910 CHAPTER 25. NONLINEAR OPERATORS

and that, therefore, ||x∗|| ≤ ||x|| and |⟨x∗,x⟩| ≤ ||x||2. Now return to 25.7.69 and let y = x.Then

⟨x∗,x⟩ ≥ 12s

[−2 ||x− sx|| ||sx||+ ||sx||2

]= −∥x∥2 (1− s)+ s∥x∥2

Letting s→ 1,⟨x∗,x⟩ ≥ ||x||2.

Since it was already shown that |⟨x∗,x⟩| ≤ ||x||2, this shows ⟨x∗,x⟩= ∥x∥2 and also ∥x∗∥ ≤∥x∥. Thus

∥x∗∥ ≥⟨

x∗x∥x∥

⟩= ∥x∥

so in fact x∗ ∈ F (x) .The next result gives conditions under which the subgradient is onto. This means that

if y∗ ∈ X ′, then there exists x ∈ X such that y∗ ∈ ∂φ (x).

Theorem 25.7.49 Suppose X is a reflexive Banach space and suppose φ : X → (−∞,∞] isconvex, proper, l.s.c., and for all y∗ ∈ X ′, x→ φ (x)−⟨y∗,x⟩ is coercive,

lim||x||→∞

φ (x)−⟨y∗,x⟩= ∞

Then ∂φ is onto.

Proof: The function x→ φ (x)− y∗ (x) ≡ ψ (x) is convex, proper, l.s.c., and coercive.Let

λ ≡ inf{φ (x)−⟨y∗,x⟩ : x ∈ X}

and let {xn} be a minimizing sequence satisfying

λ = limn→∞

φ (xn)−⟨y∗,xn⟩

By coercivity,lim||x||→∞

φ (x)−⟨y∗,x⟩= ∞

and so this minimizing sequence is bounded. By the Eberlein Smulian theorem, Theorem17.5.12, there is a weakly convergent subsequence xnk → x. By Lemma 25.7.45,

λ = φ (x)−⟨y∗,x⟩ ≤ lim infk→∞

φ(xnk

)−⟨y∗,xnk

⟩= λ

so there exists x which minimizes x→ φ (x)−⟨y∗,x⟩ ≡ ψ (x). Therefore, 0 ∈ ∂ψ (x) be-cause

ψ (y)−ψ (x)≥ 0 = ⟨0,y− x⟩

Thus, 0 ∈ ∂ψ (x) = ∂φ (x)− y∗.Now let φ be a convex proper lower semicontinuous function defined on X where X is

a reflexive Banach space with strictly convex norm. Consider ∂φ . Is it maximal monotone?