Page 338 - Advanced Linear Algebra

322 Advanced Linear Algebra

)
Z
a Show that ²4Á ³ is a metric space and that 4 is bounded under this
metric, even if it is not bounded under the metric .

)
b Show that the metric spaces ²4Á ³ and ²4Á ³ have the same open
Z
sets.
:
;
11. If and are subsets of a metric space ² 4 Á ³ , we define the distance
between and by
:
;
²:Á ;³ ~ inf ²%Á &³
%:Á!;
)
a Is it true that if and only if : ~ ; ? Is a metric?
²:Á ;³ ~
)
b Show that %  cl²:³ if and only if ²¸%¹Á :³ ~ .

12. Prove that %4 is a limit point of : ‹4 if and only if every
neighborhood of meets in a point other than itself.
%
%
:
13. Prove that %4 is a limit point of : ‹4 if and only if every open ball
)²%Á ³ contains infinitely many points of :.
,
14. Prove that limits are unique, that is, ²% ³ ¦ % ²% ³ ¦ & implies that

%~&.
15. Let be a subset of a metric space 4 . Prove that  % cl ² : ³ if and only if
:
%
there exists a sequence ²% ³ in that converges to .
:

16. Prove that the closure has the following properties:
a ) :‹ cl²:³
b)cl cl²²:³³ ~ :
c ) cl²: r ;³ ~ cl²:³ r cl²;³
d)cl²: q ;³ ‹ cl²:³ q cl²;³
Can the last part be strengthened to equality?
)
17. a Prove that the closed ball )²%Á ³ is always a closed subset.
b Find an example of a metric space in which the closure of an open ball
)
)²%Á ³ is not equal to the closed ball )²%Á ³.
18. Provide the details to show that s is separable.
19. Prove that d is separable.
20. Prove that a discrete metric space is separable if and only if it is countable.
21. Prove that the metric space 8´ Á µ of all bounded functions on ´ Á µ , with
metric
² Á ³ ~ sup ( ²%³ c ²%³(
%´ Á µ
is not separable.
Z
Z
22. Show that a function ¢ ²4Á ³ ¦ ²4 Á ³ is continuous if and only if the
inverse image of any open set is open, that is, if and only if
c ² < ³ ~ ¸ %  4 “ ² % ³  < ¹ is open in 4 whenever is an open set
<
in 4 Z .
23. Repeat the previous exercise, replacing the word open by the word closed.
Z
Z
24. Give an example to show that if ¢²4Á ³ ¦ ²4 Á ³ is a continuous
<
function and is an open set in 4 , it need not be the case that ² < ³ is
open in 4 Z .