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okay so in last lecture we discussed about
span of a set of vector and then we graduated
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to basis. so a linearly independent set of
vectors is called basis. a linearly independent
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set is called basis if it generates the entire
space.
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a linearly independent set s that belongs
to vector space x which generates the entire
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space that means the span of that set is=entire
space then it is called as a basis for that
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vector space. if the number of elements in
the basis is finite then we have a finite
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dimensional vector space and if this linearly
independent set is a infinite set then we
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have an infinite dimensional vector space.
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so number of linearly independent vectors
that generate the entire space that defines
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basis and if that number is finite we have
a finite dimensional vector space number is
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infinite. infinite dimensional vector space
example would be set of continuous functions
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over 0 to 1 or set of continuous functions
over 0 to 2 pi, set of continuous functions
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over –pi to pi.
all these vector spaces will encounter many
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such examples.
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but these 2, 3 examples i gave you just now
because we have seen something like this in
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some examples. we have look at a boundary
value problem in which we have to look at
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set of continuous function over 0 to 1 or
set of continuous functions over 0 to 2 pi.
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this is something which you have looked at
when you studied fourier series in your undergraduate
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or –pi to pi.
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you have studied fourier series and will be
anyway revisiting fourier series soon next
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week.
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so we have these 2 concepts. infinite dimensional
space and then we will be using finite as
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well as infinite dimensional spaces throughout
our study of numerical methods. so this is
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not something which i am just introducing
out of completion or something. this is something
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which we are going to use and you will see
when we start talking about transformation
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how this spaces come into play.
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we need to introduce one more concept which
is called as a product space. so
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a product space is composed of 2 vector spaces.
i can create a new vector space by combining
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2 vectors spaces or more vector spaces that
is called as a product space. while the simplest
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example of product space is rn where r is
a vector space line one dimensional vector
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space line and i can create r2/r cross r.
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i can create r3/r cross r cross r. and so
on. so likewise i can create a vector space
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by merging 2 or more vector spaces those are
called as product spaces. so if i take 2 vector
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spaces x and y both defined on field f. then
x cross y what i mean here is this is x cross
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y.
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this is also a vector space. so if i take
an element x belonging to vector space x particular
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element y belonging to vector space y then
this combination xy it belongs to the product
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space. as i said simplest example of this
is r square r cube rn or c c square, c cube
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cn n dimensional vector spaces are the simplest
example of this product spaces. see the way
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you can go about thinking about it is that
you start with r. r cross r is a vector space
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then r square cross r is a vector space.
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okay i get r cube. so r cube cross r is a
vector space so i get r to the power 4 and
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so on. r to the power not power 4 r 4 and
so on. so my vector cases could be compositions
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of multiple vector spaces, but this i can
extend to some other to create more complex
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spaces. see for example my x can be set of
real numbers and my y can be set of continuous
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functions over 0 to 1.
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my y can be set of continuous functions over
0 to 1. i can create a vector space which
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is x cross y which is
i can create a vector space which is r cross
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x cross y. it is possible to have a product
space which is defined like this. we will
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hit into these kinds of spaces when we start
talking about boundary value problems and
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so on.
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examples of this will come in ample later
when we study about transformations of mathematical
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problems into computable forms. so that is
where we are going to need this. so i am not
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going to give examples right now because many
examples will come a little later, but this
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is an important concept product spaces. now
having defined linear vector spaces sometimes
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they are called as linear spaces sometimes
people refer to it as linear vector spaces.
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words are used interchangeably. now here we
need to define more structures and as i said
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the first thing that comes to your mind is
length of a vector. so i need to define what
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is length of a vector when i go to set of
continuous functions over some interval. what
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is length of a vector now. so we have to systematically
define something equivalent to magnitude or
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length.
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what we define here is a concept called norm.
now you can have a vector space defined in
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mathematics in functional analysis. you can
have a vector space defined without definition
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of a norm or a length just set of objects
which satisfy certain criteria. so what i
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want to say here is that norm is an additional
structure that we are putting on the space.
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so we have normed vector spaces or normed
linearly spaces as they are often referred
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to normed vector spaces is a vector space
together with a definition of a norm. a vector
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space x, f. f here is the field associated
with
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the vector space. so norm vector space is
nothing, but a vector space together with
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a norm defined on it and the definition of
norm and the definition of vector space it
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is not that given a vector space there is
a unique way of defining norm. we will see
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what is a norm and then you will understand
why i am saying it is a pair.
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so for example in 3 dimension itself we will
define different norms and 3 dimensional space
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with 1 norm will give you 1 norm space 3 dimensional
space with another norm will give you another
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norm space. so it is not that the 3 dimension
space if you are working 3 dimensions means
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you have to use 1 particular norm nothing
like that. so a vector space together with
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a norm definition will give you a norm linear
space.
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and then it will have certain properties.
so what is this norm? so norm is a generalization
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of concept of length of a vector. so when
i say length of a vector all of you think
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probably from your undergraduate experience
in one direction. if i give you vector in
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3 dimensions with say 3 coordinates x, y,
z then x square+ y square+ z square whole
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raise to ½. this is what we know as norm
of the vector or length of a vector.
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we do not know any other way of looking at
it. so a vector space together with a definition
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of norm defined on it forms a norm vector
space.
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so what is a norm? norm is a real valued function
in fact it is a function that is defined from
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0 to infinity. it is not defined on the negative
side of the real line. it is a function defined
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from 0 to infinity. so given
vector space or let us wok with set of real
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numbers. so given a vector space norm of a
vector x that belongs to x. norm of x is a
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function that is defined from x to r+ set
of positive real numbers.
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well length if you look at why i am saying
this because length of a vector in 3 dimensional
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is always positive it cannot be negative.
so i need a generic function which exactly
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has similar properties as that of the concept
of length in 3 dimensions. do not forget that
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at any point that we are generalizing concept
from 3 dimensions to look at notions in higher
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dimensions. so the first property that the
norm function should satisfy is norm of x>
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or =0 for any x that belongs to x. norm x=0.
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so this norm function should be always>0.
it can be=to 0 only for one vector that vector
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is the origin it if is 0 for any other vector
which is a non 0 vector then that function
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cannot define a norm. so for the first criteria
for a function to qualify as a norm is that
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it should be a positive function and then
it should be 0 only for 0 vector should be
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non 0 for any non zero vector.
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the second quality is if i multiply the vector
x by a scalar alpha then what should happen?
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norm x should be= mod alpha norm of alpha
times x should be always= to mod alpha times
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norm x
for any scalar alpha it is very, very important
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second property. and the third property of
the length function in 3 dimensions is triangle
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in equality. so this is way of saying that
if i have given a point the shortest distance
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to the point is the straight line.
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so the third property is now triangle inequality
is a result which probably in your undergraduate
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to study in your high school. the sum of length
of 2 sides of a triangle is always> third
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side that result is being generalized. so
if i take any 2 points in the space the shortest
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distance in the straight line connecting the
2 points any other way i try to go to that
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point in the vector space by addition of 2
vectors that will be the larger part.
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so this is a very, very fundamental property
of the concept of length in the 3 dimensions
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which we are generalizing to any other vector
space. so now anything any function that satisfies
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these properties would qualify to be a norm
function. for example in 3 dimensions well
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one way to define norm is very, very common.
so my first example is r3 x corresponds to
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r3 and norm x there are 3 components x1, x2,
x3 then x1 square+ x2 square+x3 square whole
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raise is to ½.
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this is the definition of the norm. in fact
this will be called as 2 norm. so x this space
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together with this definition of norm will
give you 1 norm linear space. but this is
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not the only way to define norm all these
3 properties are very obviously satisfied
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before this. i do not know to even check.
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well my second example is x corresponds to
r3 and then my norm definition is absolute
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of x1+absolute of x2+abs of x3. you can check
whether this qualifies to be a norm. will
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it be 0 only when x1 is 0, x2 is 0, x3 is
0 and if anyone is not 0 then the value will
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be>than 0. so first property satisfied what
about the second property if i multiply a
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vector by a scalar alpha the norm will get
multiplied by alpha.
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the third property follows also in a straight
forward manner from the inequality that if
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i take any 2 scalars is always less than mod
alpha+ mod beta. this is a simple inequality
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for scalars if i look at each component as
a scalar i can apply this inequality to each
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one of them. it will follow that the third
triangle inequalities also satisfied by this
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norm. so this function which is from the space
x to r+ it will always give me a positive
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value.
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equally qualifies to be a norm of the vector
in 3 dimensions i do not have to always thinking
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this term. what i wanted to realize is that
this vector space and this vector space are
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2 separate vector spaces 2 separate norm vector
spaces because a norm space comes with a definition
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of norm on it. so do not say that if i am
working with r3 which means i have to have
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2 norm.
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i am working in 3 dimensional vector space
and i have 1 norm this is called 1 norm. so
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in general i can show that any function if
x is r3 and i can define what is called as
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a p norm x1 to power p+ x2 to power p+ absolute
of x3 to power p whole raise to 1/p where
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p is the positive integer this also forms
a vector space with p norm defined on it.
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this is i am calling xp.
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i can extend this definitions of r3 to rn
dimensional vector space. for 2 norm on n
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dimensional vector space will be x1 square+x2
square+x3 square up to xn square whole raise
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to ½. in general actually i can define this
for any p which is an integer. so 1 norm 2
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norm are some special examples of a p norm
vector space and i can very easily extend
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this definition too.
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i have written here for r3 i can extend this
to rn, i can extend this to cn and so on,
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but is important to note that these 3 are
3 different norm vector spaces. the definition
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of norm is different it is not same. what
about set of continuous functions, how do
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we define norm on it. well before i move to
that let me give me one more example which
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is in line with the p norm.
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my fourth example is going to be x. now instead
of writing r3 i will move to rn and then i
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am going to define what is called as infinite
norm. so for any element x that belongs to
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x infinite norm is defined as max over i=1,2,
to n mod x1. infinite norm this is an n dimensional
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vector space. if i give you a vector that
are n components i find out absolute of each
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components and take max of that.
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i find out absolute of each component and
then take a max of that so there are n values
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absolute values and then find out the maximum
you can show that this also forms norm it
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satisfies all the 3 properties of function
to be a norm. it will always be greater than
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0 if x is not = 0. it will be=0 only when
x=0 or the origin and then if you multiply
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a vector by a scalar the norm will get multiplied
by mod of the scalar.
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triangle inequality will be satisfied so some
of these will be exercise problem that is
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why i am not doing it on the board. my fifth
example is x corresponds to set of continuous
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functions over 0 to 1. and then i am going
to define a norm for an element say function
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ft which belongs to x is norm of ft. well
let me extend norm of ft using the earlier
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notions. so i could define an infinite norm
which is max/t belongs to 0 to 1.
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infinite norm of this function is maximum
value of the function over the interval absolute
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of the maximum value very, very important.
why absolute is required? because the norm
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has to be a positive number. when will this
be 0 when will the maximum be 0 absolute of
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the maximum when you have 0 functions. okay
if i multiply a function f by a scalar alpha
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what will happen to the norm.
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since it is mod absolute value of the scalar
will be multiplying the mod.
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so
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for example if i take a scalar alpha so what
is absolute of alpha times f of t this will
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be mod alpha* f of t. so if i put max operator
that t belongs to 0 to 1. so which is same
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as all right. so i start with this and i show
that it is mod alpha times f of t. i can prove
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the third triangle inequality. the norm is
absolute of the maximum value over interval
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0 to 1. “professor - student conversation
starts” it would all functions maximum values.
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“professor - student conversation ends”
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so given a function if i give you a vector
you find norm of that vector. so if i give
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you for example sin t where t goes from 0
to 1. you are expected to find what is the
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norm of that find the maximum value of sin
t over interval 0 to 1. find absolute of that,
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that will give you norm of that. so not overall
function it is over t the max is not over
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the functions. max is over t okay. so likewise
if i take any 2.
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what about the third thing what about the
triangle inequality. if i give you 2 functions
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say ft and gt which belong to that x set of
continuous functions over 0 to 1 then what
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about triangle inequality. what we know that
at a particular point if i fix t for a
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fixed t then i get function values. so these
are real numbers we are talking about real
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valued functions by the way. so for real numbers
what i know is ft+ for a fixed value of t
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i can write this.
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now i can use this inequality and through
the triangle inequality. so now what i am
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going to do since this holds for every t i
can argue that max it holds for all t between
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0 to 1.
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so i can write i can take a max operator.
this is nothing, but this is equal to norm
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of ft+ gt right. this will be infinite norm
for ft+ gt you agree with me, but using this
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inequality you can say that this is always
less than not equal to max of it follows from
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this inequality that initially i am looking
at these as 2 scalar numbers. if you give
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me 2 real numbers this inequality holds.
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see if i take a function over an interval
if i fixed myself to 1 t i will get one scalar
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value right. let us this is sin t this is
cos t. if i say that my t is 0.5 i will get
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some specific value of sin and cos. what we
know from a very fundamental inequality is
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that if i add 2 real numbers then that sum
is always less than or equal to sum of mod
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of 2 real numbers. now actually i want to
look at max over all t.
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so i am graduating from this inequality to
this inequality and what is this? this is
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norm ft infinity. what is this? so what i
have proved is triangle inequality that if
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i take infinite norm of sum of 2 functions
it is always less than or equal to this+ this.
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this is norm of function ft this is infinite
norm of function gt. is that okay everyone
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with me on this.
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well i can define now norms which are similar
to 1 norm or 2 norm or p norm on this basis.
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infinite norm is not the only way to go about
defining norm on this pace. so my next example
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this is sixth or seventh example.
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my sixth example is x corresponds to set of
continuous functions over 0 to 1 and my norm
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definition. so for any ft that belongs to
x my norm of ft is p norm is integral 0 to
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1. i can define a norm which is integral 0
to 1. i need a norm to be a scalar value.
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if you look at n dimensions how was the norm
defined? sum of absolute values for 1 norm,
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sum of square absolute value then square root
for the 2 norm.
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for a p norm it was defined absolute value
of each element is to p sum it and raise to
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1/p and so on. so this definition should be
logical extension because instead of sum i
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am getting integral because now t is very
continuously. in n dimensions vector space
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the index i was finite 1 to n. so we had a
summation here we have an integral. well p
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norms though they are defined we normally
use only 1 norm, 2 norm and infinite norm.
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the three norm which are very, very commonly
used are 1 norm, 2 norm.
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so ft 1 norm would be integral 0 to 1 mod
ft dt and 2 norm is integral 0 to 1 mod ft
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square dt whole raise to 1/2. so this is extension
of the ideas from 3 dimensions or from n dimensions
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to a set of continuous functions and infinite
dimension space. now it is not necessary that
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this should be limit here should be between
0 to 1. i could actually define a set of continuous
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function over any interval say a to b any
interval -5 to +10 whatever does not matter.
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my integral here will change from a to b.
my integral here will change from a to b and
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so on, but what you should realize is that
this vector space together with a definition
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of 1 specific norm will be one norm linear
space one normed space. so set of continuous
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functions with 1 norm is 1 norm linear space.
set of continuous functions with 2 norm is
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another norm linear space they are not same.
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definition of length is different. how we
measure length in 1 system is not the way
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we measure length in the other system. well
there might be advantages of this norm over
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this norm or advantages of this norm over
this infinite norm and indeed that is why
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we keep using so called 2 norm very, very
often that will become clear as we graduate
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to in a product space little later.
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so why is this guy so special, why we do not
seem to use 1 norm of infinite norm so often
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as against this 2 norm will become clear.
nevertheless do not think that 2 norm is the
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only way to define norm there are many other
ways of defining norms. now to understand
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this concept of norm i think we should have
1 or 2 more examples and then things will
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become clear. i have actually given here some
arbitrary functions which could be used to
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define norm.
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i will just talk about 1 of them right now
here and then we will move on to some other
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concept which are important. like convergence
and i mean what is the use of defining norm
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the one of the use of defining norm is to
talk about limit and convergence. now what
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happens in iterative processes is that you
have some idea about iterative process like
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newton–raphson. everyone i think knows about
newton–raphson. in newton–raphson what
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do you do you start with the guess value and
then you get another guess.
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from the new guess you concept third guess
and fourth guess. so you get what is called
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as sequence of vectors. the question that
you need to answer when you solve a numerical
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problem is this sequence converging to something.
is it convergent to my solution in fact that
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is what is in your mind, but at least before
whether to know whether it is going to the
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solution or not you would like to know whether
it is converging.
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so convergence is a very, very critical thing
in numerical analysis and when you work with
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n dimensional spaces you need to generalize
the concept of convergence. we know about
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a sequence of real numbers converging to a
point and we call it limit. all those things
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we have done in 12th standard right, but now
what is the meaning of a function sequence
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converging to a limit. we will have to talk
about these ideas so that is why we need this.
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now just to give you a feeling that the norm
can be defined in different ways. i will just
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take the same x which is set of once differentiable
continuous function. i will take set of once
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differentiable continuous function and let
me define the function you have to tell me
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whether it will define a norm or not. so what
about candidate function. so i have this function
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ft that is= that belongs to this x.
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so this is a once differentiable continuous
function. now i am defining a norm of ft as
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max over t belongs to 0 to 1 mod of d f/dt
derivative. i am taking derivative of the
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function df/dt. question is does this function
define the norm why? so what does mean? a
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non 0 vector is giving me 0 value.
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so this is
a non 0 function f of t=constant and that
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will yield 0 norm if i use this definition.
so this function cannot qualify as a norm.
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there are more such examples given listed
here you should go through them this is on
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page 12. i have listed many other functions
and i have solved systematically. if i give
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you a function and ask you to check whether
this function satisfies to be a norm of not.
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you have to systematically look at all the
3 axioms.
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first you have to look for whether non 0 element
gives you a 0 vector whether the second thing
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is alpha times original vector that will give
you mod alpha times. and a third property
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is triangle inequality. even in 3 dimensions
there are multiple ways of defining norms
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see for example you can define a norm in 3
dimensions if i have x is r3 if i give you
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a matrix w which is
symmetric and positive definite.
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then i can define a norm of vector x in 3
dimensions as x transpose wx. you can show
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that x transpose wx will also raise to ½
x transpose wx. so we call this as 2 nom with
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00:43:50,880 --> 00:43:59,339
matrix w where w is some symmetric positive
definite matrix if this matrix is singular
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which means if it is semi definite and not
semi definite. what is a semi definite matrix?.
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not all eigenvalues are positive so it means
it has some eigenvalues =0.
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so some eigenvalues are=0 we will visit this
definite semi definite many times. so let
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us wait for that right now let us wait for
that, but just think about this w cannot be
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singular if this w matrix is singular then
it will not define a norm because a singular
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matrix can take a non 0 vector to 0. yesterday
we looked at that we have a singular matrix
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with 3 columns which are linearly dependent
a singular matrix can take a non 0 vector
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to 0 vector.
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so which means a non -0 vector can give you
a 0 value for the norm which is not acceptable
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so it will not be a norm. so the next thing
that i want to talk about is convergence of
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a sequence of vectors. so as i said we are
going to deal with iterative processes.
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so let say you will have iterative processes
which will give you x k+1= some function f
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00:45:20,619 --> 00:45:43,509
of xk. we have iterative process in which
we start with a guess solution and that guess
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is used to construct the next guess and then
that guess is used to construct on next guess
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due to newton–raphson method all of you
know about this. we are going to generalize
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a newton–raphson method from one dimension
to n dimension or to a function space and
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so on.
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so that time you will have sequence of vectors
in n dimensions and i need to know whether
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they are close to each other or they are converging
how do i know about that. i need to use definitions
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of norm. if i have a sequence of functions
generated in a method how do i know whether
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this sequence of functions is converging to
solution or it is not converging. i need to
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use definition of norms that is why we need
this norm definition generalize because we
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need to talk about convergence of a sequence.
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we will talk about limit of a sequence. so
we will look at this concept in my next lecture.
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it is important to understand keep in mind
that why it is being done. it is being done
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for dealing with iterative processes. so in
the next lecture we will look at convergence
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and sequences. we will have something called
cauchy sequence which almost converges and
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a convergence sequence and we will also look
at some funny properties, funny things that
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00:47:04,509 --> 00:47:06,599
happened in infinite dimensions spaces.
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you have a sequence which converges, but the
limit is not in the space. the limit is outside
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the space and so on. i will show you some
examples where these funny things happened.