17.3. UNIFORM CONVEXITY OF Lp 451

and so J is linear. If Jx = 0, then by Lemma 17.2.9 there exists x∗ such that x∗(x) = ||x||and ||x∗||= 1. Then

0 = J(x)(x∗) = x∗(x) = ||x||.

This shows a.).To show b.), let x ∈ X and use Lemma 17.2.9 to obtain x∗ ∈ X ′ such that x∗(x) = ||x||

with ||x∗||= 1. Then

||x|| ≥ sup{|y∗(x)| : ||y∗|| ≤ 1}= sup{|J(x)(y∗)| : ||y∗|| ≤ 1}= ||Jx||≥ |J(x)(x∗)|= |x∗(x)|= ||x||

Therefore, ||Jx||= ||x|| as claimed. Therefore,

||J||= sup{||Jx|| : ||x|| ≤ 1}= sup{||x|| : ||x|| ≤ 1}= 1.

This shows b.).To verify c.), use b.). If Jxn→ y∗∗ ∈ X ′′ then by b.), xn is a Cauchy sequence converging

to some x ∈ X because||xn− xm||= ||Jxn− Jxm||

and {Jxn} is a Cauchy sequence. Then Jx = limn→∞ Jxn = y∗∗.Finally, to show the assertion about the norm of x∗, use what was just shown applied to

the James map from X ′ to X ′′′ still referred to as J.

||x∗||= sup{|x∗ (x)| : ||x|| ≤ 1}= sup{|J (x)(x∗)| : ||Jx|| ≤ 1}

≤ sup{|x∗∗ (x∗)| : ||x∗∗|| ≤ 1}= sup{|J (x∗)(x∗∗)| : ||x∗∗|| ≤ 1}

≡ ||Jx∗||= ||x∗||.

This proves the theorem.

Definition 17.2.14 When J maps X onto X ′′, X is called reflexive.

It happens the Lp spaces are reflexive whenever p > 1. This is shown later.

17.3 Uniform Convexity Of Lp

These terms refer roughly to how round the unit ball is. Here is the definition.

Definition 17.3.1 A Banach space is uniformly convex if whenever ||xn||, ||yn|| ≤ 1 and||xn + yn|| → 2, it follows that ||xn− yn|| → 0.

You can show that uniform convexity implies strict convexity. There are various otherthings which can also be shown. See the exercises for some of these. In this section, itwill be shown that the Lp spaces are examples of uniformly convex spaces. This involvessome inequalities known as Clarkson’s inequalities. Before presenting these, here are thebackwards Holder inequality and the backwards Minkowski inequality.