76.3. THE EXISTENCE OF APPROXIMATE SOLUTIONS 2541

Theorem 76.2.6 Let A : V →V ′ be type M, bounded, and coercive

lim∥u∥→∞

⟨A(u+u0) ,u⟩∥u∥

= ∞, (76.2.6)

for some u0 ∈V , where V is a separable reflexive Banach space. Then A is surjective.

In addition, there is a fundamental definition and theorem about weak derivatives whichwill be used.

Definition 76.2.7 Let f ∈ L1 (a,b,V ′) where V ′ is the dual of a Banach space V . LetD∗ (a,b) linear mappings from C∞

c (a,b) to V ′. Then we can consider f ∈ D∗ (a,b) , thelinear transformations defined on C∞

c (a,b) as follows.

f (φ)≡∫ b

af φds

This is well defined due to regularity considerations for Lebesgue measure. Then defineD f ∈D∗ (a,b) by

D f (φ)≡−∫ b

af φ′ds

To say that D f ∈ L1 (a,b,V ′) is to say that there exists g ∈ L1 (a,b,V ′) such that

D f (φ)≡−∫ b

af φ′ds =

∫ b

agφds

for all φ ∈C∞c (a,b). Note that regularity considerations imply that g is unique if it exists.

The following is Theorem 69.2.9.

Theorem 76.2.8 Suppose that f and D f are both in L1 (a,b,V ′). Then f is equal to acontinuous function a.e., still denoted by f and

f (x) = f (a)+∫ x

aD f (t)dt.

In the next section are theorems about how shifts in time relate to progressive measur-ability.

76.3 The Existence Of Approximate SolutionsThe situation is as follows. There are spaces V ⊆W where V is a reflexive separable Banachspace and W is a separable Hilbert space. It is assumed that V is dense in W. Define thespaces

V ≡ Lp ([0,T ]×Ω,V ) , W ≡ L2 ([0,T ]×Ω,W )

where in each case, the σ algebra of measurable sets will be the progressively measurablesets. Thus, from the Riesz representation theorem,

V ′ = Lp′ ([0,T ]×Ω,V ′), W ′ = L2 ([0,T ]×Ω,W ′

)