42.5 Some Imbedding Theorems
The next theorem is very useful in getting estimates in partial differential equations. It is
called Erling’s lemma.
Definition 42.5.1 Let E,W be Banach spaces such that E ⊆ Wand the injection
map from E into W is continuous. The injection map is said to be compact if every
bounded set in E has compact closure in W. In other words, if a sequence is bounded
in E it has a convergent subsequence converging in W. This is also referred to by
saying that bounded sets in E are precompact in W.
Theorem 42.5.2 Let E ⊆ W ⊆ X where the injection map is continuous from W to X
and compact from E to W. Then for every ε > 0 there exists a constant, Cε such that for
all u ∈ E,
Proof: Suppose not. Then there exists ε > 0 and for each n ∈ ℕ, un such
Now let vn = un∕
= 1 and
It follows there exists a subsequence, still denoted by vn such that vn converges to v in
W. However, the above inequality shows that
But then the
above inequality would imply that
and passing to the limit yields 0 > ε,
Definition 42.5.3 Define C
the space of functions continuous at every
having values in X.
You should verify that this is a Banach space with norm
The following theorem is an infinite dimensional version of the Ascoli Arzela theorem.
Theorem 42.5.4 Let q > 1 and let E ⊆ W ⊆ X where the injection map is continuous
from W to X and compact from E to W. Let S be defined by
Then S ⊆ C
⊆ S, there exists a subsequence,
converges to a function u ∈ C
in the following way.
Proof: First consider the issue of S being a subset of C
on Page 4829
the following holds in X
for u ∈ S.
Thus S ⊆ C
Let ε >
0 be given. Then by Theorem 42.5.2
there exists a
such that for all u ∈ W
Therefore, for all u ∈ S,
is arbitrary, it follows u ∈ C
Let D = ℚ ∩
is a countable dense subset of
compactness of the embedding of E
there exists a subsequence u
such that as
n →∞, u
converges to a point in
Now take a subsequence of this, called
such that as
converges to a point in
It follows that u
also converges to a point of
Continue this way. Now consider the diagonal sequence,
uk ≡ u
This sequence is a subsequence of u
whenever k > l.
converges for all
tj ∈ D.
be as just defined, converging at every point of
D ≡ ℚ∩
converges at every point of
Proof of claim: Let ε > 0 be given. Let t ∈
Pick tm ∈ D ∩
such that in
3. Then there exists N
such that if l,n > N,
X < ε∕
It follows that for l,n > N,
was arbitrary, this shows
is a Cauchy sequence. Since W
complete, this shows this sequence converges.
Now for t ∈
it was just shown that if ε >
0 there exists Nt
such that if
n,m > Nt,
Now let s≠t. Then
and so it follows that if δ is sufficiently small and s ∈ B
then when n,m > Nt
is compact, there are finitely many of these balls,
such that for
s ∈ B
n,m > Nti,
the above inequality holds. Let N >
Then if m,n > N
and s ∈
is arbitrary, it follows the above inequality must hold.
Therefore, this has shown the following claim.
Claim: Let ε > 0 be given. Then there exists N such that if m,n > N, then
∞,W < ε.
Now let u
Let N be in the above claim and fix n > N. Then
W ≤ ε.
is small enough,
the middle term in 42.5.23
is also smaller than ε.
Thus u is continuous. Finally, let N be as in the above claim. Then letting m,n > N, it
follows that for all t ∈
Therefore, letting m →∞, it follows that for all t ∈
∞,W ≤ ε.
is arbitrary, this proves the theorem.
The next theorem is another such imbedding theorem found in [?]. It is often used in
partial differential equations.
Theorem 42.5.5 Let E ⊆ W ⊆ X where the injection map is continuous from W to X
and compact from E to W. Let p ≥ 1, let q > 1, and define
Then S is precompact in Lp
. This means that if
n=1∞⊆ S, it has a
which converges in Lp
Proof: By Proposition 6.2.5 on Page 410 it suffices to show S has an η net in
If not, there exists η > 0 and a sequence
, such that
for all n≠m and the norm refers to Lp
The idea is to show that un approximates un well and then to argue that a subsequence
is a Cauchy sequence yielding a contradiction to
It follows from Jensen’s inequality that
From Theorems 42.5.2 and 42.1.9, if ε > 0, there exists Cε such that
This is substituted in to 42.5.25
so small that 2pεRp < ηp∕
and then choosing k
sufficiently large, it
Now use compactness of the embedding of E into W to obtain a subsequence such
is Cauchy in
and use this to contradict
. The details