42.3. SUBGRADIENTS 1393

42.3.2 Hilbert SpaceIn this section the subgradients are of a slightly different form and defined on a subset of H,a real Hilbert space. In Hilbert space the duality map is just the Riesz map defined earlierby

Rx(y)≡ (y,x).

Definition 42.3.20 dom(∂φ)≡ dom(δφ) and for x ∈ dom(∂φ),

∂φ (x)≡ R−1δφ (x).

Thus y ∈ ∂φ (x) if and only if for all z ∈ H,

Ry(z− x) = (y,z− x)≤ φ (z)−φ (x).

Recall the definition of a maximal monotone operator.

Definition 42.3.21 A mapping A : D(A) ⊆ H →P (H) is called monotone if wheneveryi ∈ Axi,

(y1− y2,x1− x2)≥ 0.

A monotone map is called maximal monotone if whenever z ∈H, there exists x ∈D(A) andy ∈ A(x) such that z = y+ x. Put more simply, I +A maps D(A) onto H.

The following lemma states, among other things, that when φ is a convex, proper, l.s.c.function defined on a Hilbert space, ∂φ is maximal monotone.

Lemma 42.3.22 If φ is a convex, proper, l.s.c. function defined on a Hilbert space, then ∂φ

is maximal monotone and (I +∂φ)−1 is a Lipschitz continuous map from H to dom(∂φ)having Lipschitz constant 1.

Proof: Let y ∈ H. Then Ry ∈ H ′ and by Corollary 42.3.19, there exists x ∈ dom(δφ)such that Rx+δφ (x) ∋ Ry. Multiplying by R−1 we see y ∈ x+∂φ (x). This shows I +∂φ

is onto. If yi ∈ ∂φ (xi), then Ryi ∈ δφ (xi) and so by the definition of subgradients,

(y1− y2,x1− x2) = R(y1− y2)(x1− x2)

= Ry1 (x1− x2)−Ry2 (x1− x2)

≥ φ (x1)−φ (x2)− (φ (x1)−φ (x2)) = 0

showing ∂φ is monotone. Now suppose xi ∈ (I +∂φ)−1 (y). Then y− xi ∈ ∂φ (xi) and bymonotonicity of ∂φ ,

−|x1− x2|2 = (y− x1− (y− x2) ,x1− x2)≥ 0

and so x1 = x2. Thus (I +∂φ)−1 is well defined. If xi = (I +∂φ)−1 (yi), then by themonotonicity of ∂φ ,

(y1− x1− (y1− x2) ,x1− x2)≥ 0