Summary.
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 Schöpf 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 \( {\Bbb R} \).
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Received: November 26, 1998; revised version: October 11, 1999.
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Ger, R., Koclega, B. Isometries and a generalized Cauchy equation. Aequat. Math. 60, 72–79 (2000). https://doi.org/10.1007/s000100050136
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DOI: https://doi.org/10.1007/s000100050136