This preview shows pages 1–2. Sign up to view the full content.
245B, Winter 2009, Assignment 6:
Notes and selected model answers
(Model solutions follow question numbers in
bold
.)
Folland Chapter 4
63.
We ﬁrst recall that since
[0
,
1]
is compact, any
f
∈
C
[0
,
1]
is bounded, and
that since
[0
,
1]
×
[0
,
1]
is compact, the continuous function
K
: [0
,
1]
×
[0
,
1]
→
C
is bounded and is actually
uniformly
continuous [this is crucial for this question].
From the second of these observations it follows that for any
ε >
0
we can choose
δ >
0
so small that

K
(
x,y
)

K
(
x
0
,y
0
)

< ε
whenever
max
{
x

x
0

,

y

y
0
}
< δ
,
and so certainly when
y
0
=
y
and

x

x
0

< δ
. Therefore given any
f
, using that
k
f
k
u
is ﬁnite, if

x

x
0

< δ
then we also have

Tf
(
x
)

Tf
(
x
0
)

=
±
±
±
Z
1
0
K
(
x,y
)
f
(
y
) d
y

Z
1
0
K
(
x
0
,y
)
f
(
y
) d
y
±
±
±
≤
Z
1
0

K
(
x,y
)

K
(
x
0
,y
)

f
(
y
)

d
y < ε
Z
1
0

f
(
y
)

d
y
≤
ε
k
f
k
∞
.
Since
ε
was chosen arbitrarily this shows that for any ﬁxed
f
the output function
Tf
is continuous.
Moreover, in case
k
f
k
u
≤
1
the above inequality specializes to the assertion that

Tf
(
x
)

Tf
(
x
0
)

< ε
whenever

x

x
0

< δ
without any further restrictions
on
f
. Since this
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 04/06/2010 for the course MATH various taught by Professor Tao/analysis during the Spring '10 term at UCLA.
 Spring '10
 tao/analysis

Click to edit the document details