25.7. MAXIMAL MONOTONE OPERATORS 883

Hence for all y,

lim infλ→0⟨Bxλ ,xλ − y⟩= lim inf

λ→0⟨Bxλ ,x− y⟩= ⟨x∗0,x− y⟩ ≥ ⟨Bx,x− y⟩

Since y is arbitrary, this shows that x∗0 = Bx and so B is onto as claimed.Again, note that it really didn’t matter about the particular duality map used, although

the usual one was featured in the argument.There are some more things which can be said about maximal monotone operators. To

include some of these, here is a very interesting lemma found in [13].

Lemma 25.7.16 Let X be a Banach space and suppose that

xn→ 0, ∥x∗n∥→ ∞

Then denoting by Dr the closed disk centered at 0 with radius r. It follows that for everyDr, there exists y0 ∈ Dr and a subsequence with index nk such that⟨

x∗nk,xnk − y0

⟩→−∞

Proof: Suppose this is not true. Then there exists Dr which has the property that for allu ∈ Dr,

⟨x∗n,xn−u⟩ ≥Cu

for all n. Now letEk ≡ {y ∈ Dr : ⟨x∗n,xn− y⟩ ≥ −k for all n}

Then this is a closed set, being the intersection of closed sets. Also, by assumption, theunion of these Ek equals Dr which is a complete metric space. Hence one of these Ek musthave nonempty interior by the Bair category theorem, say for k0. Say B(y,ε) ⊆ Dr. Thenfor all ∥u− y∥< ε,

⟨x∗n,xn−u⟩ ≥ −k0 for all n

Of course −y ∈ Dr also, and so there is C such that

⟨x∗n,xn + y⟩ ≥C for all n

Then⟨x∗n,2xn + y−u⟩ ≥C− k0 for all n

whenever ∥y−u∥ < ε. Now recall that xn→ 0. Consider only u such that ∥y−u∥ < ε/2.Therefore, for all n large enough, the expression 2xn + y− u for such u contains a smallball centered at the origin, say Dδ . (The set of all y− u for u closer to y than ε/2 is theball B(0,ε/2) and then the 2xn does not move it by much provided n is large enough.)Therefore,

⟨x∗n,v⟩ ≥C− k0

for all ∥v∥ ≤ δ . This contradicts the assumption that ∥x∗n∥→ ∞.