The representation of isometric operators on \(C ^{(1)}(X)\) (Q430025)

Let \(X\) be a connected compact set of real numbers. Let \(C^{(1)}(X)\) denote the space of continuously differentiable functions. A standard way of defining norms on this space involves the supremum norms of \(f\) and \(f'\), for example, the maximum or sum combination. In this paper, the author considers a norm induced by a regular hexagon in \(\mathbb R^2\), determined by \((1,0)\), \((\frac{1}{2},\frac{\sqrt{3}}{2})\), \((-\frac{1}{2},\frac{\sqrt{3}}{2}) \). After showing that this norm makes the space a Banach space, the author obtains a Banach-Stone type theorem, describing the surjective linear isometries on this space as composition by a homeomorphism (on \(X\)) or its negative. The proof follows the standad route of describing the extreme points of the dual unit ball and observing that such a map sends the constant function \(1\) to \(1\) or \(-1\).