ECE 580 / Math 587
SPRING 2011
Correspondence
# 4
February 7, 2011
ASSIGNMENT 2
Reading Assignment:
Text: Chp 10 (pp. 272277); Correspondence # 2.
Recommended Reading:
Curtain & Pritchard: Chapters 1 (pp. 2022), 4 (pp. 5564);
Balakrishnan: Chapter 1, and Chapter 2 (pp. 5457).
Liusternik & Sobolev: Chapter 1 (pp. 2644).
Advance Reading:
Text: Chapter 3
Problems
(to be handed in):
Due Date:
Thursday, February 17
.
11.
Consider the space
ℓ
′
p
consisting of all ordered numbers (
ξ
1
, ξ
2
, . . . , ξ
k
1
) where
k
1
is a natural
number and
ξ
i
’s are arbitrary real numbers. If
x
:= (
ξ
1
, ξ
2
, . . . , ξ
k
1
)
y
:= (
ν
1
, ν
2
, . . . , ν
k
2
)
,
k
2
≥
k
1
,
we introduce a metric by
ρ
(
x, y
) = (
k
1
∑
i
=1

ξ
i
−
ν
i

p
+
k
2
∑
j
=
k
1
+1

ν
j

p
)
1
/p
,
1
≤
p <
∞
.
Show
that
ℓ
′
p
is
not
a complete metric space.
Hint:
You have to show that there exists a Cauchy sequence in
ℓ
′
p
, whose limit is not in
ℓ
′
p
; consider, for instance, the sequence:
x
1
=
{
1
}
, x
2
=
{
1
,
1
2
}
, x
3
=
{
1
,
1
2
,
1
2
2
}
, . . . , x
n
=
{
1
,
1
2
, . . . ,
1
2
n
−
1
}
, . . .
12.
Let
f
:
C
[0
,
2]
→
R
be a functional defined by
f
(
x
) = max
0
≤
t
≤
2
x
(
t
)
(a)
Show
that
f
is
continuous
.
(b)
Is
f
uniformly
continuous?
13.
As discussed in class, the Banach space
L
p
[
a, b
] is
separable
,
for 1
≤
p <
∞
and for any
finite interval [
a, b
] (see also Example 4, on page 43 of the text). This result does not hold,
however, if the interval is infinite, that is (
−∞
,
∞
), and the norm adopted is
∥
x
∥
=
(
lim
T
→∞
1
T
∫
T
−
T

x
(
t
)

p
dt
)
1
p