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Analysis 1
Homework 07
1. True or False: The supnorm
k • k
∞
is complete on
‘
1
.
2. True or False:
C
[0
,
1] is isometrically isomorphic to a closed subspace of
‘
∞
.
3. True or False: If
Z
is a closed subspace of the Banach space
X
then for each
a
∈
X
there exists a unique
z
∈
Z
such that
k
a

z
k
=
d
(
a
,
Z
).
4. True or False: If
x
1
and
x
2
are vectors of equal norm in the Banach space
X
then
there exists an isometric isomorphism
T
:
X
→
X
such that
T
(
x
1
)
=
x
2
.
Solutions
1. F. An indirect proof: note ﬁrst that if
x
∈
‘
1
then
k
x
k
∞
6
k
x
k
1
and that
k • k
1
is
complete, whence it would follow (as a consequence of the Banach Isomorphism
Theorem) that if
k • k
∞
were complete then there would exist
K
such that
k • k
1
6
K
k • k
∞
; however, Archimedes prohibits this, for
k
e
1
+
···
+
e
n
k
∞
=
1 and yet
k
e
1
+
···
+
e
n
k
1
=
n
. A direct proof: the sequence (
∑
n
i
=
1
e
i
/
i
)
∞
n
=
1
in (
c
00
⊂
)
‘
1
is
plainly
k • k
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This note was uploaded on 07/08/2011 for the course MHF 3202 taught by Professor Larson during the Spring '09 term at University of Florida.
 Spring '09
 LARSON

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