Isometries and a generalized Cauchy equation

Abstract

A description of all continuous (resp. differentiable) solutions f mapping the real line \( {\Bbb R} \) into a real normed linear space \( (X, \Vert \cdot \Vert) \) (not necessarily strictly convex) of the functional equation¶¶\( \Vert f(x + y) \Vert = \Vert f(x) + f(y) \Vert \)¶has been presented by Peter Schopf in [10]. Looking for more readable representations we have shown that any function f of that kind fulfilling merely very mild regularity assumptions has to be proportional to an odd isometry mapping \( {\Bbb R} \) into X.¶To gain a proper proof tool we have also established an improvement of Edgar Berz's [4] result on the form of Lebesgue measurable sublinear functionals on \( {\Bbb R} \).