1392 CHAPTER 42. MAXIMAL MONOTONE OPERATORS, HILBERT SPACE

and that, therefore, ||x∗|| ≤ ||x|| and |⟨x∗,x⟩| ≤ ||x||2. Now return to 42.3.33 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∗ ∈ δφ (x).

Theorem 42.3.18 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. Then δφ 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 Theorem 18.2.11 φ is alsoweakly lower semicontinuous. Therefore,

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∈ δψ (x) because

ψ (y)−ψ (x)≥ 0 = 0(y− x)

by Theorem 42.3.14, 0 ∈ δψ (x) = δφ (x)− y∗ and this proves the theorem.

Corollary 42.3.19 Suppose X is a reflexive Banach space and φ : X → (−∞,∞] is convex,proper, and l.s.c. Then for each y∗ ∈ X ′ there exist x ∈ X, x∗1 ∈ F (x), and x∗2 ∈ δφ (x) suchthat

y∗ = x∗1 + x∗2.

Proof: Apply Theorem 42.3.18 to the convex function 12 ||x||

2+φ (x) and use Theorems42.3.14 and 42.3.17.