38.6. SOBOLEV SPACES ON MANIFOLDS 1325

a.e. x. Therefore,h∗i (uψ i) = wi

and this shows that h∗i (uψ i) ∈ Hm+s (Ui) . It remains to verify that u ∈ Hm+s (Γ) . Thisfollows from Fatou’s lemma. From 38.6.22,∣∣∣∣h∗i (u jψ i)

∣∣∣∣2Hm+s(Ui)

→ ||h∗i (uψ i)||2Hm+s(Ui)

and so

∞

∑i=1||h∗i (uψ i)||

2Hm+s(Ui)

≤ lim infj→∞

∞

∑i=1

∣∣∣∣h∗i (u jψ i)∣∣∣∣2

Hm+s(Ui)

= lim infj→∞

∣∣∣∣u j∣∣∣∣2

Hm+s(Γ)< ∞.

This proves the lemma.In fact any two such norms are equivalent. This follows from the open mapping

theorem. Suppose ||·||1 and ||·||2 are two such norms and consider the norm ||·||3 ≡max(||·||1 , ||·||2) . Then (Hm+s (Γ) , ||·||3) is also a Banach space and the identity map fromthis Banach space to (Hm+s (Γ) , ||·||i) for i = 1,2 is continuous. Therefore, by the openmapping theorem, there exist constants, C,C′ such that for all u ∈ Hm+s (Γ) ,

||u||1 ≤ ||u||3 ≤C ||u||2 ≤C ||u||3 ≤CC′ ||u||1

Therefore,||u||1 ≤C ||u||2 , ||u||2 ≤C′ ||u||1 .

This proves the following theorem.

Theorem 38.6.5 Let Γ be described above. Defining Ht (Γ) as in Definition 38.6.3, anytwo norms like those given in this definition are equivalent.

Suppose (Γ,Wi,Ui,Γi,hi,gi) are as defined above where hi,gi are Cm,1 functions. TakeW , an open set in Rp and define Γ′ ≡W ∩Γ. Then letting

W ′i ≡W ∩Wi,Γ′i ≡W ′i ∩Γ,

andU ′i ≡ gi

(Γ′i)= h−1

i(W ′i ∩Γ

),

it follows that U ′i is an open set because hi is continuous and (Γ′,W ′i ,U′i ,Γ′i,h′i,g′i) is also a

Cm,1 manifold if you define h′i to be the restriction of hi to U ′i and g′i to be the restriction ofgi to W ′i .

As a case of this, consider a Cm,1 manifold, Γ where (Γ,Wi,Ui,Γi,hi,gi) are as describedin Definition 38.6.1 and the submanifold consisting of Γi. The next lemma shows there isa simple way to define a norm on Ht (Γi) which does not depend on dragging in a partitionof unity.