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Lecture 15
We
restate
the
partition of
unity
theorem
from
last time. Let
{
U
α
:
α
∈
I
}
be a
collection of
open subsets
of
R
n
such that
±
U
=
U
α
.
(3.169)
α
∈
I
Theorem 3.30.
There
exist functions
f
i
⊆ C
0
∞
(
U
)
such that
1.
f
1
≥
0
,
2.
supp
f
i
⊆
U
α
, for
some
α
,
3. For
every
p
∈
U
, there
exists a neighborhood
U
p
of
p
such that
U
p
∪
supp
f
i
=
φ
for
all
i > N
p
,
4.
f
i
= 1
.
Remark.
Property (4)
makes
sense because of property
(3), because at each
point
it
is a
finite
sum.
Remark.
A
set
of
functions
satisfying
property
(4)
is
called
a
partition of
unity
.
Remark.
Property (2)
can be
restated
as
“the partition
of unity
is
subordinate to
the
cover
{
U
α
:
α
∈
I
}
.”
Let
us
look at
some
typical
applications
of partitions
of unity.
The
first
application is
to improper
integrals.
Let
φ
:
U
R
be a
continuous
→
map, and suppose
φ
(3.170)
U
is welldefined.
Take
a partition
of unity
f
i
=
1. The function
f
i
φ
is
continuous
and
compactly supported,
so it
bounded. Let supp
f
i
⊆
Q
i
for some rectangle
Q
i
.
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 Fall '04
 unknown
 Topology, Sets, φ, Munkres, supp fi

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