so let us begin with this is a quick review
of axioms for norm
so given the vector 
space x and of course the field f we wrote
three axioms for defining the norm on this
so this function called norm was a function
that was from x to r plus r plus is real positive
real numbers including zero so r plus a set
of positive real numbers including zero so
this is a function from an element in x which
is the compact way of writing for element
in x to r plus and we said the three axioms
one is that norm x is greater than 0 for all
x that belong to x and x not equal to 0 vector
and norm x equal to 0 if and only if x equal
to 0 vector so this was the first axiom the
second axiom was norm of alpha into x is norm
x mod alpha or absolute value of alpha into
norm x where alpha is a scalar belonging to
field f and a third axiom is triangle in equality
so this states that distance of x plus y take
any two elements from vector space x distance
of the vector x plus y is always less than
or equal to or the length of vector norm of
vector of x plus y is always less than or
equal to norm x plus norm y
this is generalization of triangle equality
that you know for one dimension or in three
dimensions for triangles generalized to any
other space so we said any function that satisfies
these criteria it should be a real positive
function it should give you a real number
it should be a nonzero real number or when
x is not 0 it should be 0 real when x is not
equal to 0 vector and so on and then we saw
couple of examples that are functions that
can be classified as a norm or that cannot
be classified as a norm
so both are important because you understand
something better when you see where these
one of these axiom fails so there are multiple
ways of defining norms not a unique way a
pair of a vector space together with or a
linear space together with a definition of
norm gives you a norm vector space so that
is take home message well why did we do this
i yesterday said that we are doing all this
because you know want to talk about point
about limits and sequences so why do i need
to talk about limits when we work in numerical
methods we are forced to look at sequences
of vector i just give you a very brief example
we will actually do this much more in detail
later let see i want to solve this equation
these two are coupled equations and i want
to solve them simultaneously i want to solve
them simultaneously these kinds of problems
i am writing it in an abstract form very often
we encounter these kind of problems well i
am going to write this in abstract way as
f function 1 xy equal to 0 and function 2
xy equal to 0 there are two functions f1 xy
equal to 0 f2 xy equal to 0 and this kind
of equations arise steady state of a cstr
concentration and temperature are linked
so first equation could be energy balance
second could be material balance and then
you get two equations into one concentration
temperature let us solve them for i am going
to define a vector this is my function vector
i am going to call this as f of x of well
let me call some new variable equal to 0 so
my eta is a vector which comprises of x and
y and then i want to solve for f eta equal
to 0 vector
this is my 0 vector i am just writing the
same thing in a different format now what
method you know for solving this how do you
solve this “professor - student conversation
starts” bijection method pardon me bjection
method bijection method is for difficult to
scale to two variables one variable well defined
bijection method is there “professor - student
conversation ends” you can have bisection
method for two variables but well let us take
a very simple iterative scheme
let us construct a very simple iterative scheme
i will write eta plus f eta i will add this
vector eta on both sides and then i construct
an iteration whether it will convergent or
it is a different story but i will construct
an iterative process so i will start with
some gas vector eta 0 that is let say well
i do not know what the solution is so i am
going to guess some solution so let say i
start with say minus 1 and 1
this is x this is y and then what i want to
do is to say that eta k plus 1 equal to eta
k plus f of eta k is it okay i have just formulated
an iteration scheme in which i start with
vector 0 i take the 0 vector substitute here
i will get vector 1 i take vector 1 substitute
here i will get vector 2 “professor - student
conversation starts” how do i know whether
this sequence of vectors is converging to
something pardon me the difference between
eta k plus 1 what is difference
see it is a two dimensional vector now i will
just further convenience it into two dimensional
vector i could have done this in n dimensions
i could have written this in an n dimensions
n equations in n unknowns very very common
chemical engineering starting trying to solve
steady state energy material balance for a
plant you can get 1000 equations and 1000
unknowns okay the difference vector pardon
me the difference vector but what of difference
vector norm of the difference vector so we
have to talk about a vector converging to
another vector a vector converging to a solution
what should happen at the solution
let us say if x star is a solution eta star
is a solution 
f eta equal to 0 what she say is correct that
one thing is that you know should be equal
to 0 of course at the solution so at eta start
so that is f eta star equal to 0 fine but
i am starting an iterative process so what
i am going to get is i am going to get this
vector sequence x eta 0 eta 1 2 and so on
and i am going to get this vector the question
is is the sequence is the sequence converging
to eta star
does this goes to eta star that is the question
i need to answer see this is the solution
if i had pluck eta k here it is not going
to be equal to zero it is not going to be
equal to zero so it is going to some other
small number probably is it small so how do
you answer this question in general n dimensional
spaces or function spaces that is where we
need to now talk about i may have scenario
where i have a sequence of functions i have
sequence of function and i will give an example
i am going to show you a small demo also sequence
of functions
“professor - student conversation ends”
so the question is is this sequence convergent
so this kind of problems are always encountered
in numerical analysis because almost every
method that you have for solving you know
most of the problems through computing is
iterative you start with the guess and you
come up with a new guess and so on so there
is this need to look at convergence of sequences
so we are going to define two notions one
is cauchy sequence
so 
i am taking a set of infinite set of sequences
or infinite set of vectors which are generated
by some process you know it could be some
iterative scheme by which you are working
or whatever it is now i want to know how do
i formally define convergence a 
sequence of vectors is said to cauchy if difference
between xn minus that is nth element in the
sequence and m filament in the sequence if
this tends to 0 norm of this tends to zero
as m and n become infinitive so more and more
elements are generating this the vectors come
closer and closer
well in one dimensional vector space so in
one dimensional vector space that is a set
of real numbers well when a sequence is cauchy
it convergence to a limit inside a set but
depends upon the space funny things can happen
if the space is not complete what is this
business of completeness we will come to that
soon before that let me define convergence
sequence so there are two different notions
one is cauchy sequence other is convergence
sequence these just for the sake of nice mathematics
these are very very relevant to computing
what is the convergence sequence
so i a m considering this sequence again in
fact this is a short hand notation for sequence
i am not going to write every time k going
from 0 to infinity or k going from 0 to n
whatever it is curly braces x superscript
k is a sequence in a norm linear space or
a norm vector space now this is said to be
convergent to a vector x star if this is said
to be convergent to an element x star if difference
between x star and x k goes to zero difference
between x star and x k goes to 0 as k goes
to infinity
so what i want to show you is that it is not
obvious that a cauchy sequence will always
be convergent it depends upon the space that
you are considering a convergence sequence
is always a cauchy sequence but vice versa
is not necessarily true a cauchy sequence
may not be convergent a convergent sequence
is always a cauchy sequence
now examples will make it clear why i am talking
of this funny things and we will also realize
that this is something that you deal with
every day when you use computers so i am going
to take a example of a vector space in which
a cauchy sequence is not convergent i am going
to take a example of a vector space in which
a cauchy sequence is not convergent
so basically i want to give an example of
this idea that convergence to a particular
element is something different when it depends
upon the space my first example here is my
space x is my first example here is a set
of rational numbers q and i am taking field
f also to be q i am taking a field also to
be q so this combination will form a vector
space and i can find very easily a sequence
in this vector space which is cauchy but not
convergent a simple example is now consider
sequence
(refer time 16:51)
whether i start index with 0 or 1 it does
not matter i am starting with 1 x2 is 1 x
1 plus 1 by 2 factorial and so on so my nth
element in this sequence is 1 by 1 plus 1
by 2 factorial plus 1 by 3 factorial i think
this is a well known series where does it
converse to e but e is it a cauchy sequence
it is known to be a cauchy sequence it is
a convergence sequence is real line on real
line where does it converse to
so this sequence xn this converges to element
e as n tends to infinity we know that this
particular element tends to e but e is not
a rational number so this element where it
converges to is outside this space so you
have funny situation you have a cauchy sequence
if you apply the definition of cauchy sequence
if you take any two elements as n and m goes
to infinity you take difference it goes to
zero that is very easy to show look at any
book on real analysis
you will see this proof it is just one or
two pages of proof that this is a cauchy sequence
but in this particular space it does not converge
it does not converge and in this space i can
find many such sequences i can find a sequence
that is almost converging to pi but pi is
irrational number pi is not there inside this
space 
so likewise you know i have this sequence
so this sequence that is 3 by 1 11 by 3 41
by 11 and so on it converges to not a rational
number i can find infinite such examples where
you have a convergence sequence you have a
cauchy sequence but not converging to an element
inside this particular space “professor
- student conversation starts” those are
rational numbers so this sequence is converging
somewhere but it is not converging inside
this space it will never converge inside the
space
so e does not belong to set of rational numbers
that is what you are saying we know that in
a real line this will converge to e see e
is not the why these are all rational number
sir we individually x is being as 1 by 1 factorial
plus 1 by 2 factorial plus 1 by 3 factorial
so you can always define one common denominator
it is a rational number if it is just 1 by
1 factorial it is a rational number
no no all these are rational numbers i think
we can talk about it little later this particular
thing these are all rational numbers they
are not irrational numbers so you mean to
say that 1 by 3 may not be expressible but
it is a summation of rational number rational
number is whether you can write it as integer
upon integer i can always write integer upon
integer whether you can express it as a continued
fraction
we are not looking at that problem right now
the true representation is integer upon integer
i can have a common denominator for this it
becomes a rational number you are confusing
between its representations in this computer
i am coming to that so do not confuse between
the two so do not confuse one third with 0.33
do not confuse that with 0.33 “professor
- student conversation ends”
if this is true about q it is also true about
qn i can define a product space which is qn
n dimensional space my x can be qn i can take
a space which is where do you get qn when
i am doing computing in a computer i can deal
only with finite dimensional vectors i can
only leave with finite dimensional vectors
and in computer you cannot represent many
of these you know irrational numbers because
computer has a finite precession
if i take 64 beat precession the resulting
number which you approximate as e actually
will be a rational number something divided
by i have to truncate right i cannot have
a representation do you understand what i
am saying in a computer whatever is the precession
64 bit you know 128 bit you go to very high
precession computer any number is actually
represented as you know using binary 1 0 1
0 1 0 sequence and there is finite number
of bits used to represent the number
so that number will always be representable
as a rational number something divided by
something i truncate it so the point which
i want to make is that incomplete spaces are
not so you know when you work with computer
you are working with incomplete spaces and
we have to bother we have cauchy sequence
which does not converge cauchy sequence this
does not converge in a computer i will have
a cauchy sequence which does not converge
to a number
no it is true value see for all practical
purposes we say that well this is almost close
to e but it is not e we take an approximation
of pi may be you know correct up to 1000 decimals
but it is not pi okay so we are working with
this incomplete spaces and then let me give
you one more example and i want to show a
demonstration here of an incomplete space
so my second example is set of continuous
functions over minus infinity set of continuous
functions over minus infinity to infinity
this is my second example and i am going to
construct a sequence in this particular vector
space and what i want to demonstrate is that
this sequence will converge to a discontinuous
function i have a sequence of continuous functions
converging to a discontinuous function so
you are trying to solve some partial differential
equation
or some problem you construct the solution
as a sequence of continuous functions or continuously
differentiable functions the sequence might
converge to a nondifferentiable noncontinuous
function so you can have funny situations
so my sequence here is this 1 by 2 plus my
sequence here is a sequence of functions these
are continuous functions defined over interval
minus infinity to plus infinity this is a
function sequence define so t goes from minus
infinity to plus infinity my k changes k would
be 1 2 3 4 5 i will get different functions
for each value of k so i will get k goes from
1 2 and so on k goes from
function sequence i just want to animate and
show you what is happening so this is for
k equal to 1 this is for k equal to 6 i am
going to increment by 5 and see what is happening
this is k equal to 11 16 and so on i just
go on right i am going closer and closer towards
this step kind of a function i am going closer
and closer to the step function 
so if you do this i have gone only up to 100
if i 
do this by incrementing k much much longer
much to a larger value this will converge
to a step function
so moral of the story is that i am starting
with a set of continuous functions i am generating
a sequence in this set but this sequence does
not converge to element in the set the sequence
does not converge to an element in the set
so there is a problem so if what is nice about
real line that every real line every sequence
which is cauchy will converge to an element
inside them ever cauchy sequence on the real
line will converge to a number on the real
line so in some sense real line is a complete
set
there is nothing outside it whereas set of
all rational numbers is incomplete there is
something outside and the sequences here seem
to converge to something which is outside
the space seem to converge or something which
is outside the space so what is nice about
real line its complete space what is nice
about because real line is the complete space
same thing is sure about r2 two dimensional
vector space
any sequence in two dimensional vector space
will converge to the point in two dimensions
any sequence in you n dimensional real rn
will converge to element in rn but in qn there
are holes you know so where the sequence be
cauchy but it will not converge so this spaces
you know in which all sequences converge within
the space are called as complete vector spaces
and these are special vector spaces
so there is something different about the
spaces in which so we move back to the black
board so we want this nice property to hold
even in the vector spaces so we call this
vector spaces which have the special property
as complete vector spaces or they are named
after a famous mathematicians banach who actually
founded this one of the founders of functional
analysis
so what is banach space so every cauchy sequence
to converge to an element is space 
this word here every is important every cauchy
sequence if i can find one sequence which
does not converge the space is not a banach
space every cauchy sequence should converge
so the real line or rn or equivalently if
you take complex numbers cn they have some
very nice property they are all complete spaces
function spaces need not be complete spaces
set of continuous function we saw is not a
complete space well in functional analysis
you talk about completion of an incomplete
space you add all the elements and then create
a new space which is complete and so on but
we do not want to go into those details right
now i just wanted to sensitize you about the
fact that even in a computer we are working
with incomplete vector spaces and then you
can get into funny situations in advance computing
because of this incomplete behaviour well
so far so good we talk about we started generalizing
notions from three dimensions do not forget
that we talked about a vector and then we
said there are essential properties of a set
which the two essential properties vector
addition and scalar multiplications so these
two things hold in a set then or if a set
is closed under vector addition and scalar
multiplication we call it a vector space any
set so we freed ourselves from the notion
of vector space which is just three dimensional
we can now talk about set of continuous functions
set of continuously differentiable functions
set of twice differentiable three differentiable
and you can so now how many such spaces are
there infinite spaces then we said well we
now that is not enough to have just generalizing
of vector space we also need notion of length
so we talk about norm right we talked about
norm
norm was in some saying generalization of
notion of magnitude of a vector and we said
there are so many ways of defining norms and
a pair of a vector space and a norm defined
on it will give you a normed vector space
or norm linear space so this up to here fine
now we need something more i need angle one
of the primary thing that you use in three
dimensions one of the most fundamental result
in our school geometry or in three dimensional
geometry pythagoras theorem and i need pythagoras
theorem in all these spaces what i am going
to do
i need pythagoras theorem so i need orthogonality
i need perpendicularity one of the most important
concepts that you use in applied mathematics
in modelling in physics in chemistry and every
where orthogonality is very very quantum chemistry
chemistry in the sense you might wonder where
in chemistry so orthogonality is very very
important and we need to generalize the notion
of orthogonality and that is where we will
start looking at in a product spaces
we will start looking at inner product spaces
so here the attempt is to generalize the concept
of dot product “professor - student conversation
starts” how do you define angle in three
dimensions well if i am given any two vectors
say x and y which belongs to r3 how do i find
the angle between them so what i do is i find
out excap which is a unit vector in this direction
normally i take a two norm here well why two
norm we will come to that why not one norm
so this is something special about this two
norm and why cap equal to and then we say
that dot product that is x cap cos theta angle
between these two vectors is just x cap transpose
y cap this is the fundamental way by which
we define angle between any two vectors in
three dimensions now can i come up with something
that we will generalize notion of angle in
three dimensions
when do you say two vectors are perpendicular
in three dimensions dot product when dot product
is 0 cos theta is 0 two vectors are perpendicular
so i am going to peg on to these ideas well
that dot product between unit vectors is used
to define angle when dot product is 0 you
call two vectors to be orthogonal and come
up with a generalization in the product spaces
of concepts of angle orthogonality and once
the orthogonality you have pythagoras theorem
i can talk about pythagoras theorem in any
n dimensional infinite dimensional space of
course it has to qualify certain properties
what are those properties those are the properties
of inner product space so now we have to start
questioning what is characteristics of an
inner product see we had three properties
of magnitude what were the three properties
of magnitude
magnitude is always nonnegative for a nonzero
vector and zero for a zero vector alpha times
you get you know mod alpha gets multiplied
to the norm and triangular equality likewise
what are the essential properties of inner
product in this which can be used to generalize
in any other vector space those vector spaces
are going to be called as inner product spaces
because we are going to define a norm vector
space in its additional structure is put called
inner product
“professor - student conversation ends”
all these spaces which are describing till
now we did not talk about inner product so
now i am going to introduce something new
which is the inner product space which will
have definition of inner product what you
release there are umpteen number of ways to
defining the product and so the way of defining
generalizing orthogonality is not unique and
so we will see from our next lecture
