Isometries of C⁽ⁿ⁾[0, 1]

V. D. Pathak

Vol. 94 (1981), No. 1, 211–222

DOI: 10.2140/pjm.1981.94.211

Abstract

By C⁽ⁿ⁾[0,1] (henceforth denoted by C⁽ⁿ⁾) we denote the Banach algebra of complex valued n times continuously differentiable functions on [0,1] with norm given by

By an isometry of C⁽ⁿ⁾ we mean a norm-preserving linear map of C⁽ⁿ⁾ onto itself.

The purpose of this article is to describe the isometries of C⁽ⁿ⁾ for any positive integer n. More precisely, we show that any isometry of C⁽ⁿ⁾ is induced by a point map of the interval [0,1] onto itself.

Mathematical Subject Classification 2000

Primary: 46E15

Milestones

Received: 2 February 1979

Revised: 27 April 1979

Published: 1 May 1981