SURJECTIVE ISOMETRIES ON $C^{1}[0,1]$ WITH RESPECT TO SEVERAL NORMS (Researches on isometries from various viewpoints)

Abstract

Let C^{1}[0, 1] be a complex linear space of all continuously differentiable complex valued functions on the unit interval [0, 1]. We give a characterization of surjective, not necessarily linear, isometries on C^{1}[0, 1] with respect to the following norms: Vert fVert_{$Sigma$}=Vert fVert_{infty}+Vert f'Vert_{infty}, displaystyle Vert fVert_{C}=sup{|f(t)|+|f'(t)| : tin[0, 1]} and Vert fVert_{$sigma$}=|f(0)|+Vert f'Vert_{infty} for fin C^{1}[0, 1], respectively.