2598 CHAPTER 77. STOCHASTIC INCLUSIONS

for all n. Also, suppose for each ω /∈ N, each subsequence of {un} has a further subse-quence that converges weakly in Lp′ ([0,T ] ;V ′) to v(·,ω) ∈ Lp′ ([0,T ] ;V ′) such that thefunction t→ v(t,ω) is weakly continuous into V ′.

Then, there exists a product measurable function u such that t→ u(t,ω)is weakly con-tinuous into V ′ and for each ω /∈ N, a subsequence un(ω) such that un(ω) (·,ω)→ u(·,ω)

weakly in Lp′ ([0,T ] ;V ′).

We prove the theorem in steps given below. Let X = ∏∞k=1 C ([0,T ]) and note that when

it is equipped with the product topology, then one can consider X as a metric space usingthe metric

d (f,g)≡∞

∑k=1

2−k ∥ fk−gk∥1+∥ fk−gk∥

,

where f = ( f1, f2, . . .),g = (g1,g2, . . .) ∈ X , and the norm is the maximum norm in thespace C ([0,T ]). With this metric, X is complete and separable.

Lemma 77.2.5 Let {fn} be a sequence in X and suppose that each one of the componentsfnk is bounded by C = C(k) in C0,1 ([0,T ]). Then, there exists a subsequence

{fn j

}that

converges to some f ∈ X as n j→ ∞. Thus, {fn} is pre-compact in X.

Proof: By the Ascoli−Arzelà theorem, there exists a subsequence {fn1} such that thesequence of the first components fn11 converges in C ([0,T ]). Then, taking a subsequence,one can obtain {n2} a subsequence of {n1} such that both the first and second componentsof fn2 converge. Continuing in this way one obtains a sequence of subsequences, eacha subsequence of the previous one such that fn j has the first j components convergingto functions in C ([0,T ]). Therefore, the diagonal subsequence has the property that ithas every component converging to a function in C ([0,T ]) . The resulting function is f ∈∏k C ([0,T ]).

Now, for m ∈ N and φ ∈V, define lm(t)≡max(0, t− (1/m)) and

ψm,φ : Lp′ ([0,T ] ;V ′)→C ([0,T ])

by

ψm,φ u(t)≡∫ T

0

⟨mφX[lm(t),t] (s) ,u(s)

⟩V ds = m

∫ t

lm(t)⟨φ ,u(s)⟩V ds.

Here, X[lm(t),t] (·) is the indicator function of the interval [lm(t), t] and ⟨·, ·⟩V = ⟨·, ·⟩V is theduality pairing between V and V ′.

Let D = {φ r}∞

r=1 denote a countable dense subset of V . Then the pairs (φ ,m) for

φ ∈ D and m ∈ N form a countable set. Let(

mk,φ rk

)denote an enumeration of the pairs

(m,φ) ∈ N×D . To simplify the notation, we set

fk (u)(t)≡ ψmk,φ rk(u)(t) = mk

∫ t

lmk (t)

⟨φ rk

,u(s)⟩

Vds.