Page 339 - Advanced Linear Algebra

Metric Spaces 323

25. Show that any convergent sequence is a Cauchy sequence.
26. If ²% ³ ¦ % in a metric space 4 , show that any subsequence ²% ³ of ²% ³

also converges to .
%
27. Suppose that ²% ³ is a Cauchy sequence in a metric space 4 and that some

subsequence ²% ³ of ²% ³ converges. Prove that ²% ³ converges to the

same limit as the subsequence.
28. Prove that if ²% ³ is a Cauchy sequence, then the set ¸% ¹ is bounded. What

about the converse? Is a bounded sequence necessarily a Cauchy sequence?
29. Let ²% ³ and ²& ³ be Cauchy sequences in a metric space 4 . Prove that the

sequence ~ ²% Á & ³ converges.

30. Show that the space of all convergent sequences of real numbers or
²
)
complex numbers is complete as a subspace of M B .
31. Let denote the metric space of all polynomials over , with metric
d
F
² Á ³ ~ sup ( ²%³ c ²%³(
%´ Á µ
Is complete?
F
(
32. Let :‹ M B be the subspace of all sequences with finite support that is,
with a finite number of nonzero terms . Is complete?
)
:
33. Prove that the metric space { of all integers, with metric
² Á ³ ~ c , is complete.
(
(
(
34. Show that the subspace of the metric space ´ * Á µ under the sup metric)
:
consisting of all functions  *´ Á µ for which ² ³ ~ ² ³ is complete.
35. If 4š 4 Z and 4 is complete, show that 4 Z is also complete.
36. Show that the metric spaces *´ Á µ and *´ Á µ , under the sup metric, are
isometric.
37. Prove Holder's inequality
¨
° °
B B B
%& (  8 ( ( % 9 8 ( ( & ( 9
~ ~ ~
as follows:
a) Show that ~! c ¬ !~ c
)
b Let and be positive real numbers and consider the rectangle in
9
#
"

,
,
s with corners ² Á ³ ²"Á ³ ² Á #³ and ²"Á #³ , with area "# . Argue
(
geometrically that is, draw a picture to show that
)
" #
"#  ! c ! b c

and so
" #
"#  b

c Now let ?~ ' ) % (  B and @ ~ '( & ( (  B . Apply the results of
)
part b to