Exercises from Appendix C
C.10
Hint: Expand
bardbl
f
+
g
bardbl
2
=
bardbl
Lf
+
Lg
bardbl
2
and
bardbl
f
+
ig
bardbl
2
=
bardbl
Lf
+
iLg
bardbl
2
using the Polar Identity.
C.14
Hint: (b) If
λ /
∈
ℓ
∞
, then there exists a subsequence (
λ
n
k
)
k
∈
N
such that

λ
n
k
 ≥
k
for each
k
. Let
c
n
k
= 1
/k
and define all other
c
n
to be zero. Show
that
f
=
∑
c
n
e
n
converges but
Lf
=
∑
λ
n
c
n
e
n
does not converge.
C.15
Hints: (a) For the case
p <
∞
, to show the inequality
bardbl
M
φ
bardbl ≤ bardbl
φ
bardbl
∞
,
choose any
ε >
0. Then
E
=
{
φ

>
bardbl
φ
bardbl
∞
−
ε
}
has positive measure, so some
set
E
k
=
E
∩
[
k, k
+ 1] must have positive (and finite) measure. Consider
f
=

E
k

−
1
/p
χ
E
k
.
(b) Suppose
p <
∞
, and assume
φ /
∈
L
∞
(
R
). The sets
E
k
=
{
k
≤ 
φ

<
k
+ 1
}
. are measurable and disjoint, and since
φ
is not in
L
∞
(
X
) there must
be infinitely many
E
k
with positive measure, say
E
n
k
for
k
∈
N
.
Choose
F
k
⊆
E
n
k
with 0
<

F
k

<
∞
, and consider
f
(
x
) =
n
k

F
k

−
1
/p
for
x
∈
F
k
,
and
f
(
x
) = 0 otherwise.
C.26
Hints: Fix
f
∈
X.
Since
Y
is dense in
X,
there exist
g
n
∈
Y
such
that
g
n
→
f
. Show that
{
Lg
n
}
n
∈
N
is Cauchy in
Z
, so there exists an
h
∈
Z
such that
Lg
n
→
h
. Show that
tildewide
Lf
=
h
is welldefined and has the required
properties.
C.24
Hints: (b) Let
{
A
n
}
n
∈
N
be a sequence in
B
(
X, Y
) that is Cauchy in
operator norm. Given
f
∈
X,
show that
{
A
n
f
}
n
∈
N
is a Cauchy sequence in
Y
and hence converges, say to
g
. Define
Af
=
g
. Use the fact that
A
n
f
→
Af
for
each
f
together with the fact that
{
A
n
}
n
∈
N
is Cauchy to show that
A
n
→
A
in operator norm.
C.31
Hint: (b) Consider a function
g
∈
C
0
(
R
) that is nonzero everywhere.
C.34
Hints: (b) If
μ
negationslash
= 0, choose
g /
∈
ker(
μ
). Let
p
be the orthogonal projection
of
g
onto ker(
μ
), and set
e
=
g
−
p
. Show that
μ
(
e
)
negationslash
= 0, and set
u
=
e/μ
(
e
).
Show that
f
−
μ
(
f
)
u
∈
ker(
μ
) for every
f
∈
H
. Use the fact that
u
⊥
ker(
μ
)
to show that
μ
=
μ
h
where
h
=
u/
bardbl
u
bardbl
2
.
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 Fall '09
 HEIL
 Linear Algebra, LP, Cauchy

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