Linear Isometries on Spaces of Continulously Differentiable and Lipschitz Continuous Functions

Abstract

We characterize the surjective linear isometries on $C^{(n)} [0, 1]$ and Lip$[0, 1]$. Here $C^{(n)} [0, 1]$ denotes the Banach space of $n$-times continuously differentiable functions on $[0, 1]$ equipped with the norm \begin{equation*} \| f \| = \sum_{k=0}^{n-1}|f^{(k)} (0)| + \sup _{x \in [0, 1]} | f^{(n)} (x) | \quad (f \in C^{(n)} [0, 1]) , \end{equation*} and Lip$[0, 1]$ denotes the Banach space of Lipschitz continuous functions on $[0, 1]$ equipped with the norm \begin{equation*} \| f \| = |f(0)| + \mathop{\operatorname{ess \, sup}} _{x \in [0, 1]} | f' (x) | \quad (f \in {\rm Lip}[0, 1]). \end{equation*}