Question

If z, w ∈ C, let (z|w) := Re(zw¯). Show that
(z|w)2 + (iz|w)2 = |z|2|w|2

and derive the Cauchy-Schwarz inequalit

(z|w)2 ≤ |z|2|w|2, z, w ∈

Answer 1

Answer:

Uniform Convexity

2.34  

As noted previously, the parallelogram law in an inner product space guarantees the uniform convexity of the corresponding norm on that space. This applies to L2(Ω). Now we will develop certain inequalities due to Clarkson [Clk] that generalize the parallelogram law and verify the uniform convexity of Lp (Ω) for 1 < p < ∞.

We begin by preparing three technical lemmas needed for the proof.

2.35 Lemma

If 0 < s < 1, then f(t) = (1 − st)/t is a decreasing function of t > 0.

Proof

f′(t) = (1/t2)(g(st) − 1) where g(r) = r − r ln r. Since 0 < st < 1 and since g′(r) = − ln r ≥ 0 for 0 < r ≤ 1, it follows that g(st) < g(1) = 1 whence f′(t) < 0.

2.36 Lemma

If 1 < p ≤ 2 and 0 ≤ t ≤ 1, then

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