24.11. SOME EMBEDDING THEOREMS 697

Thus u is continuous. Finally, let N be as in the above claim. Then letting m,n > N, itfollows that for all t ∈ [a,b] ,

∥um (t)−un (t)∥W < ε.

Therefore, letting m→ ∞, it follows that for all t ∈ [a,b] ,∥u(t)−un (t)∥W ≤ ε. and so∥u−un∥∞,W ≤ ε. ■

Here is an interesting corollary. Recall that for E a Banach space C0,α ([0,T ] ,E) is thespace of continuous functions u from [0,T ] to E such that ∥u∥

α,E ≡ ∥u∥∞,E +ρα,E (u)< ∞

where here ρα,E (u)≡ supt ̸=s∥u(t)−u(s)∥E|t−s|α

Corollary 24.11.7 Let E ⊆ W ⊆ X where the injection map is continuous from Wto X and compact from E to W. Then if γ > α, the embedding of C0,γ ([0,T ] ,E) intoC0,α ([0,T ] ,X) is compact.

Proof: Let φ ∈C0,γ ([0,T ] ,E)

∥φ (t)−φ (s)∥X|t− s|α

≤(∥φ (t)−φ (s)∥W|t− s|γ

)α/γ

∥φ (t)−φ (s)∥1−(α/γ)W

≤(∥φ (t)−φ (s)∥E|t− s|γ

)α/γ

∥φ (t)−φ (s)∥1−(α/γ)W ≤ ργ,E (φ)∥φ (t)−φ (s)∥1−(α/γ)

W

Now suppose {un} is a bounded sequence in C0,γ ([0,T ] ,E) . By Theorem 24.11.6 above,there is a subsequence still called {un} which converges in C0 ([0,T ] ,W ) . Thus from theabove inequality

∥un (t)−um (t)− (un (s)−um (s))∥X|t− s|α

≤ ργ,E (un−um)∥un (t)−um (t)− (un (s)−um (s))∥1−(α/γ)W

≤ C ({un})(

2∥un−um∥∞,W

)1−(α/γ)

which converges to 0 as n,m→ ∞. Thus, ρα,X (un−um)→ 0 as n,m→ ∞ Also

∥un−um∥∞,X → 0

as n,m→ ∞ so this sequence is a Cauchy sequence in C0,α ([0,T ] ,X). ■The next theorem is a well known result probably due to Lions, Teman, or Aubin.

Theorem 24.11.8 Let E ⊆W ⊆ X where the injection map is continuous from Wto X and compact from E to W. Let p≥ 1, let q > 1, and define

S≡ {u ∈ Lp ([a,b] ;E) : for some C, ∥u(t)−u(s)∥X ≤C |t− s|1/q and ∥u∥Lp([a,b];E) ≤ R}.

Thus S is bounded in Lp ([a,b] ;E) and Holder continuous into X. Then S is precompact inLp ([a,b] ;W ). This means that if {un}∞

n=1 ⊆ S, it has a subsequence{

unk

}which converges

in Lp ([a,b] ;W ) .