69.2. STANDARD TECHNIQUES IN EVOLUTION EQUATIONS 2359

for fixed w ∈ V . Suppose un → u in V and fix w ∈ L∞ (D) ⊆ V . Then it follows from aneasy argument using the Vitali convergence theorem and the fact that from the estimatesabove

Ψ(un)w

is uniformly integrable that

u→−∫

DΨ(u)wdx

is continuous. For general w ∈ Lp (D) , let wn→ w in Lp (D) where each wn is in L∞ (D).Then the function

u→−∫

DΨ(u)wdx≡ ⟨Au,w⟩ (69.1.7)

is the limit of the continuous functions

u→−∫

DΨ(u)wndx

and so the function 69.1.7 is Borel measurable. Now by the Pettis theorem this showsA : V →V ′ is B (V ) measurable. This shows A is an example of an operator which satisfiessome conditions which will be considered later.

69.2 Standard Techniques In Evolution EquationsIn this section, several significant theorems are presented. Unless indicated otherwise, themeasure will be Lebesgue measure. First here is a lemma.

Lemma 69.2.1 Suppose g∈ L1 ([a,b] ;X) where X is a Banach space. If∫ b

a g(t)φ (t)dt = 0for all φ ∈C∞

c (a,b) , then g(t) = 0 a.e.

Proof: Let E be a measurable subset of (a,b) and let K ⊆ E ⊆ V ⊆ (a,b) where Kis compact, V is open and m(V \K) < ε. Let K ≺ h ≺ V as in the proof of the Rieszrepresentation theorem for positive linear functionals. Enlarging K slightly and convolvingwith a mollifier, it can be assumed h ∈C∞

c (a,b) . Then∣∣∣∣∫ b

aXE (t)g(t)dt

∣∣∣∣ =

∣∣∣∣∫ b

a(XE (t)−h(t))g(t)dt

∣∣∣∣≤

∫ b

a|XE (t)−h(t)| ||g(t)||dt

≤∫

V\K||g(t)||dt.

Now let Kn ⊆ E ⊆Vn with m(Vn \Kn)< 2−n. Then from the above,∣∣∣∣∫ b

aXE (t)g(t)dt

∣∣∣∣≤ ∫ b

aXVn\Kn (t) ||g(t)||dt