69.5. SOME IMBEDDING THEOREMS 2387

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 69.5.5 Let E ⊆W ⊆ X where the injection map is continuous from W to X andcompact from E to W. Then if γ > α, the embedding of C0,γ ([0,T ] ,E) into C0,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 69.5.4 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 is a Cauchy sequence in C0,α ([0,T ] ,X).The next theorem is a well known result probably due to Lions.

Theorem 69.5.6 Let E ⊆W ⊆ X where the injection map is continuous from W to X andcompact 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