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 T_f,g : C⁽ⁿ⁾(I) → C⁽ⁿ⁾(I) such that T(f) = T_f,g(f) and T(g) = T_f,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.