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 ∥Ƥ∥n = max{ |ƒ (0) | ƒ ’ (0) |, …,;ƒ^(n-1)(0) | , ∥ ƒ ⁽ⁿ⁾ ∥∞ } (ƒ ∈ C ⁽ⁿ⁾ (ΐ) ) 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(ƒ) = Tƒ;g(ƒ) and T(g) = Tƒ;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.

Mathematics Subject Classi_cation (2010): Primary 47B38; Secondary 46J10.

Key words: 2-local isometry, continuously differentiable function, real-linear isometry.