Page 337 - Advanced Linear Algebra
P. 337
Metric Spaces 321
Exercises
1. Prove the generalized triangle inequality
²%Á % ³ ²%Á %³ b ²%Á %³ b Ä b ²% 3 c Á % ³
)
2. a Use the triangle inequality to prove that
( ( ²%Á &³ c ² Á ³ ²%Á ³ b ²&Á ³
b) Prove that
( ( ²%Á '³ c ²&Á '³ ²%Á &³
(
3. Let : M B be the subspace of all binary sequences sequences of 's and
) :'s . Describe the metric on .
4. Let 4 ~ ¸ Á ¹ be the set of all binary -tuples. Define a function
¢ : d : ¦ s by letting ²%Á &³ be the number of positions in which % and
(
& differ. For example, ´² ³Á ² ³µ ~ . Prove that is a metric. It
is called the Hamming distance function and plays an important role in
the theory of error-correcting codes.³
5. Let B .
)
a If % ~²% ³ M show that % ¦
)
b Find a sequence that converges to but is not an element of any for
M
B.
)
%
%
6. a Show that if ~²% ³ M , then M for all .
)
b Find a sequence % ~²% ³ that is in for , but is not in .
M
M
7. Show that a subset of a metric space 4 is open if and only if contains
:
:
an open neighborhood of each of its points.
8. Show that the intersection of any collection of closed sets in a metric space
is closed.
9. Let ²4Á ³ be a metric space. The diameter of a nonempty subset : 4
is
²:³ ~ sup ²%Á &³
%Á&:
A set is bounded if ² : ³ B .
:
)
a Prove that is bounded if and only if there is some % 4 and s
:
for which : )²%Á ³ .
)
b Prove that ²:³ ~ if and only if consists of a single point.
:
)
c Prove that : ; implies ²:³ ²;³ .
)
d If and are bounded, show that r : ; is also bounded.
;
:
Z
10. Let ²4Á ³ be a metric space. Let be the function defined by
²%Á &³
Z
²%Á &³ ~
b ²%Á &³
