Proof. Suppose that T has properties a and b in the theorem. For any f, g ∈ B(K),
kf − gk is a constant. Because f ≤ g + kf − gk (where the norm is the super
norm), we have:
Tf ≤ T (g + kf − gk) ≤ Tg + β kf − gk .
The first inequality follows from property a and the second inequality from property
b. Reversing the roles of f and g, we have
Tg ≤ Tf + β kf − gk .
Thus, |Tf − Tg| ≤ β kf − gk. Because this holds for all k ∈ K, kTf − Tgk ≤ β kf − gk.

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