Metric Spaces - Linear Functional Analysis - 1 | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

 Page 1


1. Metric spaces
Denition 1.1. A metric space is a pair (X;d) consisting of a non-empty
set X and a map d :XX!R such that for all x;y;z2X,
(i) d(x;y) 0
(ii) d(x;y) = 0 if and only if x =y
(iii) d(x;y) =d(y;x)
(iv) d(x;z)d(x;y) +d(y;z) (the Triangle Inequality).
We will call the elements of X points. The mapping d is called a metric
and we can think of d(x;y) as the distance between two points x and y.
Our goal is to develop a theory for metric spaces which we can apply in a
variety of dierent situations. Our rst examples of metric spaces are the
Euclidean spacesR
n
.
Page 2


1. Metric spaces
Denition 1.1. A metric space is a pair (X;d) consisting of a non-empty
set X and a map d :XX!R such that for all x;y;z2X,
(i) d(x;y) 0
(ii) d(x;y) = 0 if and only if x =y
(iii) d(x;y) =d(y;x)
(iv) d(x;z)d(x;y) +d(y;z) (the Triangle Inequality).
We will call the elements of X points. The mapping d is called a metric
and we can think of d(x;y) as the distance between two points x and y.
Our goal is to develop a theory for metric spaces which we can apply in a
variety of dierent situations. Our rst examples of metric spaces are the
Euclidean spacesR
n
.
1.1. Euclidean spaces. We denote by R
n
the set of ordered n-tuples of
real numbers,
R
n
= f(x
1
;x
2
;:::;x
n
) :x
1
;x
2
;:::;x
n
2Rg
= R
 n!



 R
R
n
is a vector space (overR) with the following operations of addition and
scalar multiplication: If x = (x
1
;:::;x
n
) and y = (y
1
;:::;y
n
) are points in
R
n
then
x + y = (x
1
+y
1
;:::;x
n
+y
n
)
If 2R is a scalar then
 x = (x
1
;:::;x
n
)
The dot product of x and y is
x
 y =x
1
y
1
+


 +x
n
y
n
The Euclidean norm of a point x = (x
1
;:::;x
n
) is
kxk =
p
x
 x
=
q
x
2
1
+


 +x
2
n
Theorem 1.2. (Cauchy-Schwarz inequality) Let x = (x
1
;:::;x
n
) and y =
(y
1
;:::;y
n
) be points inR
n
. Then
jx
 yjkxkkyk
Corollary 1.3. Let x = (x
1
;:::;x
n
) and y = (y
1
;:::;y
n
) be points inR
n
.
Then
kx + ykkxk +kyk
Page 3


1. Metric spaces
Denition 1.1. A metric space is a pair (X;d) consisting of a non-empty
set X and a map d :XX!R such that for all x;y;z2X,
(i) d(x;y) 0
(ii) d(x;y) = 0 if and only if x =y
(iii) d(x;y) =d(y;x)
(iv) d(x;z)d(x;y) +d(y;z) (the Triangle Inequality).
We will call the elements of X points. The mapping d is called a metric
and we can think of d(x;y) as the distance between two points x and y.
Our goal is to develop a theory for metric spaces which we can apply in a
variety of dierent situations. Our rst examples of metric spaces are the
Euclidean spacesR
n
.
1.1. Euclidean spaces. We denote by R
n
the set of ordered n-tuples of
real numbers,
R
n
= f(x
1
;x
2
;:::;x
n
) :x
1
;x
2
;:::;x
n
2Rg
= R
 n!



 R
R
n
is a vector space (overR) with the following operations of addition and
scalar multiplication: If x = (x
1
;:::;x
n
) and y = (y
1
;:::;y
n
) are points in
R
n
then
x + y = (x
1
+y
1
;:::;x
n
+y
n
)
If 2R is a scalar then
 x = (x
1
;:::;x
n
)
The dot product of x and y is
x
 y =x
1
y
1
+


 +x
n
y
n
The Euclidean norm of a point x = (x
1
;:::;x
n
) is
kxk =
p
x
 x
=
q
x
2
1
+


 +x
2
n
Theorem 1.2. (Cauchy-Schwarz inequality) Let x = (x
1
;:::;x
n
) and y =
(y
1
;:::;y
n
) be points inR
n
. Then
jx
 yjkxkkyk
Corollary 1.3. Let x = (x
1
;:::;x
n
) and y = (y
1
;:::;y
n
) be points inR
n
.
Then
kx + ykkxk +kyk
Corollary 1.4. The mapping d :R
n
R
n
!R dened by
d(x; y) =kx yk
is a metric onR
n
.
The metric d dened in Corollary 1.4 is called the Euclidean metric on
R
n
. We calld(x; y) the Euclidean distance between the points x and y. The
metric space (R
n
;d) will be called n-dimensional Euclidean space. Unless
otherwise stated it can be assumed thatR
n
denotesn-dimensional Euclidean
space.
Note that by expanding out the Euclidean norm we get
d(x; y) = kx yk
=
p
(x
1
y
1
)
2
+ (x
2
y
2
)
2
+


 + (x
n
y
n
)
2
Example 1.5. (a) The Euclidean metric onR. For real numbers x;y2R
the Euclidean distance is expressed in terms of absolute value
d(x;y) =jxyj
(b) The Euclidean metric on R
2
. We can think of the elements of R
2
as
coordinates for points in the plane. The Euclidean distance between
two points x = (x
1
;x
2
) and y = (y
1
;y
2
) is
d(x; y) =
p
(x
1
y
1
)
2
+ (x
2
y
2
)
2
(c) The Euclidean metric on R
3
. We can think of the elements of R
3
as
coordinates for points in space. The Euclidean distance between two
points x = (x
1
;x
2
;x
3
) and y = (y
1
;y
2
;y
3
) is
d(x; y) =
p
(x
1
y
1
)
2
+ (x
2
y
2
)
2
+ (x
3
y
3
)
2
Page 4


1. Metric spaces
Denition 1.1. A metric space is a pair (X;d) consisting of a non-empty
set X and a map d :XX!R such that for all x;y;z2X,
(i) d(x;y) 0
(ii) d(x;y) = 0 if and only if x =y
(iii) d(x;y) =d(y;x)
(iv) d(x;z)d(x;y) +d(y;z) (the Triangle Inequality).
We will call the elements of X points. The mapping d is called a metric
and we can think of d(x;y) as the distance between two points x and y.
Our goal is to develop a theory for metric spaces which we can apply in a
variety of dierent situations. Our rst examples of metric spaces are the
Euclidean spacesR
n
.
1.1. Euclidean spaces. We denote by R
n
the set of ordered n-tuples of
real numbers,
R
n
= f(x
1
;x
2
;:::;x
n
) :x
1
;x
2
;:::;x
n
2Rg
= R
 n!



 R
R
n
is a vector space (overR) with the following operations of addition and
scalar multiplication: If x = (x
1
;:::;x
n
) and y = (y
1
;:::;y
n
) are points in
R
n
then
x + y = (x
1
+y
1
;:::;x
n
+y
n
)
If 2R is a scalar then
 x = (x
1
;:::;x
n
)
The dot product of x and y is
x
 y =x
1
y
1
+


 +x
n
y
n
The Euclidean norm of a point x = (x
1
;:::;x
n
) is
kxk =
p
x
 x
=
q
x
2
1
+


 +x
2
n
Theorem 1.2. (Cauchy-Schwarz inequality) Let x = (x
1
;:::;x
n
) and y =
(y
1
;:::;y
n
) be points inR
n
. Then
jx
 yjkxkkyk
Corollary 1.3. Let x = (x
1
;:::;x
n
) and y = (y
1
;:::;y
n
) be points inR
n
.
Then
kx + ykkxk +kyk
Corollary 1.4. The mapping d :R
n
R
n
!R dened by
d(x; y) =kx yk
is a metric onR
n
.
The metric d dened in Corollary 1.4 is called the Euclidean metric on
R
n
. We calld(x; y) the Euclidean distance between the points x and y. The
metric space (R
n
;d) will be called n-dimensional Euclidean space. Unless
otherwise stated it can be assumed thatR
n
denotesn-dimensional Euclidean
space.
Note that by expanding out the Euclidean norm we get
d(x; y) = kx yk
=
p
(x
1
y
1
)
2
+ (x
2
y
2
)
2
+


 + (x
n
y
n
)
2
Example 1.5. (a) The Euclidean metric onR. For real numbers x;y2R
the Euclidean distance is expressed in terms of absolute value
d(x;y) =jxyj
(b) The Euclidean metric on R
2
. We can think of the elements of R
2
as
coordinates for points in the plane. The Euclidean distance between
two points x = (x
1
;x
2
) and y = (y
1
;y
2
) is
d(x; y) =
p
(x
1
y
1
)
2
+ (x
2
y
2
)
2
(c) The Euclidean metric on R
3
. We can think of the elements of R
3
as
coordinates for points in space. The Euclidean distance between two
points x = (x
1
;x
2
;x
3
) and y = (y
1
;y
2
;y
3
) is
d(x; y) =
p
(x
1
y
1
)
2
+ (x
2
y
2
)
2
+ (x
3
y
3
)
2
1.2. More examples of metric spaces.
Example 1.6. Let X be any non-empty set.
The discrete metric on X is dened by
d(x;y) =
8
<
:
0 if x =y
1 if x6=y
for all points x;y in X.
Example 1.7. Let x = (x
1
;x
2
) and y = (y
1
;y
2
) be points inR
2
.
(a) The taxi-cab metric onR
2
is dened by
d(x; y) =jx
1
y
1
j +jx
2
y
2
j
(b) The Irish rail metric onR
2
is dened by
d
0
(x; y) =
8
<
:
0 if x = y
d(x; 0) +d(0; y) if x6= y
where 0 = (0; 0) is the origin inR
2
andd is the Euclidean metric onR
2
.
Example 1.8. The complex numbers.
(C;d) is a metric space where d is dened by
d(z;w) =jzwj; 8z;w2C
Example 1.9. A function space.
Let C[0; 1] be the set of all continuous functions f : [0; 1]!R.
The following dene two dierent metrics on C[0; 1],
(a) d(f;g) = sup
x2[0;1]
jf(x)g(x)j
(b) d(f;g) =
R
1
0
jf(x)g(x)jdx
for all f;g2C[0; 1].
Page 5


1. Metric spaces
Denition 1.1. A metric space is a pair (X;d) consisting of a non-empty
set X and a map d :XX!R such that for all x;y;z2X,
(i) d(x;y) 0
(ii) d(x;y) = 0 if and only if x =y
(iii) d(x;y) =d(y;x)
(iv) d(x;z)d(x;y) +d(y;z) (the Triangle Inequality).
We will call the elements of X points. The mapping d is called a metric
and we can think of d(x;y) as the distance between two points x and y.
Our goal is to develop a theory for metric spaces which we can apply in a
variety of dierent situations. Our rst examples of metric spaces are the
Euclidean spacesR
n
.
1.1. Euclidean spaces. We denote by R
n
the set of ordered n-tuples of
real numbers,
R
n
= f(x
1
;x
2
;:::;x
n
) :x
1
;x
2
;:::;x
n
2Rg
= R
 n!



 R
R
n
is a vector space (overR) with the following operations of addition and
scalar multiplication: If x = (x
1
;:::;x
n
) and y = (y
1
;:::;y
n
) are points in
R
n
then
x + y = (x
1
+y
1
;:::;x
n
+y
n
)
If 2R is a scalar then
 x = (x
1
;:::;x
n
)
The dot product of x and y is
x
 y =x
1
y
1
+


 +x
n
y
n
The Euclidean norm of a point x = (x
1
;:::;x
n
) is
kxk =
p
x
 x
=
q
x
2
1
+


 +x
2
n
Theorem 1.2. (Cauchy-Schwarz inequality) Let x = (x
1
;:::;x
n
) and y =
(y
1
;:::;y
n
) be points inR
n
. Then
jx
 yjkxkkyk
Corollary 1.3. Let x = (x
1
;:::;x
n
) and y = (y
1
;:::;y
n
) be points inR
n
.
Then
kx + ykkxk +kyk
Corollary 1.4. The mapping d :R
n
R
n
!R dened by
d(x; y) =kx yk
is a metric onR
n
.
The metric d dened in Corollary 1.4 is called the Euclidean metric on
R
n
. We calld(x; y) the Euclidean distance between the points x and y. The
metric space (R
n
;d) will be called n-dimensional Euclidean space. Unless
otherwise stated it can be assumed thatR
n
denotesn-dimensional Euclidean
space.
Note that by expanding out the Euclidean norm we get
d(x; y) = kx yk
=
p
(x
1
y
1
)
2
+ (x
2
y
2
)
2
+


 + (x
n
y
n
)
2
Example 1.5. (a) The Euclidean metric onR. For real numbers x;y2R
the Euclidean distance is expressed in terms of absolute value
d(x;y) =jxyj
(b) The Euclidean metric on R
2
. We can think of the elements of R
2
as
coordinates for points in the plane. The Euclidean distance between
two points x = (x
1
;x
2
) and y = (y
1
;y
2
) is
d(x; y) =
p
(x
1
y
1
)
2
+ (x
2
y
2
)
2
(c) The Euclidean metric on R
3
. We can think of the elements of R
3
as
coordinates for points in space. The Euclidean distance between two
points x = (x
1
;x
2
;x
3
) and y = (y
1
;y
2
;y
3
) is
d(x; y) =
p
(x
1
y
1
)
2
+ (x
2
y
2
)
2
+ (x
3
y
3
)
2
1.2. More examples of metric spaces.
Example 1.6. Let X be any non-empty set.
The discrete metric on X is dened by
d(x;y) =
8
<
:
0 if x =y
1 if x6=y
for all points x;y in X.
Example 1.7. Let x = (x
1
;x
2
) and y = (y
1
;y
2
) be points inR
2
.
(a) The taxi-cab metric onR
2
is dened by
d(x; y) =jx
1
y
1
j +jx
2
y
2
j
(b) The Irish rail metric onR
2
is dened by
d
0
(x; y) =
8
<
:
0 if x = y
d(x; 0) +d(0; y) if x6= y
where 0 = (0; 0) is the origin inR
2
andd is the Euclidean metric onR
2
.
Example 1.8. The complex numbers.
(C;d) is a metric space where d is dened by
d(z;w) =jzwj; 8z;w2C
Example 1.9. A function space.
Let C[0; 1] be the set of all continuous functions f : [0; 1]!R.
The following dene two dierent metrics on C[0; 1],
(a) d(f;g) = sup
x2[0;1]
jf(x)g(x)j
(b) d(f;g) =
R
1
0
jf(x)g(x)jdx
for all f;g2C[0; 1].
Example 1.10. A sequence space.
Letc
0
be the set of all sequences (x
k
)
1
k=1
of real numbers which converge to
0. Then (c
0
;d) is a metric space where we dene
d(x; y) = sup
k
jx
k
y
k
j
for all points x = (x
k
)
1
k=1
and y = (y
k
)
1
k=1
in c
0
.
Example 1.11. Subspaces.
If (X;d) is a metric space and A is a subset of X then (A;d
A
) is a metric
space where we dene
d
A
(x;y) =d(x;y) 8x;y2A
(A;d
A
) is called a subspace of (X;d) and d
A
is called the induced metric.