instead of taking the sequences
from n to R, suppose I have taken sequences
from n to c, then also you can define absolute
convergence. All that in the usual way that
would become an example of a complex vector
space all right. Now, the next question why
exactly we are discussing all these things.
What is the idea of discussing this norm linear
spaces. To understand that let us again look
at our definition of the convergence of a
sequence.
How did we define the convergence of a sequence.
Suppose x n is a sequence in R, suppose we
take a sequence in R, then when did we say
x n converges to x we say that x n converges
to x, x n converges to x. If you remember
what will this we said that this bend for
every epsilon bigger than 0 there exists n
0 in n, such that n bigger nor equal that
is, if n is bigger than or equal to n 0 n
bigger n 0 implies mod x n minus x less than
epsilon. In other words the concept of convergence
of a sequence depends on this function mod,
x n minus x.
We proved several theorems about the convergent
sequences about the real numbers, using this
properties of this absolute value function
and of course, some theorems using the all
the completeness of the real numbers etcetera,
but you can say that sequence can be defined
on any set ok. After all what is a sequence,
sequence is a function whose domain is the
set of all natural numbers, core domain can
be anything. So, instead of take considering
sequence of real numbers I can consider the
sequence of any objects right. Sequence of
say elements in R 2 or R n, sequence of vectors,
sequence of matrices sequence of functions.
Suppose I want to ask the question, how do
we define what is meant by such a sequence
converges. Suppose you are given sequence
of matrices ok. Suppose that sequence is a
n each a n is a matrix of some fix order,
let us say 3 by 3 and I want to say that this
sequence an converges to a what is the meaning
of that or whole this one define. We can say
that, if we had some notation is like this
norm hundred. Then I put on simply imitate
this will simply change to norm of x n minus
x.
So, instead of sequences in a real line sequences,
in a real line I can take sequence in any
norm linear space and define what is meant
by the sequence converges in that norm radius
space right or more generally. This is one
idea, that is the reason for discussing norm
radius spaces. More generally see by mod x
n minus x is the thing, but a distance between
these two numbers x and x n x and x n and.
So, if you remember what we had said all the
time saying that sequence converges means
distance between x n and x becomes small as
n becomes large that was the idea.
So, one can say similarly, that if we have
a concept of distance in any set. Suppose
we have the concept of distance, then we can
talk about the convergence of a sequence in
that set. All that we need is the concept
of distance right. Similarly, for example,
other concept of limits continuity etcetera
all those concepts depend in some sense. either
on this like absolute value function or on
the concept of distance. So, we can also develop
all those concepts in more general sets like
that. What is the advantage, see now we have
proved let us say some theorems about convergent
sequences.
For example, we have proved that every convergent
sequences cauchy or every convergent sequences
bounded we proved all these things for the
sequences of real numbers. Let us say some
time letter we talk of sequences in R 2 or
R 3 or R n or sequences of matrices or sequences
of functions. Then again we can define what
is meant by convergence and again we may have
to again separately prove, that every convergent
sequences is cauchy or every convergent sequences
is bounded and things like that. That means
essentially you will be repeating the same
proof again in various different context.
What is the way to avoid that instead of avoid
this repetition, that is the method is what
is called abstraction and very commonly used
in mathematics. You may heard this word at
mathematics is a very extract subject. People
use it in a some sort of a negative way that
mathematics is an extract subject, but extract
subject is a very powerful tool. It is used
in all sciences and as I said because of this
abstraction, we can avoid these repetitions.
It saves lot of time and energy and it is
more efficient way of doing this. So, what
we do is that we see for example, what you
have done, we have this long linear space
is an abstraction ok. Abstraction of what
a real line and that l 1.
So, many spaces whatever common to all those
spaces those properties we have taken and
defined that as a norm linear space. So, similarly
we will do about a distance and then follow
the idea. Then after that we shall just develop
all the theory in those particular either
in norm linear spaces or those new x objects
likewise let me just tell what this new objects
are called those are called metric spaces.
Once we develop in metric spaces it can applied
to any different any of this other specific
examples, R r 2 R n l 1 and all those things
ok.
Now, let us come to this what is what is a
metric space or what is a metric. This is
something more general than norm linear spaces.
Here what you have seen we shall subsequently
show that every norm linear space is also
metric space, but before that yeah which is
basically same as saying that metric space
is a more general concept because norm is
defined only on a metric space. Starting point
has to be a vector space, we the starting
point has to be a vector space whereas, metric
10 can be defined on any set. So, we take
x as a x as any non empty set.
X as any non empty set then what is a metric
it is nothing but a function which says something.
Suppose we take two points x and y, in that
it says what is the distance between those
two points. So, that function which satisfies
the property which we normally associate with
the distance between the two functions. Whatever
we commonly associate some of those properties
are taken and those are taken as a definition
of a distance or definition of metric space.
So, obviously we talk about the distance between
the two points. So, it means it is a function
from the pair of points, it will associate
some real numbers to appear of points. So,
we will say that a function, it is a function
d from x cross x 2 R, function d from x cross
x 2 R is called a metric. If it satisfies
some properties.
What are those properties, those properties
are again very similar to this. First property
is that suppose you take two points. If you
take distance between it is called a metric
and metric is a must distance this is just
a different word met. So, d x y distance that
is a distance between in fact strictly speaking,
I should write one more bracket here because
it is d of this some element in x cross x,
that element is x comma y. So, strictly speaking
I should use this definition, but I will just
what we will understand we mean is this.
So, d this is let me just remove this, just
for the convenience. So, d x y this is bigger
nor equal to 0 for every x y in x. That is
what we normally expected the distance between
any two function in non negative number and
it should be 0 only when or it should if I
that x and x distance between the points should
be 0 and the distance between two points are
0, those two points must coincide. So, which
is same as saying this and d x y is equal
to 0, if and only if x is equal to y ok. Then
second property is that distance between x
and y this should be same as distance between
y and x. It should not matter whether I call
distance from x to y or from y to x, that
should be the same. This should be true for
every x y in x.
Lastly whether this property has a name in
fact it is an obvious name this is called
symmetry. So, we explain this by saying that
distance is a symmetric function. This property
is called symmetry. Then last property suppose
we take 3 points x y and z, then we want to
compare the distance between x and z and the
distance between x and y and distance between
y and z, suppose we take 3 points. You imagine
that those 3 points form a triangle, then
the distance between x and z is the length
of one side and distance between x y is are
the other two sides. So, what we should expect
is that, this should be less nor equal to
that this should be less nor equal to that.
So, this is true for every x y z in x and
because of the comma is which might. Just
now this last property is called triangle
inequality. This last property is called triangle
inequality. By the way similarly, in this
definition of a norm this property 2 is also
called triangle inequality, this is also called
triangle inequality. We will there is a reason
for this a little later.
So, that is about a metric and. So, what is
a metric space, again in a similar way metric
space, is a pair x d where x is a non empty
set and d is a metric defined on it. So, let
us just recall it once again.
So, metric space 
is a pair x d where x is a non empty set.
Here also I should have said is called a metric
on x, this is called metric on x. So, coming
back to this a metric space is a pair x d,
where x is a non empty set and d is a metric
on x. 
Again why pair again because when one can
define several metric on the same set x. So,
for example, I can define say d 1 as 1 metric
d 2 as the arbitrary d 3 as. So, the set underline
set with the same, but a metrics may be different.
So, in this case those become different metric
spaces. Now, you can see that all these actions
which we have written here or the properties
which you have. So, said with the usual concept
of distance and those are the ones which are
taken for defining the distance. Now, you
may ask there are some many other things also
which we associate with a distance. For example,
we also know that given two points we can
talk of something like a midpoint of the two.
Then that has not come here in the actions,
but again which of the property is to be chosen
for making definition.
That is a matter of convenience and also matter
of history because this definitions like this
arise after several years of efforts from
various mathematicians by trying various axioms.
Which were better etcetera and ultimately
it will decided which exactly the things that
go into the definition. So, let us not into
that end of history right. Now, let us see
some examples of the metric spaces. So, in
example is what it should be some non empty
set and function defined like this. Usually
ah with given function like this to check
whether it found a metric or not. Usually
these two properties are very easy to check.
In fact by trivial and if at all anything
takes some time it is this last property triangle
inequality. There is one very famous example
where this a metric which you can define on
any set. Suppose x is any non empty set and
suppose you define set d x y is equal to 0
if x equal to y and one if x not equal to
one ok. It is easy to see that it is only
this last property will take some time to
check as I said other two properties are trivial.
So, this is also a well known metric, it is
called a discrete metric. The space is called
as a discrete metric space it is called discrete
metric space.
The main use of this discrete metric space
is basically for understanding. It is not
much of practical importance you do not come
across discrete metric in any applications,
but in order to understand the various concepts
in metric spaces and to check whether you
have understood or not, this example is very
useful all right. Then the next obvious example
is that of a real line, you can take the real
line and define d x y as mod x minus y distance
between x. That is the usual distance between
the two real numbers. Again it is easy to
see that that satisfies all this three properties
ok.
Now, let us come back to this, will just finish
this. So, we have seen that this function
norm is nothing but the generalization of
the function of the absolute value. So, we
can use this idea in any norm linear space.
Suppose I take this instead of taking x and
y as two real numbers. Suppose I take x and
y as 2 elements in a vector space and define
distance between x and y as norm of x minus
y. Then that should also satisfy all this
properties because those are basically followed
from the properties of the absolute value.
Let us just quickly see how this happens and
then we will stop with this.
Now, suppose let us say v is a norm linear
space and take any 2 x and y in the. Define
d x y s norm of x minus y right. Then we will
just quickly verify these properties one by
one, but in the first thing that we require.
The distance between x and y should be bigger
nor equal to 0 right. Is it true norm of and
this should be 0, if and only x equal to is
does the norm true that follows from this
pattern norm of x minus y will be 0, if and
only x minus which is same as x equal to y
all right. What about this distance between
x y is equal to distance between y x.
So, distance between y x will be norm y minus
x right. By this definition are these two
things same y, what it follows from what you,
it is nothing but minus 1 time is this. That
follows from this last property, if you take
alpha is equal to minus that is the basically
form norm of minus x is same as norm of x
for every x. So, this symmetry follows from
this property 3 right. What about the triangle
inequality, this is norm of x minus z this
is norm of x minus y and that is norm of y
minus z. Is that true that norm of x minus
z is less nor equal to norm of x minus y plus
norm of y minus z. Again you see you can first
for example, suppose you take a as x minus
z, b as x minus y and c as y minus z, then
you can say that norm of a plus b is less
nor equal to norm a plus norm b. It will give
this.
So, this property 2 implies this triangle
inequality here and that is why that is also
called triangle inequality right. So, what
it what follows from it is that, every norm
linear space can be made into a metric space.
Every norm can will lead to a metric on that
factor space and. So, this is big class of
examples of metric spaces and that is what.
Most important in applications most metric
spaces which are important from the point
of few applications are basically norm linear
spaces. I think let us stop with that, we
shall see more examples of this in the next
class.
