Finite codimensional linear isometries on spaces of differentiable and Lipschitz functions (Q632277)

A linear isometry on a Banach space is said to be finite codimensional if the range is a subspace of finite codimension. In this paper, the author studies such isometries for the space of differentiable functions, \(C^{(n)}[0,1]\), equipped with the norm \(\|f\|= \max\{|f(0)|,|f'(0)|,\dots,|f^{(n-1)}(0)|,\|f^{(n)}\|_{\infty}\}\). Also considered is the space of Lipschitz continuous functions, with the norm \(\|f\|=\max\{|f(0)|,\|f'\|_{\infty}\}\). The technique involved in both cases is to identify them as a continuous function space \(C(X)\), for an appropriate compact set \(X\). Here, the author shows that any finite codimensional isometry is indeed surjective.