75.2. DUALITY MAPS 2531

75.2 Duality MapsThe duality map is an attempt to duplicate some of the features of the Riesz map in Hilbertspace which is discussed in the chapter on Hilbert space.

Definition 75.2.1 A Banach space is said to be strictly convex if whenever ||x||= ||y|| andx ̸= y, then ∣∣∣∣∣∣∣∣x+ y

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∣∣∣∣∣∣∣∣< ||x||.F : X→ X ′ is said to be a duality map if it satisfies the following: a.) ||F(x)||= ||x||p−1. b.)F(x)(x) = ||x||p, where p > 1.

Duality maps exist. Here is why. Let

F (x)≡{

x∗ : ||x∗|| ≤ ||x||p−1 and x∗ (x) = ||x||p}

Then F (x) is not empty because you can let f (αx) = α ||x||p . Then f is linear and definedon a subspace of X . Also

sup||αx||≤1

| f (αx)|= sup||αx||≤1

|α| ||x||p ≤ ||x||p−1

Also from the definition,f (x) = ||x||p

and so, letting x∗ be a Hahn Banach extension, it follows x∗ ∈ F (x). Also, F (x) is closedand convex. It is clearly closed because if x∗n→ x∗, the condition on the norm clearly holdsand also the other one does too. It is convex because

||x∗λ +(1−λ )y∗|| ≤ λ ||x∗||+(1−λ ) ||y∗|| ≤ λ ||x||p−1 +(1−λ ) ||x||p−1

If the conditions hold for x∗, then we can show that in fact ||x∗|| = ||x||p−1. This isbecause

||x∗|| ≥∣∣∣∣x∗( x

||x||

)∣∣∣∣= 1||x|||x∗ (x)|= ||x||p−1 .

Now how many things are in F (x) assuming the norm on X ′ is strictly convex? Supposex∗1, and x∗2 are two things in F (x) . Then by convexity, so is (x∗1 + x∗2)/2. Hence by strictconvexity, if the two are different, then∣∣∣∣∣∣∣∣x∗1 + x∗2

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∣∣∣∣∣∣∣∣= ||x||p−1 <12||x∗1||+

12||x∗2||= ||x||

p−1

which is a contradiction. Therefore, F is an actual mapping.What are some of its properties? First is one which is similar to the Cauchy Schwarz

inequality. Since p−1 = p/p′,

sup||y||≤1

|⟨Fx,y⟩|= ||x||p/p′