Lec. 2 - Metric and Normed Spaces

8/10/2019 Lec. 2 - Metric and Normed Spaces

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METRIC AND NORMEDSPACES

Week 2


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INTRODUCTION

Week 2

Lecture 2.1


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What we will learn this week:

What is a distance?

What is a metric space?

What is a converging sequence in a metric space?

What is a Cauchy sequence?

What is a normed space?

What is a converging sequence in a normed space?

What are equivalent norms?

An example of Normed spaces: Lp

Density


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DISTANCE FUNCTION

Week 2

Lecture 2.2


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Definition: Distance Function

Let E be a set and d : ExE!Rbe a function.

d is a distance functionon E if


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i. 0y)d(x,E,Ey)x,( !"#$


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i.

ii.

yx0y)d(x,E,Ey)(x, =!="#$

0y)d(x,E,Ey)x,( !"#$


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i.

ii.

iii.

yx0y)d(x,E,Ey)(x, =!="#$

0y)d(x,E,Ey)x,( !"#$

x)d(y,y)d(x,E,Ey)(x, =!"#


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i.

ii.

iii.

iv.

yx0y)d(x,E,Ey)(x, =!="#$

0y)d(x,E,Ey)x,( !"#$


y)d(z,z)d(x,y)d(x,E,EEz)y,(x, +!""#$


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y

x


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y

x


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y

z

x


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i.

ii.

iii.

iv.

yx0y)d(x,E,Ey)(x, =!="#$

0y)d(x,E,Ey)x,( !"#$




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i.

ii.

iii.

iv.

(E,d) is a metric space.

yx0y)d(x,E,Ey)(x, =!="#$

0y)d(x,E,Ey)x,( !"#$




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Definition: Pseudodistance Function


d is apseudodistance functionon E if

i.

ii.

iii.

iv.

(E,d) is apseudometric space.

yx0y)d(x,E,Ey)(x, =!="#$

0y)d(x,E,Ey)x,( !"#$




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Definition: Quasidistance Function


d is a quasidistance functionon E if

i.

ii.

iii.

iv.

(E,d) is a quasimetric space.

yx0y)d(x,E,Ey)(x, =!="#$

0y)d(x,E,Ey)x,( !"#$




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Examples

Consider E=Rnwith n in N* and p in [1,"[

pd (X,Y) =p

iX! iYi=

1

n

"p

!d (X,Y) = maxi"[1,n]#N iX $ iY


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d2is a distance:

the Euclidean Distance Function

( )! "=

=

n

1i

2

2 YXd iiY)(X,

Example

Consider E=Rnand


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Example

Consider E=Rnwith n=2

Photo Credit Google Inc.


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Example




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Example


!= "=

n

1iii1 YXd Y)(X,

and

d1is a distance.



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Example


!= "=

n

1iii1 YXd Y)(X,

and

d1is a distance.



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The metro in Paris

Consider:

E = {metro stations inParis}

d(x,y) = average time toget from x in E to y in E,

using the fastest way.

Is d a distance?

Photo Credit Rgie Autonome des Transports Parisiens


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The metro in Paris

Consider:




Is d a distance?



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The metro in Paris

Consider:




Is d a distance?

It is a quasidistance only:

d(Botzaris,Danube) !d(Danube,Botzaris)



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In-Video Quiz

Consider E = { functions from Rto Rdefined in 0 }

For f and g in E, define d(f,g) = |g(0)-f(0)|

Is d a distance function on E?


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In-Video Quiz


For f and g in E, define d(f,g) = |g(0)-f(0)|


No, it is a pseudodistance only:

d(x!x2,x!x3)=0


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In-Video Quiz

Consider a set E


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In-Video Quiz

Consider a set E

For all x and y in E, define

d(x,y)=0 if x=yd(x,y)=1 if x!y


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In-Video Quiz

Consider a set E





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In-Video Quiz

Consider a set E



Is d a distance function on E?Yes.


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Distance from a Point to a Set

Let E be a metric space with distance d.

Let aE.

Let XE.


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UNDERLYING TOPOLOGYIN A METRIC SPACE

Week 2

Lecture 2.3


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From a Metric to a Topological Space

Let (E,d) be a metric space.


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Given x in E,

define the open ball around x with radius r>0 by:

r}y)d(x,|E{y(x)Br


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Given x in E,

define the open ball around x with radius r>0 by:

Define a topology on E by

r}y)d(x,|E{y(x)Br

O}(x)B0,rO,x|E{OT r !>"#$!=


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In-Quiz Video

Prove T is a topology

(use a pen and paper)


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In-Quiz Video

O}(x)B0,rO,x|E{OT r !>"#$!=


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In-Quiz Video

O}(x)B0,rO,x|E{OT r !>"#$!=

i. The empty set and X are elements of T


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In-Quiz Video

O}(x)B0,rO,x|E{OT r !>"#$!=

i. The empty set and X are elements of Tii. Any union of elements of T is in T

iii.

Any finite intersection of elements of T is in T


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Given a metric space,

we can derive an associated topological space.


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It is a Normal Hausdorff Space.


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Given a topological space,

we may not always find a distance

from which the topology derives.


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Given a topological space,

we may not always find a distance

from which the topology derives.

When it is possible, the space is metrizable.


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In-Video Quiz

Find a topology which is not metrizable.


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In-Video Quiz

The trivial topology is not metrizable.


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CONVERGENCE IN A METRICSPACE & COMPLETENESS

Week 2

Lecture 2.4


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Converging Sequences

Let (X,d) be metric space

(now also a Hausdorff topological space)

the limit l is unique.


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Let (xn) be a sequence of elements of X.


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We say that (xn) convergesto l if

It is equivalent to

VxNnN,N(l),V n!"#!$!% V

(l)BxNnN,N0,!!n !"#!$>%


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It is equivalent to


!l),d(xNnN,N0,! n


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It is equivalent to


!l),d(xNnN,N0,! n


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Example

xn=1/n2

Prove (xn) converges to 0


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Example

xn=1/n2


Let ">0

Let N=[1/"1/2]=[1/"1/2]+1


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Example

xn=1/n2


Let ">0

Let N=[1/"1/2]=[1/"1/2]+1

Then n>N implies n>1/"1/2

Thus 1/n2< "


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Example

xn=1/n2


Let ">0

Let N=[1/"1/2]=[1/"1/2]+1

Then n>N implies n>1/"1/2

Thus 1/n2< "Thus d(xn,0) < "


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Definition: Completeness

In a metric space,

we call Cauchy sequence, a sequence (un) s.t.

!)u,ud(Nnm0,N0,! nm >>">#


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In a metric space,


A metric space X is called complete

if all Cauchy sequences of elements of X converge.

!)u,ud(Nnm0,N0,! nm >>">#


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In a metric space,


A metric space X is called complete

if all Cauchy sequences of elements of X converge.

Ris complete. Qisnt.

!)u,ud(Nnm0,N0,! nm >>">#


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Example

xn=1/n2

Prove (xn) is a Cauchy sequence


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Example

xn=1/n2


Let ">0


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Example

xn=1/n2


Let ">0Let N=[1/(2")]=[1/(2")]+1


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Example

xn=1/n2


Let ">0Let N=[1/(2")]=[1/(2")]+1

Then q>p>N implies q>p>1/(2")


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Example

xn=1/n2


Let ">0Let N=[1/(2")]=[1/(2")]+1


Thus (1/q-1/p)(1/q+1/p) #2 (1/q-1/p) < 2 x 1/(2")


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Example

xn=1/n2


Let ">0Let N=[1/(2")]=[1/(2")]+1


Thus (1/q-1/p)(1/q+1/p) #2 (1/q-1/p) < 2 x 1/(2")Thus 1/q2 1/p2< "


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Example

xn=1/n2


Let ">0Let N=[1/(2")]=[1/(2")]+1


Thus (1/q-1/p)(1/q+1/p) #2 (1/q-1/p) < 2 x 1/(2")Thus 1/q2 1/p2< "

Thus d(xp,xq) < "


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Example

Let xn= [10n $2] / 10n

x0= 1

x1= 1.4

x2= 1.41

x3= 1.414

(xn) is a Cauchy sequence

Its limit is $2


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NORMED VECTOR SPACES

Week 2

Lecture 2.5


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Definition: Norm

Let E be a vector space and N : E!Ra function.


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Definition: Norm


N is a norm on E if

D fi i i N


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Definition: Norm


N is a norm on E if

i.

ii.

!x " E, N(x) = 0# x = 0

D fi iti N


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Definition: Norm


N is a norm on E if

i.

ii.

!x " E, N(x) = 0# x = 0

N(x)||x)N(R,E)(x, !!! ="#$

D fi iti N


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Definition: Norm


N is a norm on E if

i.

ii.

iii.

!x " E, N(x) = 0# x = 0

N(x)||x)N(R,E)(x, !!! ="#$N(y)N(x)y)N(xE,Ey)(x, +!+"#$

D fi iti N


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Definition: Norm


N is a norm on E if

i.

ii.

iii.

Assertions (ii) and (iii) imply N(x) %0 for all x in E.

!x " E, N(x) = 0# x = 0


D fi iti N


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Definition: Norm


N is a norm on E if

i.

ii.

iii.

Assertions (ii) and (iii) imply N(x) %0 for all x in E.

(E,N) is a normed vector space.

N(x) is usually noted || x ||Eor simply || x ||

!x " E, N(x) = 0# x = 0


D fi iti S i


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Definition: Seminorm


N is a seminorm on E if

i.

ii.

iii.N(x)||x)N(R,E)(x, !!! ="#$

N(y)N(x)y)N(xE,Ey)(x, +!+"#$

!x " E, N(x) = 0# x = 0

D fi iti S i


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i.

ii.

iii.

Nevertheless, assertion (ii) implies N(0)=0.


!x " E, N(x) = 0# x = 0

D fi iti S i


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i.

ii.

iii.

Nevertheless, assertion (ii) implies N(0)=0.

(E,N) is a seminormed vector space.


!x " E, N(x) = 0# x = 0

E l f


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Examples of norms

Consider E = #"= { bounded sequences }

For u in E, define N(u) = sup { |ui|, i&N}

E l f


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Examples of norms



N(u) = 0 iff u=0

E amples of norms


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Examples of norms



N(u) = 0 iff u=0N($u) = |$|N(u), for any real number $

Examples of norms


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Examples of norms




N(u+v) %N(u)+N(v)

Examples of norms


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Examples of norms




N(u+v) %N(u)+N(v)

(E,N) is a normed space.

Examples of norms


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Examples of norms


(X,0)di p

p

n

1i

p

p |X|||X|| == !=

(X,0)d|X|max iNn][1,i

||X|| !"#!

==

Examples of norms


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Examples of norms


These are norms on E.

(X,0)di p

p

n

1i

p

p |X|||X|| == !=

(X,0)d|X|max iNn][1,i

||X|| !"#!

==

In Video Quiz


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In-Video Quiz


For f in E, define N(f) = d(f,0) = |f(0)|

In Video Quiz


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In-Video Quiz



Is N a norm on E?

In Video Quiz


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In-Video Quiz



Is N a norm on E?No. It is a seminorm only:

||x!x2||=0


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UNDERLYING METRIC ANDTOPOLOGY IN A NORMED

SPACE

Week 2

Lecture 2.6

From a Normed to a Metric Space


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Let (E,N) be a normed space.



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Given x and y in E,

define d(x,y) = N(y-x)





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Given x and y in E,



The unit open ball associated to the norm is


1}N(x)|E{yB(0,1)B


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Given x and y in E,





1}N(x)|E{yB(0,1)B

Norm 2



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Given x and y in E,





1}N(x)|E{yB(0,1)B

Norm '


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Given a normed space,we can derive an associated metric space.

d(x,y)=||x-y||



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d(x,y)=||x-y||


It may not be a linear space.



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d(x,y)=||x-y||



Even if it is a linear space,There may be no norm inducing the distance.



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d(x,y)=||x-y||



Even if it is a linear space,There may be no norm inducing the distance.

For example: d(x,x)=0 and d(x,y)=1 for x&y

Topological, Metric and Normed Vector Spaces


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Topological


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opo og ca , et c a d o ed ecto Spaces

Normed Vector

Metric

Topological


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Given a normed spacewe can derive an associated metric space

Given a metric space,It may not be a linear space.

Even if it is a linear space,

There may be no norm inducing the distance.For example: d(x,x)=0 and d(x,y)=1 for x&y



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g g q

Let (X,N) be a normed space.



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g g q

Let (X,N) be a normed space.Let (xn) be a sequence of elements of X.



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g g q





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g g q


We say that (xn) convergesto l ifVxNnN,N(l),V n!"#!$!% V


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g g q


We say that (xn) convergesto l if!!> 0, "N# N, n $N%|| x

n& l ||< !

Remark


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Remark


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Remember last week?

The norm is a continuous function.


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Remark


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Remember last week?

The norm is a continuous function.

| N(xn) N(l) | #N(xn-l) = d(xn,l)

If (xn) converges to l then N(xn) converges to N(l)

Definition: Strength of a Norm


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Let E be a vector space.



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Nais stronger than Nbif

there exists a non-negative constant Casuch that for all x in E,

Nb(x) %CaNa(x)



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Nais stronger than Nbif

there exists a non-negative constant Casuch that for all x in E,

Nb(x) %CaNa(x)

The balls of Na can be included in the balls of Nb

(after a possible homothetic transformation)


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Definition: Norm Equivalence


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Naand Nbare equivalentif

there exist two non-negative constants C1and C2

such that for all x in E,

C1Na(x) %Nb(x) %C2Na(x)



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Norms are equivalent iff:

associated balls can be included in one another(after a possible homothetic transformation).



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Norms are equivalent iff:

associated balls can be included in one another(after a possible homothetic transformation).


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Norm Equivalence


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Theorem

Let E be a finite-dimensional vector space.

All norms on E are equivalent.

Norm Equivalence


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Theorem

Let E be a finite-dimensional vector space.

All norms on E are equivalent.

Corollary

Let E be a finite-dimensional vector space.There is only one topology induced by the norms.

The usual topology of Rn


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There is only one topology induced by the norms.It is called the usual topology of Rn.

What do the open sets look like?



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Norm Equivalence


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Recall

A stronger norm will provide a stronger topology.

O}(x)B0,rO,x|E{OT r !>"#$!=

r}||y-x|||X{y(x)Br


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AN EXAMPLE OF A NORMED

SPACE: LP

Week 2

Lecture 2.7

Integration


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The Rieman Integral:Subdivision of the x-axis

Integration


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Adding up the surface area

of rectangles

Integration


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of rectangles

Integration


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of rectangles


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Integration


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of rectangles

The Lebesgue Integral:

Subdivision of the y-axis

The inverse image ismeasuredand added up

Integration


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of rectangles




Integration


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of rectangles




Integration


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of rectangles




Integration


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of rectangles




Integration


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of rectangles




Integration


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of rectangles




Integration


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of rectangles




Integration


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of rectangles




Measure


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Measure


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Measure


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Measure


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Integration


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We will consider (an open set of RN

equipped with the Lebesgue measure.

The set of Lebesgue-integrable functions from

(to Rwill be noted L1(() or simply L1when

no confusion is possible. Functions that are equal

almost everywhere are identified.

We note fL1 = f(x) dx

!! = f!

Definition


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Let p '[1,"[.

We note Lp(() the set of measurable functions

from(

to Rwhose p-th power belongs to L1

((

).


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Definition


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Let p '[1,"[.


from(


((

).Functions equal almost everywhere are identified.

We note Lpwhen no confusion is possible.

Definition


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Let p '[1,"[.


from(


((

).Functions equal almost everywhere are identified.

We note Lpwhen no confusion is possible.

We note pp !! === !p

!

ppL

|f|dx|f(x)|ff p

Definition


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We note L"

(() the set of measurable functionsfrom (to Rfor which there exists a real number C

s.t. for almost every x in (, |f(x)|%C.

Definition


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We note L"



Functions equal almost everywhere are identified.

Definition


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We note L"



Functions equal almost everywhere are identified.We note L"when no confusion is possible.

Definition


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We note L"



Functions equal almost everywhere are identified.We note L"when no confusion is possible.

We note }!ona.e.C|f(x)|C,{Infff L !== ""

Definition


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Let p '[1,"].

A function f belongs to Lploc(() when

f 1Kbelongs to Lp

((

) for every compact K

(

(1Kis the characteristic function of K:

1K(x)=1 if x'K and 0 otherwise)

Definition


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Let p '[1,"].

We call Hlder conjugate (or dual index) of p,

the number p = 1 + 1/(p-1) so that 1/p + 1/p = 1

Definition


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Let p '[1,"].


the number p = 1 + 1/(p-1) so that 1/p + 1/p = 1(if p=1 then p="and p="then p=1)

Definition


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Let p '[1,"].


the number p = 1 + 1/(p-1) so that 1/p + 1/p = 1(if p=1 then p="and p="then p=1)

Note that the Hlder conjugate of 2 is 2.

Norm on Lp


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Proposition (Hlders Inequality)Let p '[1,"] and p be its Hlder conjugate.

Let f 'Lpand g 'Lp

Then f g 'L1

and || f g ||1%|| f ||p|| g ||p

Norm on Lp


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Proposition (Hlders Inequality)Let p '[1,"] and p be its Hlder conjugate.

Let f 'Lpand g 'Lp

Then f g 'L1

and || f g ||1%|| f ||p|| g ||p

Corollary

Let p '[1,"]

|| (||pis a norm on Lp

Interpolation Inequality


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PropositionLet { fi, i 'I } be a family of functions with fi'L

pi

and 1/p = )1/pi%1.

then*

fi'L

p

((

) and ||*

fi ||p%*

|| fi||pi

Interpolation Inequality


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PropositionLet { fi, i 'I } be a family of functions with fi'L

pi

and 1/p = )1/pi%1.

then*

fi'L

p

((

) and ||*

fi ||p%*

|| fi||pi

Corollary

If f 'Lp )Lq

then f 'Lr for any r s.t. p%r %q


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DENSITY

Week 2

Lecture 2.8


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Approximation of +


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+= 3.14159265358979323846264

33832

Approximation of +


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+= 3.14159265358979323846264

33832

+ Q

Johann Heinrich Lambert

Photo Credit: Wikimedia Commons


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Approximation of +


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+= 3.14159265358979323846264

33832

+ Q


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Approximation of +


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+= 3.14159265358979323846264

33832

+ Q

0 1 2 3 4+

+

Approximation of +


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+= 3.14159265358979323846264

33832

+ Q

0 1 2 3 4+

+a b

Approximation of +


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+= 3.14159265358979323846264

33832

+ Q

3.14

Lec. 2 - Metric and Normed Spaces

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Transcript of Lec. 2 - Metric and Normed Spaces