878 CHAPTER 25. NONLINEAR OPERATORS

Then taking another subsequence, written with index k, it can be assumed that

Bxk/∥Bxk∥→ y∗ ∈ E ′,∥y∗∥E ′ = 1

Hence,⟨y∗,xk− x⟩ ≥ 0

for all x∈E, but this requires that y∗= 0, a contradiction. Thus B is monotone, hemicontin-uous, and bounded. It follows from Theorem 25.1.4 which says that monotone and hemi-continuous operators are pseudomonotone and Proposition 25.1.6 which says that boundedpseudomonotone operators are demicontinuous that B is demicontinuous, hence continuousbecause, as just noted above, weak and strong convergence are the same for finite dimen-sional spaces. In case B is bounded, then this follows from Proposition 25.1.6 above. It ispseudomonotone and bounded hence demicontinuous and weak and strong convergence isthe same in finite dimensions.

Proof of Theorem 25.7.9: Let {Xn} be an increasing sequence of finite dimensionalsubspaces. Let  be maximal monotone on ∪nXn and extending A. By this is meant thatthe graph of  contains the graph of A restricted to ∪nXn,  is monotone and there is noother larger graph with these properties. See the above observation. Let jn : Xn → X bethe inclusion map and j∗n : X ′ → X ′n be the dual map. Then j∗n jn ≡ An and j∗nB jn ≡ Bnhave monotone graphs from Xn to P (X ′n) with Bn being continuous and single valued.This follows from the hemicontinuity and the above lemma which states that on finitedimensional spaces, hemicontinuity and monotonicity imply continuity. Then

[u,v] ∈ G (An)

meansu ∈ D(A)∩Xn and v ∈ j∗n jn (u) = j∗nÂ(u) since u ∈ Xn

Then from Lemma 25.7.7, there exists xn ∈ Xn such that

⟨Bnxn + vn,un− xn⟩X ′,X ≥ 0 all [un,vn] ∈ G (An)

That is, there exists xn ∈ K∩Xn such that for all u ∈ D(Â)∩Xn, [u,v] ∈ G

(Â)

⟨Bxn + v,u− xn⟩X ′,X ≥ 0 (25.7.48)

Then⟨v,u− xn⟩ ≥ ⟨Bxn,xn−u⟩ (25.7.49)

From the assumption that 0 ∈ D(Â), one can let u = 0 and then pick v0 ∈ Â0. Then the

above reduces to⟨v0,−xn⟩ ≥ ⟨Bxn,xn⟩

By coercivity of B, these xn are all bounded and so by the Eberlien Smulian theorem, thereis a subsequence {xn} which satisfies

xn → x weakly in X

Bxn → y weakly in X ′