25.8. PERTURBATION THEOREMS 921

For such u, it follows that on a dense subset of V, namely D(L) ,v→⟨Lz,u⟩ is a continuouslinear map. Hence there exists a unique element of V ′, denoted as L∗u such that for allv ∈ D(L) ,

⟨Lv,u⟩V ′,V = ⟨L∗u,v⟩V ′,VThus

L : D(L)⊆V →V ′

L∗ : D(L∗)⊆V →V ′

There is an interesting description of L∗ in terms of L which will be quite useful.

Proposition 25.8.2 Let τ : V ×V ′→V ′×V be given by τ (a,b)≡ (−b,a) . Also for S⊆ Xa reflexive Banach space,

S⊥ ≡{

z∗ ∈ X ′ : ⟨z∗,s⟩= 0 for all s ∈ S}

Also denote by G (L)≡ {(x,Lx) : x ∈ D(L)}. Then

G (L∗) = (τG (L))⊥

Proof: Let (x,L∗x) ∈ G (L∗) . This means that

|⟨Ly,x⟩| ≤C∥y∥ for all y ∈ D(L)

and ⟨Ly,x⟩= ⟨L∗x,y⟩ for all y ∈D(L) . Let (y,Ly)∈ G (L) . Then τ (y,Ly) = (−Ly,y) . Then

⟨(x,L∗x) ,(−Ly,y)⟩= ⟨x,−Ly⟩+ ⟨L∗x,y⟩=−⟨x,Ly⟩+ ⟨x,L∗y⟩= 0

Thus G (L∗)⊆ (τG (L))⊥ . Next suppose (x,y∗) ∈ (τG (L))⊥ . This means that if (u,Lu) ∈G (L) , then

⟨(x,y∗) ,(−Lu,u)⟩ ≡ ⟨x,−Lu⟩+ ⟨y∗,u⟩= 0

and so for all u ∈ D(L) ,⟨y∗,u⟩= ⟨x,Lu⟩

and so x ∈ D(L∗) . Hence for all u ∈ D(L) ,

⟨y∗,u⟩= ⟨x,Lu⟩= ⟨L∗x,u⟩

Then, since D(L) is dense, it follows that y∗ = L∗x and so (x,y) ∈ G (L∗) . Thus these arethe same.

Theorem 25.5.4 is a very nice surjectivity result for set valued pseudomonotone opera-tors. We recall what it said here. Recall the meaning of coercive.

lim∥v∥→∞

inf{⟨z∗,v⟩||v||

: z∗ ∈ T v}= ∞

In this section, we use the convenient notation ⟨z∗,x⟩V ′,V ≡ z∗ (x).