690 CHAPTER 24. THE BOCHNER INTEGRAL

Proposition 24.10.1 Let E be a Banach space and let {un} be a sequence in L2 (Γ,E)and let G(x) be a weakly compact set in E, and un (x) ∈ G(x) a.e. for each n. Letlimsup{un (x)} denote the set of all weak limits of subsequences of {un (x)} and let H (x)be the closure of the convex hull of limsup{un (x)}. Then if un → u weakly in L2 (Γ,E) ,then u(x) ∈ H (x) for a.e. x.

Proof: Let H ={

w ∈ L2 (Γ,E) : w(x) ∈ H (x) a.e.}

. Then H is convex. If you havewi ∈ H, then since each H (x) is convex, it follows that λw1 (x)+ (1−λ )w2 (x) ∈ H fora.e. x and λ ∈ [0,1]. Is H closed? Suppose you have wn ∈ H and wn → w in L2 (Γ,E).Then there is a subsequence such that pointwise convergence happens a.e. and so since His closed, you have w(x) ∈ H for a.e. x. Hence H is also weakly closed in L2 (Γ,H). Thusif u is the weak limit of {un} in L2 (Γ,E) , it must be the case that u(x) ∈ H (x) a.e. ■

As a case of this which might be pretty interesting, suppose G(x) is not just weaklycompact but also convex. Then H (x) = G(x) and you can say that u(x) ∈H (x) a.e. when-ever it is a weak limit in L2 (Γ,E) of functions un for which un (x) ∈ G(x).

24.11 Some Embedding TheoremsThe next lemma is a very useful little result which involves embeddings of Banach spaces.

Lemma 24.11.1 Suppose V ⊆W and the injection map is compact, hence continuous.Suppose also that W ⊆ U with continuous injection. Then for any ε > 0 there exists Cε

such that for all v ∈V,∥v∥W ≤ ε ∥v∥V +Cε ∥v∥U

Proof: Suppose not. Then there exists ε > 0 for which things don’t work out. Thusthere exists vn ∈ V such that ∥vn∥W > ε ∥vn∥V + n∥vn∥U . Dividing by ∥vn∥V , it can alsobe assumed that ∥vn∥V = 1. Thus ∥vn∥W > ε + n∥vn∥U , and so ∥vn∥U → 0. However, vnis contained in the closed unit ball of V which is, by assumption precompact in W . Hence,there exists a subsequence, still denoted as {vn} such that vn → v in W . But it was justdetermined that v = 0 and so 0≥ limsupn→∞ (ε +n∥vn∥U )≥ ε which is a contradiction. ■

Recall the following definition, this time for the space of continuous functions definedon a compact set with values in a Banach space.

Definition 24.11.2 Let A ⊆C (K;V ) where the last symbol denotes the continuousfunctions defined on a compact set K ⊆ X a metric space having values in V a Banachspace. Then A is equicontinuous if for every ε > 0, there exists δ > 0 such that for everyf ∈A , if d (x,y)< δ , then ∥ f (x)− f (y)∥V < ε. Also A ⊆C (K;V ) is uniformly boundedmeans

supf∈A∥ f∥

∞,V < ∞ where ∥ f∥∞,V ≡max

x∈K∥ f (x)∥V .

Here is a general version of the Ascoli Arzela theorem valid for Banach spaces.

Theorem 24.11.3 Let V ⊆W ⊆U where the injection map of V into W is compactand W embedds continuously into U, these being Banach spaces. Assume:

1. A ⊆C (K;U) where K is compact and A is equicontinuous.

2. sup f∈A ∥ f∥∞,V < ∞ where ∥ f∥

∞,V ≡maxx∈K ∥ f (x)∥V .

Then