Download - Lec. 2 - Metric and Normed Spaces
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
1/207
METRIC AND NORMEDSPACES
Week 2
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
2/207
INTRODUCTION
Week 2
Lecture 2.1
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
3/207
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
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
4/207
DISTANCE FUNCTION
Week 2
Lecture 2.2
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
5/207
Definition: Distance Function
Let E be a set and d : ExE!Rbe a function.
d is a distance functionon E if
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
6/207
Definition: Distance Function
Let E be a set and d : ExE!Rbe a function.
d is a distance functionon E if
i. 0y)d(x,E,Ey)x,( !"#$
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
7/207
Definition: Distance Function
Let E be a set and d : ExE!Rbe a function.
d is a distance functionon E if
i.
ii.
yx0y)d(x,E,Ey)(x, =!="#$
0y)d(x,E,Ey)x,( !"#$
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
8/207
Definition: Distance Function
Let E be a set and d : ExE!Rbe a function.
d is a distance functionon E if
i.
ii.
iii.
yx0y)d(x,E,Ey)(x, =!="#$
0y)d(x,E,Ey)x,( !"#$
x)d(y,y)d(x,E,Ey)(x, =!"#
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
9/207
Definition: Distance Function
Let E be a set and d : ExE!Rbe a function.
d is a distance functionon E if
i.
ii.
iii.
iv.
yx0y)d(x,E,Ey)(x, =!="#$
0y)d(x,E,Ey)x,( !"#$
x)d(y,y)d(x,E,Ey)(x, =!"#
y)d(z,z)d(x,y)d(x,E,EEz)y,(x, +!""#$
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
10/207
y
x
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
11/207
y
x
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
12/207
y
z
x
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
13/207
Definition: Distance Function
Let E be a set and d : ExE!Rbe a function.
d is a distance functionon E if
i.
ii.
iii.
iv.
yx0y)d(x,E,Ey)(x, =!="#$
0y)d(x,E,Ey)x,( !"#$
x)d(y,y)d(x,E,Ey)(x, =!"#
y)d(z,z)d(x,y)d(x,E,EEz)y,(x, +!""#$
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
14/207
Definition: Distance Function
Let E be a set and d : ExE!Rbe a function.
d is a distance functionon E if
i.
ii.
iii.
iv.
(E,d) is a metric space.
yx0y)d(x,E,Ey)(x, =!="#$
0y)d(x,E,Ey)x,( !"#$
x)d(y,y)d(x,E,Ey)(x, =!"#
y)d(z,z)d(x,y)d(x,E,EEz)y,(x, +!""#$
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
15/207
Definition: Pseudodistance Function
Let E be a set and d : ExE!Rbe a 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,( !"#$
x)d(y,y)d(x,E,Ey)(x, =!"#
y)d(z,z)d(x,y)d(x,E,EEz)y,(x, +!""#$
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
16/207
Definition: Quasidistance Function
Let E be a set and d : ExE!Rbe a 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,( !"#$
x)d(y,y)d(x,E,Ey)(x, =!"#
y)d(z,z)d(x,y)d(x,E,EEz)y,(x, +!""#$
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
17/207
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
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
18/207
d2is a distance:
the Euclidean Distance Function
( )! "=
=
n
1i
2
2 YXd iiY)(X,
Example
Consider E=Rnand
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
19/207
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
20/207
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
21/207
Example
Consider E=Rnwith n=2
Photo Credit Google Inc.
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
22/207
Example
Consider E=Rnwith n=2
Photo Credit Google Inc.
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
23/207
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
24/207
Example
Consider E=Rnwith n=2
!= "=
n
1iii1 YXd Y)(X,
and
d1is a distance.
Photo Credit Google Inc.
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
25/207
Example
Consider E=Rnwith n=2
!= "=
n
1iii1 YXd Y)(X,
and
d1is a distance.
Photo Credit Google Inc.
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
26/207
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
27/207
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
28/207
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
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
29/207
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
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
30/207
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?
It is a quasidistance only:
d(Botzaris,Danube) !d(Danube,Botzaris)
Photo Credit Rgie Autonome des Transports Parisiens
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
31/207
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
32/207
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?
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
33/207
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?
No, it is a pseudodistance only:
d(x!x2,x!x3)=0
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
34/207
In-Video Quiz
Consider a set E
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
35/207
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
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
36/207
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
Is d a distance function on E?
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
37/207
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
Is d a distance function on E?Yes.
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
38/207
Distance from a Point to a Set
Let E be a metric space with distance d.
Let aE.
Let XE.
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
39/207
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
40/207
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
41/207
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
42/207
UNDERLYING TOPOLOGYIN A METRIC SPACE
Week 2
Lecture 2.3
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
43/207
From a Metric to a Topological Space
Let (E,d) be a metric space.
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
44/207
From a Metric to a Topological Space
Let (E,d) be a metric space.
Given x in E,
define the open ball around x with radius r>0 by:
r}y)d(x,|E{y(x)Br
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
45/207
From a Metric to a Topological Space
Let (E,d) be a metric space.
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 !>"#$!=
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
46/207
In-Quiz Video
Prove T is a topology
(use a pen and paper)
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
47/207
In-Quiz Video
O}(x)B0,rO,x|E{OT r !>"#$!=
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
48/207
In-Quiz Video
O}(x)B0,rO,x|E{OT r !>"#$!=
i. The empty set and X are elements of T
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
49/207
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
50/207
In-Quiz Video
O}(x)B0,rO,x|E{OT r !>"#$!=
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
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
51/207
From a Metric to a Topological Space
Given a metric space,
we can derive an associated topological space.
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
52/207
From a Metric to a Topological Space
Given a metric space,
we can derive an associated topological space.
It is a Normal Hausdorff Space.
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
53/207
From a Metric to a Topological Space
Given a metric space,
we can derive an associated topological space.
It is a Normal Hausdorff Space.
Given a topological space,
we may not always find a distance
from which the topology derives.
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
54/207
From a Metric to a Topological Space
Given a metric space,
we can derive an associated topological space.
It is a Normal Hausdorff Space.
Given a topological space,
we may not always find a distance
from which the topology derives.
When it is possible, the space is metrizable.
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
55/207
In-Video Quiz
Find a topology which is not metrizable.
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
56/207
In-Video Quiz
The trivial topology is not metrizable.
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
57/207
CONVERGENCE IN A METRICSPACE & COMPLETENESS
Week 2
Lecture 2.4
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
58/207
Converging Sequences
Let (X,d) be metric space
(now also a Hausdorff topological space)
the limit l is unique.
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
59/207
Converging Sequences
Let (X,d) be metric space
(now also a Hausdorff topological space)
the limit l is unique.
Let (xn) be a sequence of elements of X.
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
60/207
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
61/207
Converging Sequences
Let (X,d) be metric space
(now also a Hausdorff topological space)
the limit l is unique.
Let (xn) be a sequence of elements of X.
We say that (xn) convergesto l if
It is equivalent to
VxNnN,N(l),V n!"#!$!% V
(l)BxNnN,N0,!!n !"#!$>%
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
62/207
Converging Sequences
Let (X,d) be metric space
(now also a Hausdorff topological space)
the limit l is unique.
Let (xn) be a sequence of elements of X.
We say that (xn) convergesto l if
It is equivalent to
VxNnN,N(l),V n!"#!$!% V
!l),d(xNnN,N0,! n
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
63/207
Converging Sequences
Let (X,d) be metric space
(now also a Hausdorff topological space)
the limit l is unique.
Let (xn) be a sequence of elements of X.
We say that (xn) convergesto l if
It is equivalent to
VxNnN,N(l),V n!"#!$!% V
!l),d(xNnN,N0,! n
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
64/207
Example
xn=1/n2
Prove (xn) converges to 0
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
65/207
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
66/207
Example
xn=1/n2
Prove (xn) converges to 0
Let ">0
Let N=[1/"1/2]=[1/"1/2]+1
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
67/207
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
68/207
Example
xn=1/n2
Prove (xn) converges to 0
Let ">0
Let N=[1/"1/2]=[1/"1/2]+1
Then n>N implies n>1/"1/2
Thus 1/n2< "
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
69/207
Example
xn=1/n2
Prove (xn) converges to 0
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) < "
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
70/207
Definition: Completeness
In a metric space,
we call Cauchy sequence, a sequence (un) s.t.
!)u,ud(Nnm0,N0,! nm >>">#
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
71/207
Definition: Completeness
In a metric space,
we call Cauchy sequence, a sequence (un) s.t.
A metric space X is called complete
if all Cauchy sequences of elements of X converge.
!)u,ud(Nnm0,N0,! nm >>">#
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
72/207
Definition: Completeness
In a metric space,
we call Cauchy sequence, a sequence (un) s.t.
A metric space X is called complete
if all Cauchy sequences of elements of X converge.
Ris complete. Qisnt.
!)u,ud(Nnm0,N0,! nm >>">#
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
73/207
Example
xn=1/n2
Prove (xn) is a Cauchy sequence
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
74/207
Example
xn=1/n2
Prove (xn) is a Cauchy sequence
Let ">0
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
75/207
Example
xn=1/n2
Prove (xn) is a Cauchy sequence
Let ">0Let N=[1/(2")]=[1/(2")]+1
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
76/207
Example
xn=1/n2
Prove (xn) is a Cauchy sequence
Let ">0Let N=[1/(2")]=[1/(2")]+1
Then q>p>N implies q>p>1/(2")
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
77/207
Example
xn=1/n2
Prove (xn) is a Cauchy sequence
Let ">0Let N=[1/(2")]=[1/(2")]+1
Then q>p>N implies q>p>1/(2")
Thus (1/q-1/p)(1/q+1/p) #2 (1/q-1/p) < 2 x 1/(2")
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
78/207
Example
xn=1/n2
Prove (xn) is a Cauchy sequence
Let ">0Let N=[1/(2")]=[1/(2")]+1
Then q>p>N implies q>p>1/(2")
Thus (1/q-1/p)(1/q+1/p) #2 (1/q-1/p) < 2 x 1/(2")Thus 1/q2 1/p2< "
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
79/207
Example
xn=1/n2
Prove (xn) is a Cauchy sequence
Let ">0Let N=[1/(2")]=[1/(2")]+1
Then q>p>N implies q>p>1/(2")
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) < "
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
80/207
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 $2
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
81/207
NORMED VECTOR SPACES
Week 2
Lecture 2.5
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
82/207
Definition: Norm
Let E be a vector space and N : E!Ra function.
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
83/207
Definition: Norm
Let E be a vector space and N : E!Ra function.
N is a norm on E if
D fi i i N
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
84/207
Definition: Norm
Let E be a vector space and N : E!Ra function.
N is a norm on E if
i.
ii.
!x " E, N(x) = 0# x = 0
D fi iti N
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
85/207
Definition: Norm
Let E be a vector space and N : E!Ra function.
N is a norm on E if
i.
ii.
!x " E, N(x) = 0# x = 0
N(x)||x)N(R,E)(x, !!! ="#$
D fi iti N
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
86/207
Definition: Norm
Let E be a vector space and N : E!Ra function.
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
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
87/207
Definition: Norm
Let E be a vector space and N : E!Ra function.
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
N(x)||x)N(R,E)(x, !!! ="#$N(y)N(x)y)N(xE,Ey)(x, +!+"#$
D fi iti N
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
88/207
Definition: Norm
Let E be a vector space and N : E!Ra function.
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
N(x)||x)N(R,E)(x, !!! ="#$N(y)N(x)y)N(xE,Ey)(x, +!+"#$
D fi iti S i
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
89/207
Definition: Seminorm
Let E be a vector space and N : E!Ra function.
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
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
90/207
Definition: Seminorm
Let E be a vector space and N : E!Ra function.
N is a seminorm on E if
i.
ii.
iii.
Nevertheless, assertion (ii) implies N(0)=0.
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
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
91/207
Definition: Seminorm
Let E be a vector space and N : E!Ra function.
N is a seminorm on E if
i.
ii.
iii.
Nevertheless, assertion (ii) implies N(0)=0.
(E,N) is a seminormed vector space.
N(x)||x)N(R,E)(x, !!! ="#$N(y)N(x)y)N(xE,Ey)(x, +!+"#$
!x " E, N(x) = 0# x = 0
E l f
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
92/207
Examples of norms
Consider E = #"= { bounded sequences }
For u in E, define N(u) = sup { |ui|, i&N}
E l f
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
93/207
Examples of norms
Consider E = #"= { bounded sequences }
For u in E, define N(u) = sup { |ui|, i&N}
N(u) = 0 iff u=0
E amples of norms
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
94/207
Examples of norms
Consider E = #"= { bounded sequences }
For u in E, define N(u) = sup { |ui|, i&N}
N(u) = 0 iff u=0N($u) = |$|N(u), for any real number $
Examples of norms
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
95/207
Examples of norms
Consider E = #"= { bounded sequences }
For u in E, define N(u) = sup { |ui|, i&N}
N(u) = 0 iff u=0N($u) = |$|N(u), for any real number $
N(u+v) %N(u)+N(v)
Examples of norms
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
96/207
Examples of norms
Consider E = #"= { bounded sequences }
For u in E, define N(u) = sup { |ui|, i&N}
N(u) = 0 iff u=0N($u) = |$|N(u), for any real number $
N(u+v) %N(u)+N(v)
(E,N) is a normed space.
Examples of norms
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
97/207
Examples of norms
Consider E=Rnwith n in N* and p in [1,"[
(X,0)di p
p
n
1i
p
p |X|||X|| == !=
(X,0)d|X|max iNn][1,i
||X|| !"#!
==
Examples of norms
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
98/207
Examples of norms
Consider E=Rnwith n in N* and p in [1,"[
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
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
99/207
In-Video Quiz
Consider E = { functions from Rto Rdefined in 0 }
For f in E, define N(f) = d(f,0) = |f(0)|
In Video Quiz
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
100/207
In-Video Quiz
Consider E = { functions from Rto Rdefined in 0 }
For f in E, define N(f) = d(f,0) = |f(0)|
Is N a norm on E?
In Video Quiz
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
101/207
In-Video Quiz
Consider E = { functions from Rto Rdefined in 0 }
For f in E, define N(f) = d(f,0) = |f(0)|
Is N a norm on E?No. It is a seminorm only:
||x!x2||=0
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
102/207
UNDERLYING METRIC ANDTOPOLOGY IN A NORMED
SPACE
Week 2
Lecture 2.6
From a Normed to a Metric Space
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
103/207
Let (E,N) be a normed space.
From a Normed to a Metric Space
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
104/207
From a Normed to a Metric Space
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
105/207
Let (E,N) be a normed space.
Given x and y in E,
define d(x,y) = N(y-x)
(E,d) is a metric space.
From a Normed to a Metric Space
From a Normed to a Metric Space
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
106/207
Let (E,N) be a normed space.
Given x and y in E,
define d(x,y) = N(y-x)
(E,d) is a metric space.
The unit open ball associated to the norm is
From a Normed to a Metric Space
1}N(x)|E{yB(0,1)B
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
107/207
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
108/207
From a Normed to a Metric Space
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
109/207
Let (E,N) be a normed space.
Given x and y in E,
define d(x,y) = N(y-x)
(E,d) is a metric space.
The unit open ball associated to the norm is
From a Normed to a Metric Space
1}N(x)|E{yB(0,1)B
Norm 2
From a Normed to a Metric Space
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
110/207
Let (E,N) be a normed space.
Given x and y in E,
define d(x,y) = N(y-x)
(E,d) is a metric space.
The unit open ball associated to the norm is
From a Normed to a Metric Space
1}N(x)|E{yB(0,1)B
Norm '
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
111/207
From a Normed to a Metric Space
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
112/207
From a Normed to a Metric Space
Given a normed space,we can derive an associated metric space.
d(x,y)=||x-y||
From a Normed to a Metric Space
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
113/207
From a Normed to a Metric Space
Given a normed space,we can derive an associated metric space.
d(x,y)=||x-y||
Given a metric space,
It may not be a linear space.
From a Normed to a Metric Space
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
114/207
From a Normed to a Metric Space
Given a normed space,we can derive an associated metric space.
d(x,y)=||x-y||
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.
From a Normed to a Metric Space
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
115/207
From a Normed to a Metric Space
Given a normed space,we can derive an associated metric space.
d(x,y)=||x-y||
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
Topological, Metric and Normed Vector Spaces
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
116/207
Topological, Metric and Normed Vector Spaces
Topological, Metric and Normed Vector Spaces
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
117/207
Topological, Metric and Normed Vector Spaces
Topological
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
118/207
Topological, Metric and Normed Vector Spaces
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
119/207
opo og ca , et c a d o ed ecto Spaces
Normed Vector
Metric
Topological
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
120/207
From a Normed to a Metric Space
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
121/207
From a Normed to a Metric Space
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
Converging Sequences
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
122/207
g g q
Let (X,N) be a normed space.
Converging Sequences
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
123/207
g g q
Let (X,N) be a normed space.Let (xn) be a sequence of elements of X.
Converging Sequences
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
124/207
g g q
Let (X,N) be a normed space.Let (xn) be a sequence of elements of X.
We say that (xn) convergesto l if
Converging Sequences
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
125/207
g g q
Let (X,N) be a normed space.Let (xn) be a sequence of elements of X.
We say that (xn) convergesto l ifVxNnN,N(l),V n!"#!$!% V
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
126/207
Converging Sequences
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
127/207
g g q
Let (X,N) be a normed space.Let (xn) be a sequence of elements of X.
We say that (xn) convergesto l if!!> 0, "N# N, n $N%|| x
n& l ||< !
Remark
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
128/207
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
129/207
Remark
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
130/207
Remember last week?
The norm is a continuous function.
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
131/207
Remark
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
132/207
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
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
133/207
Let E be a vector space.
Definition: Strength of a Norm
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
134/207
Let E be a vector space.
Nais stronger than Nbif
there exists a non-negative constant Casuch that for all x in E,
Nb(x) %CaNa(x)
Definition: Strength of a Norm
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
135/207
Let E be a vector space.
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)
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
136/207
Definition: Norm Equivalence
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
137/207
Let E be a vector space.
Naand Nbare equivalentif
there exist two non-negative constants C1and C2
such that for all x in E,
C1Na(x) %Nb(x) %C2Na(x)
Definition: Norm Equivalence
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
138/207
Let E be a vector space.
Naand Nbare equivalentif
there exist two non-negative constants C1and C2
such that for all x in E,
C1Na(x) %Nb(x) %C2Na(x)
Norms are equivalent iff:
associated balls can be included in one another(after a possible homothetic transformation).
Definition: Norm Equivalence
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
139/207
Let E be a vector space.
Naand Nbare equivalentif
there exist two non-negative constants C1and C2
such that for all x in E,
C1Na(x) %Nb(x) %C2Na(x)
Norms are equivalent iff:
associated balls can be included in one another(after a possible homothetic transformation).
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
140/207
Norm Equivalence
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
141/207
Theorem
Let E be a finite-dimensional vector space.
All norms on E are equivalent.
Norm Equivalence
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
142/207
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
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
143/207
There is only one topology induced by the norms.It is called the usual topology of Rn.
What do the open sets look like?
The usual topology of Rn
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
144/207
There is only one topology induced by the norms.It is called the usual topology of Rn.
What do the open sets look like?
The usual topology of Rn
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
145/207
There is only one topology induced by the norms.It is called the usual topology of Rn.
What do the open sets look like?
The usual topology of Rn
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
146/207
There is only one topology induced by the norms.It is called the usual topology of Rn.
What do the open sets look like?
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
147/207
Norm Equivalence
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
148/207
Recall
A stronger norm will provide a stronger topology.
O}(x)B0,rO,x|E{OT r !>"#$!=
r}||y-x|||X{y(x)Br
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
149/207
AN EXAMPLE OF A NORMED
SPACE: LP
Week 2
Lecture 2.7
Integration
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
150/207
The Rieman Integral:Subdivision of the x-axis
Integration
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
151/207
The Rieman Integral:Subdivision of the x-axis
Adding up the surface area
of rectangles
Integration
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
152/207
The Rieman Integral:Subdivision of the x-axis
Adding up the surface area
of rectangles
Integration
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
153/207
The Rieman Integral:Subdivision of the x-axis
Adding up the surface area
of rectangles
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
154/207
Integration
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
155/207
The Rieman Integral:Subdivision of the x-axis
Adding up the surface area
of rectangles
The Lebesgue Integral:
Subdivision of the y-axis
The inverse image ismeasuredand added up
Integration
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
156/207
The Rieman Integral:Subdivision of the x-axis
Adding up the surface area
of rectangles
The Lebesgue Integral:
Subdivision of the y-axis
The inverse image ismeasuredand added up
Integration
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
157/207
The Rieman Integral:Subdivision of the x-axis
Adding up the surface area
of rectangles
The Lebesgue Integral:
Subdivision of the y-axis
The inverse image ismeasuredand added up
Integration
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
158/207
The Rieman Integral:Subdivision of the x-axis
Adding up the surface area
of rectangles
The Lebesgue Integral:
Subdivision of the y-axis
The inverse image ismeasuredand added up
Integration
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
159/207
The Rieman Integral:Subdivision of the x-axis
Adding up the surface area
of rectangles
The Lebesgue Integral:
Subdivision of the y-axis
The inverse image ismeasuredand added up
Integration
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
160/207
The Rieman Integral:Subdivision of the x-axis
Adding up the surface area
of rectangles
The Lebesgue Integral:
Subdivision of the y-axis
The inverse image ismeasuredand added up
Integration
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
161/207
The Rieman Integral:Subdivision of the x-axis
Adding up the surface area
of rectangles
The Lebesgue Integral:
Subdivision of the y-axis
The inverse image ismeasuredand added up
Integration
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
162/207
The Rieman Integral:Subdivision of the x-axis
Adding up the surface area
of rectangles
The Lebesgue Integral:
Subdivision of the y-axis
The inverse image ismeasuredand added up
Integration
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
163/207
The Rieman Integral:Subdivision of the x-axis
Adding up the surface area
of rectangles
The Lebesgue Integral:
Subdivision of the y-axis
The inverse image ismeasuredand added up
Measure
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
164/207
Measure
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
165/207
Measure
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
166/207
Measure
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
167/207
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
168/207
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
169/207
Integration
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
170/207
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
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
171/207
Let p '[1,"[.
We note Lp(() the set of measurable functions
from(
to Rwhose p-th power belongs to L1
((
).
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
172/207
Definition
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
173/207
Let p '[1,"[.
We note Lp(() the set of measurable functions
from(
to Rwhose p-th power belongs to L1
((
).Functions equal almost everywhere are identified.
We note Lpwhen no confusion is possible.
Definition
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
174/207
Let p '[1,"[.
We note Lp(() the set of measurable functions
from(
to Rwhose p-th power belongs to L1
((
).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
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
175/207
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
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
176/207
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.
Functions equal almost everywhere are identified.
Definition
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
177/207
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.
Functions equal almost everywhere are identified.We note L"when no confusion is possible.
Definition
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
178/207
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.
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
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
179/207
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
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
180/207
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
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
181/207
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(if p=1 then p="and p="then p=1)
Definition
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
182/207
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(if p=1 then p="and p="then p=1)
Note that the Hlder conjugate of 2 is 2.
Norm on Lp
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
183/207
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
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
184/207
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
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
185/207
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
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
186/207
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
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
187/207
DENSITY
Week 2
Lecture 2.8
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
188/207
Approximation of +
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
189/207
+= 3.14159265358979323846264
33832
Approximation of +
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
190/207
+= 3.14159265358979323846264
33832
+ Q
Johann Heinrich Lambert
Photo Credit: Wikimedia Commons
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
191/207
Approximation of +
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
192/207
+= 3.14159265358979323846264
33832
+ Q
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
193/207
Approximation of +
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
194/207
+= 3.14159265358979323846264
33832
+ Q
0 1 2 3 4+
+
Approximation of +
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
195/207
+= 3.14159265358979323846264
33832
+ Q
0 1 2 3 4+
+a b
Approximation of +
-
8/10/2019 Lec. 2 - Metric and Normed Spaces
196/207
+= 3.14159265358979323846264
33832
+ Q
3.14