Generalized 2-local isometries of spaces of continuously differentiable functions

Abstract

Let C(n) (I) denote the Banach space of n-times continuously differentiable functions on I = [0, 1], equipped with the norm ||f||n = max { |f(0)|, |fʹ(0)|, … , |f(n–1)(0)|, ||f(n)||∞} (f ) ∈ C(n) (I)), where ||·||∞ is the supremum norm. We call a map T : C(n) (I) → C(n)(I) a 2-local real-linear isometry if for each pair f, g in C(n)(I), there exists a surjective real-linear isometry Tf,g : C(n)(I) → C(n)(I) such that T(f) = Tf,g(f) and T(g) = Tf,g(g). In this paper we show that every 2-local real-linear isometry of C(n)(I) is a surjective real-linear isometry. Moreover, a complete description of such maps is presented.