2384 CHAPTER 69. GELFAND TRIPLES

valid for t ∈ D, it follows that the same formula holds for all t. This formula impliest → ⟨BX (t) ,X (t)⟩ is continuous. Also recall that t → BX (t) was shown to be weaklycontinuous into W ′. Then

⟨B(X (t)−X (s)) ,X (t)−X (s)⟩= ⟨BX (t) ,X (t)⟩−2⟨BX (t) ,X (s)⟩+ ⟨BX (s) ,X (s)⟩

From this, it follows that t→ BX (t) is continuous into W ′ because limt→s of the right sidegives 0 and so the same is true of the left. Hence,

|⟨B(X (t)−X (s)) ,y⟩| ≤ ⟨By,y⟩1/2 ⟨B(X (t)−X (s)) ,X (t)−X (s)⟩1/2

≤ ∥B∥1/2 ⟨B(X (t)−X (s)) ,X (t)−X (s)⟩1/2 ∥y∥

so∥B(X (t)−X (s))∥W ′ ≤ ∥B∥

1/2 ⟨B(X (t)−X (s)) ,X (t)−X (s)⟩1/2

which converges to 0 as t→ s.

69.5 Some Imbedding TheoremsThe next theorem is very useful in getting estimates in partial differential equations. It iscalled Erling’s lemma.

Definition 69.5.1 Let E,W be Banach spaces such that E ⊆Wand the injection map fromE into W is continuous. The injection map is said to be compact if every bounded set in Ehas compact closure in W. In other words, if a sequence is bounded in E it has a convergentsubsequence converging in W. This is also referred to by saying that bounded sets in E areprecompact in W.

Theorem 69.5.2 Let E ⊆W ⊆ X where the injection map is continuous from W to X andcompact from E to W. Then for every ε > 0 there exists a constant, Cε such that for allu ∈ E,

||u||W ≤ ε ||u||E +Cε ||u||X

Proof: Suppose not. Then there exists ε > 0 and for each n ∈ N, un such that

||un||W > ε ||un||E +n ||un||X

Now let vn = un/ ||un||E . Therefore, ||vn||E = 1 and

||vn||W > ε +n ||vn||X

It follows there exists a subsequence, still denoted by vn such that vn converges to v in W.However, the above inequality shows that ||vn||X → 0. Therefore, v = 0. But then the aboveinequality would imply that ||vn||> ε and passing to the limit yields 0 > ε, a contradiction.