1390 CHAPTER 42. MAXIMAL MONOTONE OPERATORS, HILBERT SPACE

by similar reasoning so letting y = x,

z∗ (x)+

 =0︷ ︸︸ ︷φ 1 (x)− x∗ (x− x)−φ 1 (x)

≥ z∗ (y1)+φ 2 (x)−φ 2 (y1).

Therefore,z∗ (y1− x)≤ φ 2 (y1)−φ 2 (x)

for all y1 and so z∗ ∈ δφ 2 (x). Now let y1 = x in 42.3.28 and using 42.3.26 and 42.3.27, itfollows

z∗ (y)+φ 1 (y)− x∗ (y− x)−φ 1 (x)≥ z∗ (x)

φ 1 (y)−φ 1 (x)≥ x∗ (y− x)− z∗ (y− x)

and so x∗ − z∗ ∈ δφ 1 (x) so x∗ = z∗ + (x∗− z∗) ∈ δφ 2 (x) + δφ 1 (x) and this proves thetheorem.

Next is a very important example known as the duality map from a Banach space to itsdual space. Before doing this, consider a Hilbert space H. Define a map R from H to H ′,called the Riesz map, by the rule

R(x)(y)≡ (y,x).

By the Riesz representation theorem, this map is onto and one to one with the properties

R(x)(x) = ||x||2 , and ||Rx||2 = ||x||2.

The duality map from a Banach space to its dual is an attempt to generalize this notion ofRiesz map to an arbitrary Banach space.

Definition 42.3.15 For X a Banach space define F : X →P (X ′) by

F (x)≡{

x∗ ∈ X ′ : x∗ (x) = ||x||2 , ||x∗|| ≤ ||x||}. (42.3.29)

Lemma 42.3.16 With F (x) defined as above, it follows that

F (x) ={

x∗ ∈ X ′ : x∗ (x) = ||x||2 , ||x∗||= ||x||}

and F (x) is a closed, nonempty, convex subset of X ′.

Proof: If x∗ is in the set described in 42.3.29,

x∗(

x||x||

)= ||x||

and so ||x∗|| ≥ ||x||. Therefore

x∗ ∈{

x∗ ∈ X ′ : x∗ (x) = ||x||2 , ||x∗||= ||x||}.