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so let us begin with this is a quick review
of axioms for norm
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so given the vector
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space x and of course the field f we wrote
three axioms for defining the norm on this
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so this function called norm was a function
that was from x to r plus r plus is real positive
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real numbers including zero so r plus a set
of positive real numbers including zero so
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this is a function from an element in x which
is the compact way of writing for element
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in x to r plus and we said the three axioms
one is that norm x is greater than 0 for all
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x that belong to x and x not equal to 0 vector
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and norm x equal to 0 if and only if x equal
to 0 vector so this was the first axiom the
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second axiom was norm of alpha into x is norm
x mod alpha or absolute value of alpha into
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norm x where alpha is a scalar belonging to
field f and a third axiom is triangle in equality
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so this states that distance of x plus y take
any two elements from vector space x distance
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of the vector x plus y is always less than
or equal to or the length of vector norm of
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vector of x plus y is always less than or
equal to norm x plus norm y
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this is generalization of triangle equality
that you know for one dimension or in three
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dimensions for triangles generalized to any
other space so we said any function that satisfies
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these criteria it should be a real positive
function it should give you a real number
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it should be a nonzero real number or when
x is not 0 it should be 0 real when x is not
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equal to 0 vector and so on and then we saw
couple of examples that are functions that
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can be classified as a norm or that cannot
be classified as a norm
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so both are important because you understand
something better when you see where these
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one of these axiom fails so there are multiple
ways of defining norms not a unique way a
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pair of a vector space together with or a
linear space together with a definition of
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norm gives you a norm vector space so that
is take home message well why did we do this
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i yesterday said that we are doing all this
because you know want to talk about point
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about limits and sequences so why do i need
to talk about limits when we work in numerical
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methods we are forced to look at sequences
of vector i just give you a very brief example
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we will actually do this much more in detail
later let see i want to solve this equation
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these two are coupled equations and i want
to solve them simultaneously i want to solve
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them simultaneously these kinds of problems
i am writing it in an abstract form very often
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we encounter these kind of problems well i
am going to write this in abstract way as
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f function 1 xy equal to 0 and function 2
xy equal to 0 there are two functions f1 xy
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equal to 0 f2 xy equal to 0 and this kind
of equations arise steady state of a cstr
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concentration and temperature are linked
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so first equation could be energy balance
second could be material balance and then
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you get two equations into one concentration
temperature let us solve them for i am going
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to define a vector this is my function vector
i am going to call this as f of x of well
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let me call some new variable equal to 0 so
my eta is a vector which comprises of x and
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y and then i want to solve for f eta equal
to 0 vector
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this is my 0 vector i am just writing the
same thing in a different format now what
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method you know for solving this how do you
solve this “professor - student conversation
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starts” bijection method pardon me bjection
method bijection method is for difficult to
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scale to two variables one variable well defined
bijection method is there “professor - student
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conversation ends” you can have bisection
method for two variables but well let us take
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a very simple iterative scheme
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let us construct a very simple iterative scheme
i will write eta plus f eta i will add this
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vector eta on both sides and then i construct
an iteration whether it will convergent or
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it is a different story but i will construct
an iterative process so i will start with
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some gas vector eta 0 that is let say well
i do not know what the solution is so i am
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going to guess some solution so let say i
start with say minus 1 and 1
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this is x this is y and then what i want to
do is to say that eta k plus 1 equal to eta
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k plus f of eta k is it okay i have just formulated
an iteration scheme in which i start with
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vector 0 i take the 0 vector substitute here
i will get vector 1 i take vector 1 substitute
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here i will get vector 2 “professor - student
conversation starts” how do i know whether
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this sequence of vectors is converging to
something pardon me the difference between
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eta k plus 1 what is difference
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see it is a two dimensional vector now i will
just further convenience it into two dimensional
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vector i could have done this in n dimensions
i could have written this in an n dimensions
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n equations in n unknowns very very common
chemical engineering starting trying to solve
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steady state energy material balance for a
plant you can get 1000 equations and 1000
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unknowns okay the difference vector pardon
me the difference vector but what of difference
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vector norm of the difference vector so we
have to talk about a vector converging to
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another vector a vector converging to a solution
what should happen at the solution
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let us say if x star is a solution eta star
is a solution
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f eta equal to 0 what she say is correct that
one thing is that you know should be equal
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to 0 of course at the solution so at eta start
so that is f eta star equal to 0 fine but
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i am starting an iterative process so what
i am going to get is i am going to get this
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vector sequence x eta 0 eta 1 2 and so on
and i am going to get this vector the question
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is is the sequence is the sequence converging
to eta star
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does this goes to eta star that is the question
i need to answer see this is the solution
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if i had pluck eta k here it is not going
to be equal to zero it is not going to be
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equal to zero so it is going to some other
small number probably is it small so how do
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you answer this question in general n dimensional
spaces or function spaces that is where we
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need to now talk about i may have scenario
where i have a sequence of functions i have
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sequence of function and i will give an example
i am going to show you a small demo also sequence
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of functions
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“professor - student conversation ends”
so the question is is this sequence convergent
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so this kind of problems are always encountered
in numerical analysis because almost every
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method that you have for solving you know
most of the problems through computing is
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iterative you start with the guess and you
come up with a new guess and so on so there
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is this need to look at convergence of sequences
so we are going to define two notions one
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is cauchy sequence
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so
i am taking a set of infinite set of sequences
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or infinite set of vectors which are generated
by some process you know it could be some
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iterative scheme by which you are working
or whatever it is now i want to know how do
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i formally define convergence a
sequence of vectors is said to cauchy if difference
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between xn minus that is nth element in the
sequence and m filament in the sequence if
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this tends to 0 norm of this tends to zero
as m and n become infinitive so more and more
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elements are generating this the vectors come
closer and closer
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well in one dimensional vector space so in
one dimensional vector space that is a set
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of real numbers well when a sequence is cauchy
it convergence to a limit inside a set but
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depends upon the space funny things can happen
if the space is not complete what is this
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business of completeness we will come to that
soon before that let me define convergence
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sequence so there are two different notions
one is cauchy sequence other is convergence
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sequence these just for the sake of nice mathematics
these are very very relevant to computing
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what is the convergence sequence
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so i a m considering this sequence again in
fact this is a short hand notation for sequence
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i am not going to write every time k going
from 0 to infinity or k going from 0 to n
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whatever it is curly braces x superscript
k is a sequence in a norm linear space or
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a norm vector space now this is said to be
convergent to a vector x star if this is said
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to be convergent to an element x star if difference
between x star and x k goes to zero difference
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between x star and x k goes to 0 as k goes
to infinity
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so what i want to show you is that it is not
obvious that a cauchy sequence will always
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be convergent it depends upon the space that
you are considering a convergence sequence
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is always a cauchy sequence but vice versa
is not necessarily true a cauchy sequence
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may not be convergent a convergent sequence
is always a cauchy sequence
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now examples will make it clear why i am talking
of this funny things and we will also realize
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that this is something that you deal with
every day when you use computers so i am going
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to take a example of a vector space in which
a cauchy sequence is not convergent i am going
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to take a example of a vector space in which
a cauchy sequence is not convergent
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so basically i want to give an example of
this idea that convergence to a particular
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element is something different when it depends
upon the space my first example here is my
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space x is my first example here is a set
of rational numbers q and i am taking field
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f also to be q i am taking a field also to
be q so this combination will form a vector
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space and i can find very easily a sequence
in this vector space which is cauchy but not
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convergent a simple example is now consider
sequence
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(refer time 16:51)
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whether i start index with 0 or 1 it does
not matter i am starting with 1 x2 is 1 x
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1 plus 1 by 2 factorial and so on so my nth
element in this sequence is 1 by 1 plus 1
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by 2 factorial plus 1 by 3 factorial i think
this is a well known series where does it
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converse to e but e is it a cauchy sequence
it is known to be a cauchy sequence it is
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a convergence sequence is real line on real
line where does it converse to
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so this sequence xn this converges to element
e as n tends to infinity we know that this
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particular element tends to e but e is not
a rational number so this element where it
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converges to is outside this space so you
have funny situation you have a cauchy sequence
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if you apply the definition of cauchy sequence
if you take any two elements as n and m goes
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to infinity you take difference it goes to
zero that is very easy to show look at any
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book on real analysis
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you will see this proof it is just one or
two pages of proof that this is a cauchy sequence
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but in this particular space it does not converge
it does not converge and in this space i can
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find many such sequences i can find a sequence
that is almost converging to pi but pi is
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irrational number pi is not there inside this
space
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so likewise you know i have this sequence
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so this sequence that is 3 by 1 11 by 3 41
by 11 and so on it converges to not a rational
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number i can find infinite such examples where
you have a convergence sequence you have a
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cauchy sequence but not converging to an element
inside this particular space “professor
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- student conversation starts” those are
rational numbers so this sequence is converging
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somewhere but it is not converging inside
this space it will never converge inside the
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space
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so e does not belong to set of rational numbers
that is what you are saying we know that in
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a real line this will converge to e see e
is not the why these are all rational number
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sir we individually x is being as 1 by 1 factorial
plus 1 by 2 factorial plus 1 by 3 factorial
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so you can always define one common denominator
it is a rational number if it is just 1 by
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1 factorial it is a rational number
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no no all these are rational numbers i think
we can talk about it little later this particular
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thing these are all rational numbers they
are not irrational numbers so you mean to
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say that 1 by 3 may not be expressible but
it is a summation of rational number rational
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number is whether you can write it as integer
upon integer i can always write integer upon
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integer whether you can express it as a continued
fraction
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we are not looking at that problem right now
the true representation is integer upon integer
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i can have a common denominator for this it
becomes a rational number you are confusing
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between its representations in this computer
i am coming to that so do not confuse between
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the two so do not confuse one third with 0.33
do not confuse that with 0.33 “professor
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- student conversation ends”
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if this is true about q it is also true about
qn i can define a product space which is qn
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n dimensional space my x can be qn i can take
a space which is where do you get qn when
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i am doing computing in a computer i can deal
only with finite dimensional vectors i can
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only leave with finite dimensional vectors
and in computer you cannot represent many
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of these you know irrational numbers because
computer has a finite precession
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if i take 64 beat precession the resulting
number which you approximate as e actually
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will be a rational number something divided
by i have to truncate right i cannot have
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a representation do you understand what i
am saying in a computer whatever is the precession
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64 bit you know 128 bit you go to very high
precession computer any number is actually
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represented as you know using binary 1 0 1
0 1 0 sequence and there is finite number
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of bits used to represent the number
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so that number will always be representable
as a rational number something divided by
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something i truncate it so the point which
i want to make is that incomplete spaces are
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not so you know when you work with computer
you are working with incomplete spaces and
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we have to bother we have cauchy sequence
which does not converge cauchy sequence this
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does not converge in a computer i will have
a cauchy sequence which does not converge
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to a number
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no it is true value see for all practical
purposes we say that well this is almost close
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to e but it is not e we take an approximation
of pi may be you know correct up to 1000 decimals
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but it is not pi okay so we are working with
this incomplete spaces and then let me give
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you one more example and i want to show a
demonstration here of an incomplete space
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so my second example is set of continuous
functions over minus infinity set of continuous
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functions over minus infinity to infinity
this is my second example and i am going to
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construct a sequence in this particular vector
space and what i want to demonstrate is that
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this sequence will converge to a discontinuous
function i have a sequence of continuous functions
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converging to a discontinuous function so
you are trying to solve some partial differential
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00:24:51,700 --> 00:24:52,700
equation
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or some problem you construct the solution
as a sequence of continuous functions or continuously
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differentiable functions the sequence might
converge to a nondifferentiable noncontinuous
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function so you can have funny situations
so my sequence here is this 1 by 2 plus my
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sequence here is a sequence of functions these
are continuous functions defined over interval
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minus infinity to plus infinity this is a
function sequence define so t goes from minus
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infinity to plus infinity my k changes k would
be 1 2 3 4 5 i will get different functions
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for each value of k so i will get k goes from
1 2 and so on k goes from
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function sequence i just want to animate and
show you what is happening so this is for
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k equal to 1 this is for k equal to 6 i am
going to increment by 5 and see what is happening
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this is k equal to 11 16 and so on i just
go on right i am going closer and closer towards
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this step kind of a function i am going closer
and closer to the step function
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so if you do this i have gone only up to 100
if i
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do this by incrementing k much much longer
much to a larger value this will converge
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to a step function
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so moral of the story is that i am starting
with a set of continuous functions i am generating
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a sequence in this set but this sequence does
not converge to element in the set the sequence
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does not converge to an element in the set
so there is a problem so if what is nice about
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real line that every real line every sequence
which is cauchy will converge to an element
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inside them ever cauchy sequence on the real
line will converge to a number on the real
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line so in some sense real line is a complete
set
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there is nothing outside it whereas set of
all rational numbers is incomplete there is
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something outside and the sequences here seem
to converge to something which is outside
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the space seem to converge or something which
is outside the space so what is nice about
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real line its complete space what is nice
about because real line is the complete space
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same thing is sure about r2 two dimensional
vector space
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any sequence in two dimensional vector space
will converge to the point in two dimensions
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any sequence in you n dimensional real rn
will converge to element in rn but in qn there
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are holes you know so where the sequence be
cauchy but it will not converge so this spaces
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you know in which all sequences converge within
the space are called as complete vector spaces
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and these are special vector spaces
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so there is something different about the
spaces in which so we move back to the black
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board so we want this nice property to hold
even in the vector spaces so we call this
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vector spaces which have the special property
as complete vector spaces or they are named
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after a famous mathematicians banach who actually
founded this one of the founders of functional
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analysis
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so what is banach space so every cauchy sequence
to converge to an element is space
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this word here every is important every cauchy
sequence if i can find one sequence which
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does not converge the space is not a banach
space every cauchy sequence should converge
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so the real line or rn or equivalently if
you take complex numbers cn they have some
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very nice property they are all complete spaces
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function spaces need not be complete spaces
set of continuous function we saw is not a
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complete space well in functional analysis
you talk about completion of an incomplete
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space you add all the elements and then create
a new space which is complete and so on but
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we do not want to go into those details right
now i just wanted to sensitize you about the
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fact that even in a computer we are working
with incomplete vector spaces and then you
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can get into funny situations in advance computing
because of this incomplete behaviour well
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so far so good we talk about we started generalizing
notions from three dimensions do not forget
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that we talked about a vector and then we
said there are essential properties of a set
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which the two essential properties vector
addition and scalar multiplications so these
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two things hold in a set then or if a set
is closed under vector addition and scalar
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multiplication we call it a vector space any
set so we freed ourselves from the notion
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of vector space which is just three dimensional
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we can now talk about set of continuous functions
set of continuously differentiable functions
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set of twice differentiable three differentiable
and you can so now how many such spaces are
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there infinite spaces then we said well we
now that is not enough to have just generalizing
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of vector space we also need notion of length
so we talk about norm right we talked about
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norm
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norm was in some saying generalization of
notion of magnitude of a vector and we said
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there are so many ways of defining norms and
a pair of a vector space and a norm defined
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on it will give you a normed vector space
or norm linear space so this up to here fine
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now we need something more i need angle one
of the primary thing that you use in three
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dimensions one of the most fundamental result
in our school geometry or in three dimensional
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geometry pythagoras theorem and i need pythagoras
theorem in all these spaces what i am going
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to do
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i need pythagoras theorem so i need orthogonality
i need perpendicularity one of the most important
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concepts that you use in applied mathematics
in modelling in physics in chemistry and every
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where orthogonality is very very quantum chemistry
chemistry in the sense you might wonder where
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in chemistry so orthogonality is very very
important and we need to generalize the notion
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of orthogonality and that is where we will
start looking at in a product spaces
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we will start looking at inner product spaces
so here the attempt is to generalize the concept
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of dot product “professor - student conversation
starts” how do you define angle in three
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dimensions well if i am given any two vectors
say x and y which belongs to r3 how do i find
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the angle between them so what i do is i find
out excap which is a unit vector in this direction
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normally i take a two norm here well why two
norm we will come to that why not one norm
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so this is something special about this two
norm and why cap equal to and then we say
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that dot product that is x cap cos theta angle
between these two vectors is just x cap transpose
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y cap this is the fundamental way by which
we define angle between any two vectors in
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three dimensions now can i come up with something
that we will generalize notion of angle in
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three dimensions
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when do you say two vectors are perpendicular
in three dimensions dot product when dot product
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is 0 cos theta is 0 two vectors are perpendicular
so i am going to peg on to these ideas well
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that dot product between unit vectors is used
to define angle when dot product is 0 you
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call two vectors to be orthogonal and come
up with a generalization in the product spaces
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of concepts of angle orthogonality and once
the orthogonality you have pythagoras theorem
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i can talk about pythagoras theorem in any
n dimensional infinite dimensional space of
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course it has to qualify certain properties
what are those properties those are the properties
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of inner product space so now we have to start
questioning what is characteristics of an
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inner product see we had three properties
of magnitude what were the three properties
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of magnitude
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magnitude is always nonnegative for a nonzero
vector and zero for a zero vector alpha times
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you get you know mod alpha gets multiplied
to the norm and triangular equality likewise
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what are the essential properties of inner
product in this which can be used to generalize
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in any other vector space those vector spaces
are going to be called as inner product spaces
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because we are going to define a norm vector
space in its additional structure is put called
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inner product
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“professor - student conversation ends”
all these spaces which are describing till
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now we did not talk about inner product so
now i am going to introduce something new
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which is the inner product space which will
have definition of inner product what you
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release there are umpteen number of ways to
defining the product and so the way of defining
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generalizing orthogonality is not unique and
so we will see from our next lecture