36.1 Partitions Of Unity
This material has already been mostly discussed starting on Page 3336. However, that
was a long time ago and it seems like it might be good to go over it again and so, for the
sake of convenience, here it is again.
Definition 36.1.1 Let ℭ be a set whose elements are subsets of ℝn.
Then ℭ is said to be locally finite if for every x ∈ ℝn, there exists an open set, Ux
containing x such that Ux has nonempty intersection with only finitely many sets
Lemma 36.1.2 Let ℭ be a set whose elements are open subsets of ℝn and suppose
∪ℭ ⊇ H, a closed set. Then there exists a countable list of open sets,
that each Ui is bounded, each Ui is a subset of some set of ℭ, and ∪i=1∞Ui ⊇ H.
Proof: Let Wk ≡ B
For each x ∈ H ∩Wk
there exists an
open set, Ux
such that Ux
is a subset of some set of ℭ
and Ux ⊆ Wk+1 ∖Wk−1.
since H ∩Wk
is compact, there exist finitely many of these sets,
union contains H ∩Wk
. If H ∩Wk
= 0 and there are no such sets
obtained.The desired countable list of open sets is
Each open set
in this list is bounded. Furthermore, if x ∈ ℝn,
then x ∈ Wk
first positive integer with x ∈ Wk.
Then Wk ∖Wk−1
is an open set containing
and this open set can have nonempty intersection only with with a set of
a finite list of sets. Therefore, ∪k=1∞
is said to be a locally finite cover of H.
The following lemma gives
some important reasons why a locally finite list of sets is so significant. First of
all consider the rational numbers,
each rational number is a closed
The set of rational numbers is definitely not locally finite.
Lemma 36.1.3 Let ℭ be locally finite. Then
Next suppose the elements of ℭ are open sets and that for each U ∈ℭ, there exists a
differentiable function, ψU having spt
⊆ U. Then you can define the following finite
sum for each x ∈ ℝn
Furthermore, f is also a differentiable
Proof: Let p be a limit point of ∪ℭ and let W be an open set which intersects only
finitely many sets of ℭ. Then p must be a limit point of one of these sets. It follows
The inclusion in the other direction is
Now consider the second assertion. Letting x ∈ ℝn, there exists an open set, W
intersecting only finitely many open sets of ℭ, U1,U2,
Then for all
y ∈ W,
and so the desired result is obvious. It merely says that a finite sum of differentiable
functions is differentiable. Recall the following definition.
Definition 36.1.4 Let K be a closed subset of an open set, U. K ≺ f ≺ U if f is
continuous, has values in
1 on K, and has compact support contained
Lemma 36.1.5 Let U be a bounded open set and let K be a closed subset of
U. Then there exist an open set, W, such that W ⊆ W ⊆ U and a function,
f ∈ Cc∞
such that K ≺ f ≺ U.
Proof: The set, K is compact so is at a positive distance from UC. Let
Then it is clear
Now consider the function,
Since W is compact it is at a positive distance from W1C and so h is a well defined
continuous function which has compact support contained in W1, equals 1 on W, and has
Now let ϕk
be a mollifier. Letting
it follows that for such k,the function, h∗ϕk ∈ Cc∞
has values in
1 on K.
= h ∗ ϕk
The above lemma is used repeatedly in the following.
Lemma 36.1.6 Let K be a closed set and let
i=1∞ be a locally finite list of
bounded open sets whose union contains K. Then there exist functions, ψi ∈ Cc∞
such that for all x ∈ K,
and the function f
is in C∞
Proof: Let K1 = K ∖∪i=2∞V i. Thus K1 is compact because K1 ⊆ V 1.
Thus W1,V 2,
and W1 ⊆ V 1
. Suppose W1,
have been defined
such that Wi ⊆ V i
for each i
, and W1,
It follows Kr+1 is compact because Kr+1 ⊆ V r+1. Let Wr+1 satisfy
Continuing this way defines a sequence of open sets,
with the property
is locally finite because the original list,
was locally finite.
Now let Ui
be open sets which satisfy
is locally finite.
Since the set,
is locally finite, it follows ∪i=1∞Wi
and so it
is possible to define ϕi
infinitely differentiable functions having compact support
If x is such that ∑
j=1∞ϕj(x) = 0, then x
equals one on Ui
= 0 for all
thanks to the fact that ∪i=1∞Ui
closed and so ψi
) = 0 for all y
. Hence ψi
is infinitely differentiable at
. If ∑
0, this situation persists near x
because each ϕj
continuous and so ψi
is infinitely differentiable at such points also thanks to Lemma
. Therefore ψi
is infinitely differentiable. If x ∈ K
, then γ
= 1 and so
) = 1. Clearly 0 ≤ ψi
1 and spt
) ⊆ V j
. This proves the
The method of proof of this lemma easily implies the following useful corollary.
Corollary 36.1.7 If H is a compact subset of V i for some V i there exists a
partition of unity such that ψi
for all x ∈ H in addition to the conclusion
of Lemma 36.1.6.
Proof: Keep V i the same but replace V j with
≡ V j ∖H
. Now in the proof above,
applied to this modified collection of open sets, if j≠i,ϕj
= 0 whenever
x ∈ H
= 1 on
Theorem 36.1.8 Let H be any closed set and let ℭ be any open cover of H.
Then there exist functions
i=1∞ such that spt
is contained in some set
of ℭ and ψi is infinitely differentiable having values in
such that on H,
. Furthermore, the function, f
differentiable on ℝn. Also, spt
⊆ Ui where Ui is a bounded open set with the
i=1∞ is locally finite and each Ui is contained in some set of ℭ.
Proof: By Lemma 36.1.2 there exists an open cover of H composed of bounded
open sets, Ui such that each Ui is a subset of some set of ℭ and the collection,
is locally finite. Then the result follows from Lemma 36.1.6
Corollary 36.1.9 Let H be any closed set and let
i=1m be a finite open
cover of H. Then there exist functions
i=1m such that spt
⊆ V i and ϕi is
infinitely differentiable having values in
such that on H, ∑
Proof: By Theorem 36.1.8 there exists a set of functions,
properties listed in this theorem relative to the open covering,
equal the sum of all
⊆ V 1.
Next let ϕ2
equal the sum of all
which have not already been included and for which
⊆ V 2.
this manner. Since the open sets,
mentioned in Theorem 36.1.8
finite, it follows from Lemma 36.1.3
that each ϕi
is infinitely differentiable having support
in V i.
This proves the corollary.