Metric Spaces - Linear Functional Analysis - 2 | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

 Page 1


1.4. Bounded sets.
Denition 1.16. Let (X;d) be a metric space and letA be a subset ofX.
Dene the diameter of A to be
diam(A) = sup
x;y2A
d(x;y)
or if this supremum does not exist then diam(A) =1.
Theorem 1.17. LetB(x;r) be an open ball in a metric space (X;d). Then
the diameter of B(x;r) is 2r.
Example 1.18. The diameter of an open ball B(x;r) in Euclidean space
R
n
is 2r.
Example 1.19. Let d be the discrete metric on a set X which contains at
least two points. Then for eachx2X the open ballB(x;r) has diameter 0
if r 1 and diameter 1 if r> 1.
Denition 1.20. Let (X;d) be a metric space and letA be a subset ofX.
Then A is called a bounded set if there exists an open ball B(x;r) which
contains A.
Theorem 1.21. Let (X;d) be a metric space and let A be a subset of X.
Then A is a bounded set if and only if A has nite diameter.
Page 2


1.4. Bounded sets.
Denition 1.16. Let (X;d) be a metric space and letA be a subset ofX.
Dene the diameter of A to be
diam(A) = sup
x;y2A
d(x;y)
or if this supremum does not exist then diam(A) =1.
Theorem 1.17. LetB(x;r) be an open ball in a metric space (X;d). Then
the diameter of B(x;r) is 2r.
Example 1.18. The diameter of an open ball B(x;r) in Euclidean space
R
n
is 2r.
Example 1.19. Let d be the discrete metric on a set X which contains at
least two points. Then for eachx2X the open ballB(x;r) has diameter 0
if r 1 and diameter 1 if r> 1.
Denition 1.20. Let (X;d) be a metric space and letA be a subset ofX.
Then A is called a bounded set if there exists an open ball B(x;r) which
contains A.
Theorem 1.21. Let (X;d) be a metric space and let A be a subset of X.
Then A is a bounded set if and only if A has nite diameter.
1.5. Open sets.
Denition 1.22. Let (X;d) be a metric space and letA be a subset ofX.
A point x2 A is called an interior point of A if there exists an open ball
B(x;r) with centre x which is contained in A.
A subset A of X is called an open set if every point in A is an interior
point of A.
Theorem 1.23. Let (X;d) be a metric space. Every open ball in (X;d) is
an open set.
Theorem 1.24. Let (X;d) be a metric space. Then
(i); and X are open sets,
(ii) the union of any collection of open sets is an open set,
(iii) the intersection of any nite collection of open sets is an open set.
Example 1.25. Part (iii) of Theorem 1.24 does not extend to innite
collections. For example, consider the 1-dimensional Euclidean space R.
For each n, the open interval (
1
n
;
1
n
) is an open set. However,
1
\
n=1


1
n
;
1
n

=f0g
is not an open set.
The interior ofA, denotedint(A) orA

, is dened as the set of all interior
points ofA. The collection of all open sets in a metric space (X;d) is called
the metric topology on X.
Example 1.26. InR every open interval (a;b) is an open set. Intervals of
the form (a;b], [a;b), [a;b] are not open sets. A set consisting of a single
point is not an open set. The interior of the closed interval [a;b] is the open
interval (a;b).
Page 3


1.4. Bounded sets.
Denition 1.16. Let (X;d) be a metric space and letA be a subset ofX.
Dene the diameter of A to be
diam(A) = sup
x;y2A
d(x;y)
or if this supremum does not exist then diam(A) =1.
Theorem 1.17. LetB(x;r) be an open ball in a metric space (X;d). Then
the diameter of B(x;r) is 2r.
Example 1.18. The diameter of an open ball B(x;r) in Euclidean space
R
n
is 2r.
Example 1.19. Let d be the discrete metric on a set X which contains at
least two points. Then for eachx2X the open ballB(x;r) has diameter 0
if r 1 and diameter 1 if r> 1.
Denition 1.20. Let (X;d) be a metric space and letA be a subset ofX.
Then A is called a bounded set if there exists an open ball B(x;r) which
contains A.
Theorem 1.21. Let (X;d) be a metric space and let A be a subset of X.
Then A is a bounded set if and only if A has nite diameter.
1.5. Open sets.
Denition 1.22. Let (X;d) be a metric space and letA be a subset ofX.
A point x2 A is called an interior point of A if there exists an open ball
B(x;r) with centre x which is contained in A.
A subset A of X is called an open set if every point in A is an interior
point of A.
Theorem 1.23. Let (X;d) be a metric space. Every open ball in (X;d) is
an open set.
Theorem 1.24. Let (X;d) be a metric space. Then
(i); and X are open sets,
(ii) the union of any collection of open sets is an open set,
(iii) the intersection of any nite collection of open sets is an open set.
Example 1.25. Part (iii) of Theorem 1.24 does not extend to innite
collections. For example, consider the 1-dimensional Euclidean space R.
For each n, the open interval (
1
n
;
1
n
) is an open set. However,
1
\
n=1


1
n
;
1
n

=f0g
is not an open set.
The interior ofA, denotedint(A) orA

, is dened as the set of all interior
points ofA. The collection of all open sets in a metric space (X;d) is called
the metric topology on X.
Example 1.26. InR every open interval (a;b) is an open set. Intervals of
the form (a;b], [a;b), [a;b] are not open sets. A set consisting of a single
point is not an open set. The interior of the closed interval [a;b] is the open
interval (a;b).
1.6. Closed sets.
Denition 1.27. Let (X;d) be a metric space. A sequence in X is a
mapping s :N! X and is usually written as (x
n
) or (x
n
)
1
n=1
where x
n
=
s(n) for each n2N.
A sequence (x
n
) is said to converge to a pointx2X if given any positive
real number > 0 there exists N2N such that
d(x
n
;x) for all nN
The pointx is called the limit of the sequence and we write lim
n!1
x
n
=x.
A sequence (x
n
) is said to be bounded if the setfx
n
:n2Ng is a bounded
set in (X;d).
Theorem 1.28. Every convergent sequence in a metric space is bounded
and has a unique limit.
Example 1.29. A sequence (x
j
) inR
m
converges to a point x2R
m
if and
only if each coordinate sequence converges inR.
Denition 1.30. Let (X;d) be a metric space and letA be a subset ofX.
A point x2X is called a limit point ofA if there exists a sequence (x
n
) in
Anfxg which converges to x.
A is called a closed set if it contains all of its limit points.
Theorem 1.31. Let (X;d) be a metric space and let A be a subset of X.
Then A is a closed set if and only if XnA is an open set.
Theorem 1.32. Let (X;d) be a metric space. Then
(i); and X are closed sets,
(ii) the intersection of any collection of closed sets is a closed set,
(iii) the union of nitely many closed sets is a closed set.
Page 4


1.4. Bounded sets.
Denition 1.16. Let (X;d) be a metric space and letA be a subset ofX.
Dene the diameter of A to be
diam(A) = sup
x;y2A
d(x;y)
or if this supremum does not exist then diam(A) =1.
Theorem 1.17. LetB(x;r) be an open ball in a metric space (X;d). Then
the diameter of B(x;r) is 2r.
Example 1.18. The diameter of an open ball B(x;r) in Euclidean space
R
n
is 2r.
Example 1.19. Let d be the discrete metric on a set X which contains at
least two points. Then for eachx2X the open ballB(x;r) has diameter 0
if r 1 and diameter 1 if r> 1.
Denition 1.20. Let (X;d) be a metric space and letA be a subset ofX.
Then A is called a bounded set if there exists an open ball B(x;r) which
contains A.
Theorem 1.21. Let (X;d) be a metric space and let A be a subset of X.
Then A is a bounded set if and only if A has nite diameter.
1.5. Open sets.
Denition 1.22. Let (X;d) be a metric space and letA be a subset ofX.
A point x2 A is called an interior point of A if there exists an open ball
B(x;r) with centre x which is contained in A.
A subset A of X is called an open set if every point in A is an interior
point of A.
Theorem 1.23. Let (X;d) be a metric space. Every open ball in (X;d) is
an open set.
Theorem 1.24. Let (X;d) be a metric space. Then
(i); and X are open sets,
(ii) the union of any collection of open sets is an open set,
(iii) the intersection of any nite collection of open sets is an open set.
Example 1.25. Part (iii) of Theorem 1.24 does not extend to innite
collections. For example, consider the 1-dimensional Euclidean space R.
For each n, the open interval (
1
n
;
1
n
) is an open set. However,
1
\
n=1


1
n
;
1
n

=f0g
is not an open set.
The interior ofA, denotedint(A) orA

, is dened as the set of all interior
points ofA. The collection of all open sets in a metric space (X;d) is called
the metric topology on X.
Example 1.26. InR every open interval (a;b) is an open set. Intervals of
the form (a;b], [a;b), [a;b] are not open sets. A set consisting of a single
point is not an open set. The interior of the closed interval [a;b] is the open
interval (a;b).
1.6. Closed sets.
Denition 1.27. Let (X;d) be a metric space. A sequence in X is a
mapping s :N! X and is usually written as (x
n
) or (x
n
)
1
n=1
where x
n
=
s(n) for each n2N.
A sequence (x
n
) is said to converge to a pointx2X if given any positive
real number > 0 there exists N2N such that
d(x
n
;x) for all nN
The pointx is called the limit of the sequence and we write lim
n!1
x
n
=x.
A sequence (x
n
) is said to be bounded if the setfx
n
:n2Ng is a bounded
set in (X;d).
Theorem 1.28. Every convergent sequence in a metric space is bounded
and has a unique limit.
Example 1.29. A sequence (x
j
) inR
m
converges to a point x2R
m
if and
only if each coordinate sequence converges inR.
Denition 1.30. Let (X;d) be a metric space and letA be a subset ofX.
A point x2X is called a limit point ofA if there exists a sequence (x
n
) in
Anfxg which converges to x.
A is called a closed set if it contains all of its limit points.
Theorem 1.31. Let (X;d) be a metric space and let A be a subset of X.
Then A is a closed set if and only if XnA is an open set.
Theorem 1.32. Let (X;d) be a metric space. Then
(i); and X are closed sets,
(ii) the intersection of any collection of closed sets is a closed set,
(iii) the union of nitely many closed sets is a closed set.
Example 1.33. InR every closed interval [a;b] is a closed set. The interval
(0; 1] is not closed since 0 is a limit point which is not contained in the set.
Intervals of the form (a;b], [a;b), (a;b) are not closed sets. A setfxg
consisting of a single point is a closed set since it has no limit points.
The closure ofA, denoted

A, is the union ofA and the set of limit points
of A.
Example 1.34. InR, the closure of each of the intervals (a;b], [a;b) and
(a;b) is [a;b]. The closure ofQ isR.
Page 5


1.4. Bounded sets.
Denition 1.16. Let (X;d) be a metric space and letA be a subset ofX.
Dene the diameter of A to be
diam(A) = sup
x;y2A
d(x;y)
or if this supremum does not exist then diam(A) =1.
Theorem 1.17. LetB(x;r) be an open ball in a metric space (X;d). Then
the diameter of B(x;r) is 2r.
Example 1.18. The diameter of an open ball B(x;r) in Euclidean space
R
n
is 2r.
Example 1.19. Let d be the discrete metric on a set X which contains at
least two points. Then for eachx2X the open ballB(x;r) has diameter 0
if r 1 and diameter 1 if r> 1.
Denition 1.20. Let (X;d) be a metric space and letA be a subset ofX.
Then A is called a bounded set if there exists an open ball B(x;r) which
contains A.
Theorem 1.21. Let (X;d) be a metric space and let A be a subset of X.
Then A is a bounded set if and only if A has nite diameter.
1.5. Open sets.
Denition 1.22. Let (X;d) be a metric space and letA be a subset ofX.
A point x2 A is called an interior point of A if there exists an open ball
B(x;r) with centre x which is contained in A.
A subset A of X is called an open set if every point in A is an interior
point of A.
Theorem 1.23. Let (X;d) be a metric space. Every open ball in (X;d) is
an open set.
Theorem 1.24. Let (X;d) be a metric space. Then
(i); and X are open sets,
(ii) the union of any collection of open sets is an open set,
(iii) the intersection of any nite collection of open sets is an open set.
Example 1.25. Part (iii) of Theorem 1.24 does not extend to innite
collections. For example, consider the 1-dimensional Euclidean space R.
For each n, the open interval (
1
n
;
1
n
) is an open set. However,
1
\
n=1


1
n
;
1
n

=f0g
is not an open set.
The interior ofA, denotedint(A) orA

, is dened as the set of all interior
points ofA. The collection of all open sets in a metric space (X;d) is called
the metric topology on X.
Example 1.26. InR every open interval (a;b) is an open set. Intervals of
the form (a;b], [a;b), [a;b] are not open sets. A set consisting of a single
point is not an open set. The interior of the closed interval [a;b] is the open
interval (a;b).
1.6. Closed sets.
Denition 1.27. Let (X;d) be a metric space. A sequence in X is a
mapping s :N! X and is usually written as (x
n
) or (x
n
)
1
n=1
where x
n
=
s(n) for each n2N.
A sequence (x
n
) is said to converge to a pointx2X if given any positive
real number > 0 there exists N2N such that
d(x
n
;x) for all nN
The pointx is called the limit of the sequence and we write lim
n!1
x
n
=x.
A sequence (x
n
) is said to be bounded if the setfx
n
:n2Ng is a bounded
set in (X;d).
Theorem 1.28. Every convergent sequence in a metric space is bounded
and has a unique limit.
Example 1.29. A sequence (x
j
) inR
m
converges to a point x2R
m
if and
only if each coordinate sequence converges inR.
Denition 1.30. Let (X;d) be a metric space and letA be a subset ofX.
A point x2X is called a limit point ofA if there exists a sequence (x
n
) in
Anfxg which converges to x.
A is called a closed set if it contains all of its limit points.
Theorem 1.31. Let (X;d) be a metric space and let A be a subset of X.
Then A is a closed set if and only if XnA is an open set.
Theorem 1.32. Let (X;d) be a metric space. Then
(i); and X are closed sets,
(ii) the intersection of any collection of closed sets is a closed set,
(iii) the union of nitely many closed sets is a closed set.
Example 1.33. InR every closed interval [a;b] is a closed set. The interval
(0; 1] is not closed since 0 is a limit point which is not contained in the set.
Intervals of the form (a;b], [a;b), (a;b) are not closed sets. A setfxg
consisting of a single point is a closed set since it has no limit points.
The closure ofA, denoted

A, is the union ofA and the set of limit points
of A.
Example 1.34. InR, the closure of each of the intervals (a;b], [a;b) and
(a;b) is [a;b]. The closure ofQ isR.
1.7. Continuous mappings.
Denition 1.35. Let (X;d) and (Y;d
0
) be metric spaces. A mapping T :
X! Y is a called continuous at a point x
0
2 X if given any  > 0 there
exists > 0 such that
d(x;x
0
)< =) d
0
(T (x);T(x
0
))<:
T is called continuous if it is continuous at every point of X.
Theorem 1.36. Let (X;d) and (Y;d
0
) be metric spaces. A mapping T :
X ! Y is a continuous mapping if and only if for every sequence (x
n
)
converging to a point x in (X;d), the sequence (T (x
n
)) converges to T (x)
in (Y;d
0
).
Theorem 1.37. Let (X;d) and (Y;d
0
) be metric spaces. A mapping T :
X!Y is continuous if and only if the preimage
T
1
(U) =fx2X :T (x)2Ug
is open in (X;d) for each U open in (Y;d
0
).
Theorem 1.38. The composition of two continuous mappings is a contin-
uous mapping.
Denition 1.39. Let (X;d) and (Y;d
0
) be metric spaces. A mapping T :
X!Y is called an isometry if
d
0
(T (x);T(y)) =d(x;y) for all x;y2X
(i.e. T preserves distances).
Proposition 1.40. Let (X;d) and (Y;d
0
) be metric spaces and letT :X!
Y be an isometry. Then
(i) T is one-to-one,