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4
SPRING 2012
2.
Linear Maps, Operator Norm
It’s not obvious from the deFnition that
°
T
°
is Fnite for all
T
∈
L
(
R
n
,
R
m
). If [
T
]=(
a
ij
)
is the matrix representation of
T
with respect to the standard bases, deFne the 2
norm
of
T
to be
°
T
°
2
=
°
±
±
²
m
³
i
=1
n
³
j
=1
a
2
ij
.
Then for each
x
∈
R
n
,wehave
°
T
(
x
)
°
2
=
°
[
T
]
x
°
2
=
m
³
i
=1
´
n
³
j
=1
a
ij
x
j
µ
2
≤
m
³
i
=1
´
n
³
j
=1
a
2
ij
n
³
k
=1
x
2
k
µ
=
´
m
³
i
=1
n
³
j
=1
a
2
ij
µ
n
³
k
=1
x
2
k
=
°
T
°
2
2
°
x
°
2
.
(The inequality follows from the CauchySchwarz inequality.) Thus
°
T
°
2
is an upper bound
for the set
{°
T
(
x
)
°°
x
°
=1
}
, so
°
T
°≤°
T
°
2
, and in particular
°
T
°
is Fnite.
Example 2.1.
±or the identity map Id:
R
n
→
R
n
,wehave
°
Id
°
2
=
···
=
√
n.
However, for any
x
∈
R
n
with
°
x
°
=1
,wehave
°
Id(
x
)
°
=
°
x
°
=1
,
and it follows that
°
Id
°
= 1.
Exercise.
Prove that every
T
∈
L
(
R
n
,
R
m
) is uniformly continuous.
Lemma 2.2.
For each
S
∈
L
(
R
m
,
R
°
)
and
T
∈
L
(
R
n
,
R
m
)
,
°
S
◦
T
°≤°
S
°°
T
°
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This document was uploaded on 03/11/2012.
 Spring '09

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