21.10. SOME EMBEDDING THEOREMS 699

Now let u(t) = limk→∞ uk (t) .

||u(t)−u(s)||W ≤ ||u(t)−un (t)||W + ||un (t)−un (s)||W + ||un (s)−u(s)||W (21.10.48)

Let N be in the above claim and fix n > N. Then

||u(t)−un (t)||W = limm→∞||um (t)−un (t)||W ≤ ε

and similarly, ||un (s)−u(s)||W ≤ ε. Then if |t− s| is small enough, 21.10.47 shows themiddle term in 21.10.48 is also smaller than ε. Therefore, if |t− s| is small enough,

||u(t)−u(s)||W < 3ε.

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 21.10.7 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 21.10.6 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)