SCHOOL OF MATHEMATICS AND STATISTICS
Autumn Semester 20089
PMA307 Metric Spaces
2 hours 30 minutes
Answer four questions. You are advised
do, only your best four will be counted.
1
(i)
(a)
(b)
to answer more than four questions: if you
not
Dene the supremum metric d1 on the set C [0; 1] of continuous
functions on [0; 1] and prove that it is a metric.
Show that if
2 [0; 1].
fn
x
!f
in
(C [0; 1]; d1 )
then
fn (x )
! f (x ) for any
(12 marks)
(ii)
(a)
Let
(X; d )
(b)
Let
a; b: [0; 1]
Da;b
be a metric space. What is a closed subset of
(X; d )?
! R be functions and dene
= ff 2 C [0; 1] j a(x ) 6 f (x ) 6 b(x ) for all x 2 [0; 1]g:
Use (i)(b) to show that
Da;b
is a closed subset of
(C [0; 1]; d1 ).
(8 marks)
(iii)
Let
(fn )
be a sequence in
x2
and suppose that
fn
(C [0; 1]; d1 )
6 f (x ) 6 x 2
n
with the property that
for all x
2 [0; 1]
! f . Show that
6
3
1
Z 1
0
f (x )d x
6 13 :
(5 marks)
1
Turn Over
2
(i)
Let
(xn )
be a sequence in a metric space
(a)
Show that
(b)
Let
xnk
(xn )
has at most one limit.
be a subsequence of
also.
(xnk )
!x
(X; d ).
(xn ).
Show that if
xn
!
then
x
(10 marks)
(ii)
Now let R be equipped with its usual metric and consider the sequence (x
dened by
n
xn = cos
n
2
+
2n
)
:
[Hint: In the following parts you may use the fact that cos(y ) ! 1 whenever (y ) is a sequence that converges to 0. You may also use the fact
that cos(2n + ) = cos() and cos((2n + 1) + ) = cos().]
n
n
(a)
Show that the subsequence
(x4k )
converges to 1.
(b)
Show that the subsequence
(x4k +2 )
(c)
Use parts (i)(b), (ii)(a) and (ii)(b) to show that (x
a limit.
converges to
1.
n
)
does not have
(11 marks)
(iii)
Show carefully that no subsequence of the sequence (x
converges to any limit.
n
)
dened by x
n
=n
(4 marks)
3
Throughout this question R2 is given the usual Euclidean metric d2 .
(i)
(a)
Let (x1 ; y1 ); (x2 ; y2 ); (x3 ; y3 ); : : : be a sequence in (R2 ; d2 ). Show
that (x ; y ) ! (x ; y ) if and only if x ! x and y ! y .
n
(b)
n
n
n
Use part (i)(a)
to show, in terms of and N , that the sequence
n
1 1
;
n+1 n
converges to (1,0).
(15 marks)
(ii)
(a)
Dene, in terms of convergence of sequences, what it means for a
function f : (X; d ) ! (Y; d ) to be continuous.
Let f : R ! R2 be dened by
X
(b)
Y
f (x ) = (f1 (x ); f2 (x ))
for functions f1 ; f2 : R ! R. Use part (i )(a) to show that
continuous if and only if f1 and f2 are continuous.
f
is
(10 marks)
2
Continued
4
(i)
(a)
(b)
(c)
What does it mean for a sequence (x ) in a metric space (X; d ) to
be Cauchy ?
What does it mean for (X; d ) to be complete?
State without explanation which of the metric spaces (C [0; 1]; d1 ),
(C [0; 1]; d1 ) are complete.
n
(6 marks)
(ii)
Let
(fn )
be the sequence in
dened by
(C [0; 1]; d1 )
fn (x ) = 1 +
x
2
1
x2
+
+
22
xn
+ 2
1
n
n.
(a)
Show that d1 (f
(b)
Show that
(c)
1 ! 0.]
Indicate briey why the sequence (f ) converges.
n
; fm ) =
1
2n
2m
n
for
:
m
is a Cauchy sequence.
(fn )
[Hint: You may assume that
1
2
n
n
(10 marks)
(iii)
Let
(gn )
be the sequence in
gn (x ) =




(C [0; 1]; d1 )
2n
x



(a)
Show that d1 (g
n
; gm ) =
0
1
2
1
dened by
6 x 6 21 21 ;
1 1 6 x 6 1;
2 2
2
1 6 x 6 1:
2
0
+
1
2n
1
1
4n
4m
n
for
n.
g (x )j = g (x )
m
[Hint: You may assume that jg (x )
m n. You might nd it useful to sketch the graph of
Show that (g ) is a Cauchy sequence.
n
(b)
n
m
n
gm (x )
gn .]
for
n
(9 marks)
5
(i)
(ii)
(a)
(b)
What is a contraction of a metric space (X; d )?
State and prove the Contraction Mapping Principle.
Consider the function
(a)
(b)
(c)
(18 marks)
1) ! [1; 1) dened by f (x ) = x + x1 .
f (y )j < jx
y j for all x ; y 2 [1; 1).
f : [1;
Show that jf (x )
Show that f does not have a xed point.
Indicate briey why this does not contradict the contraction mapping principle.
(7 marks)
End of Question Paper
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