25.7. MAXIMAL MONOTONE OPERATORS 915

Corollary 25.7.52 Let φ : X → (−∞,∞] be convex, proper, and lower semicontinuous.Here X is a Banach space. Then ∂φ is maximal monotone.

Proof: Let ψ (x) = 12 ∥x∥

2. There exists x∗ and some number b such that φ (x) ≥b+ ⟨x∗,x⟩ . Therefore, ψ + φ is convex, lower semicontinuous, and bounded. It follows∂ (ψ +φ) is onto by Theorem 25.7.49. However, ψ is continuous everywhere, in particu-lar at every point of the domain of φ . Therefore, ∂ψ + ∂φ = ∂ (φ +ψ) and by Theorem25.7.48, this shows that F +∂φ is onto.

It seems to me that the above are the most important results about convex properlower semicontinuous functions. However, there are many other very interesting propertiesknown.

Proposition 25.7.53 Let φ : X → (−∞,∞] be convex proper and lower semicontinuous.Then D(∂φ) is dense in D(φ) and so D(∂φ) = D(φ).

Proof: Let xλ be the solution to 0 ∈ F (xλ − x)+λ∂φ (xλ ) . Here x ∈ D(φ). Say u∗λ∈

∂φ (xλ ) such that the inclusion becomes an equality. Then

0 =⟨F (xλ − x)+λu∗

λ,xλ − x

⟩= ∥xλ − x∥2−λ

⟨u∗

λ,x− xλ

⟩≥ ∥xλ − x∥2−λ (φ (x)−φ (xλ ))

Hence, letting z∗,b be such that φ (y)≥ b+ ⟨z∗,y− x⟩ ,

λ (φ (x)− [b+ ⟨z∗,xλ − x⟩])≥ λ (φ (x)−φ (xλ ))≥ ∥xλ − x∥2

λφ (x)−λb≥ ∥xλ − x∥2−λ ∥z∗∥∥xλ − x∥

≥ ∥xλ − x∥2−λ

(∥z∗∥2

2+∥xλ − x∥2

2

)Thus

λφ (x)−λb+λ∥z∗∥2

2≥(

1− λ

2

)∥xλ − x∥2

It follows that xλ → x. This shows that D(φ) ⊆ D(∂φ) and so D(φ) ⊆ D(∂φ) ⊆ D(φ).

There is a really amazing theorem, Moreau’s theorem. It is in [24], [13] and [116]. Itinvolves approximating a convex function with one which is differentiable, at least in thecase where you have a Hilbert space. In the general case considered in this chapter, thefunction is continuous.

Theorem 25.7.54 Let φ be a convex lower semicontinuous proper function defined on X.Define A≡ ∂φ ,Aλ = (∂φ)

λ

φ λ (x)≡miny∈X

(1

2λ∥x− y∥2 +φ (y)

)