Small into isomorphisms on uniformly smooth spaces ✩

Let X be a uniformly smooth infinite dimensional Banach space, and (Ω,Σ,μ) be a σ -finite measure space. Suppose that T :X→L∞(Ω,Σ,μ) satisfies (1 − ε) x T x x , ∀x ∈ X, for some positive number ε <1/2 with δX∗ (2−2ε) > 13/14. Then T is close to an isometry U :X→ L∞(Ω,Σ,μ) such that T − U 16 1 − δX∗ (2 −2ε) + 1 2 ε, where δX∗ (t ) is the modulus of convexity of the conjugate space X∗.  2003 Elsevier Inc. All rights reserved