690 CHAPTER 21. THE BOCHNER INTEGRAL

which appears to show that C ([0,T ] ,H) is weak ∗ dense in L∞ (0,T,H). However, this lastspace is obviously separable in terms of the norm topology. Let D be a countable densesubset of C ([0,T ] ,H). For f ∈ L∞ (0,T,H) let g ∈C ([0,T ] ,H) such that d (f,g)< ε

4 . Thenlet h ∈ D be so close to g in C ([0,T ] ,H) that

M

∑k=1

2−k

∣∣∣⟨h−g,zk⟩L∞,L1

∣∣∣1+∣∣∣⟨h−g,zk⟩L∞,L1

∣∣∣ < ε

2

Thend (f,h)≤ d (f,g)+d (g,h)<

ε

4+

ε

2+

ε

4= ε

It appears that D is dense in B in the weak ∗ topology.

21.9 Pointwise Behavior, Weakly Convergent SequencesThere is an interesting little result which relates to weak limits in L2 (Γ,E) for E a Banachspace. I am not sure where to put this thing but think that this would be a good place for it.It obviously generalizes to Lp spaces.

Proposition 21.9.1 Let E be a Banach space and let {un} be a sequence in L2 (Γ,E) and letG(x) be a weakly compact set in E, and un (x) ∈ G(x) a.e. for each n. Let limsup{un (x)}denote the set of all weak limits of subsequences of {un (x)} and let H (x) be the closure ofthe 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).

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

Lemma 21.10.1 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