2-Local real-linear isometries on C⁽¹⁾ ([0; 1])

Abstract

Let C(1)([0; 1]) be the Banach space of continuously differentiable functions on the closed unit interval [0; 1] equipped with the norm ∥f∥α = lf(0)l+∥f’∥∞, where ∥g∥∞ = sup{lg(t)l : t E [0; 1]g for g. If T : C ⁽¹⁾([0; 1]) →C⁽¹⁾([0; 1]) is a 2-local real-linear isometry, then T is a surjective real-linear isometry.