Isometries and a generalized Cauchy equation (Q1587839)

The functional equation \(\|f(x+y)\|=\|f(x)+f(y)\|\) \((x,y\in\mathbb{R})\) has a rich literature, beginning with \textit{P. Fischer} and \textit{G. Muszély} [Can. Math. Bull. 10, 197-205 (1967; Zbl 0157.22303)], mainly comparing it to the Cauchy equation \(f(x+y)=f(x)+f(y)\). The present authors offer, in an effort to make a result of \textit{P. Schöpf} [Math. Pannonica 8, No. 1, 117-127 (1997; Zbl 0881.39012)] ``more readable'', all solutions \(f:\mathbb{R}\to X\) \(((X,\|. \|)\) is a real normed linear space) of the first equation above, say bounded from above on a set of positive measure; in particular they establish a connection to isometries.